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\newcommand{\C}{\mathbb C}   
\newcommand{\R}{\mathbb R}
\newcommand{\Q}{\mathbb Q}
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\newcommand{\fcw}{\omega}
\newcommand{\dPhi}{\Phi^*}
\newcommand{\ddPhi}{\Pi^*}
\newcommand{\RL}{M}
\newcommand{\CL}{N}
\newcommand{\SimR}{\Pi}
\newcommand{\rnk}{n}
\newcommand{\coi}{i}
\newcommand{\parafir}{c}
\newcommand{\parasec}{a}
\newcommand{\parathr}{b}
\newcommand{\parafor}{y}
\newcommand{\A}{S}
\newcommand{\intnum}[1]{(\mu_X,{#1})}
\newcommand{\invdiv}[1]{D_{#1}}
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\newcommand{\imatcomp}[2]{\mathcal{I}^{#1}_{\ #2}}
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\title{\bf Young diagrams and intersection numbers \\ for toric manifolds associated with Weyl chambers}

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\author{Hiraku Abe\\
\small Osaka City University Advanced Mathematical Institute\\[-0.8ex]
\small 3-3-138 Sugimoto, Sumiyoshi-ku\\[-0.8ex] 
\small Osaka 558-8585 JAPAN\\
\small\tt hirakuabe@globe.ocn.ne.jp
}

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\date{\dateline{Apr 22, 2014}{Feb 20, 2014}\\
\small Mathematics Subject Classifications: 14M25, 17B22, 13F55}

\begin{document}

\maketitle

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\begin{abstract}
  We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. 
We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold. 

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Young diagrams; intersection numbers; toric varieties; structure constants
\end{abstract}

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\section{Introduction}
Let $\Phi$ be a root system in the $\rnk$-dimensional Euclidean space $\Euc$ with its inner product. We denote by $\Delta(\Phi)$ the fan determined by the collection of Weyl chambers in $E^*$, and consider the toric manifold $X$ associated with $\Delta(\Phi)$.
This toric manifold arises as the closure of a general orbit in the flag variety with respect to the standard torus action which makes $X$ a regular semisimple Hessenberg variety (\cite{De Mari-Procesi-Shayman}). 
The Weyl group $W$ naturally acts on the Weyl chambers and hence also on $X$.
The representation of $W$ on the cohomology $H^*(X;\C)$ has been extensively studied by Procesi \cite{Procesi}, Dolgachev-Lunts \cite{Dolgachev-Lunts}, and Stembridge \cite{Stembridge}. 
For the classical root system of type $A_{\rnk}$, Losev-Manin \cite{Losev-Manin} described $X$ as the moduli space of stable $(\rnk+1)$-pointed chains of projective lines (cf. Batyrev-Blume \cite{Batyrev-Blume}).

Let $\Pi=\{\alpha_1,\cdots,\alpha_{\rnk}\}\subset \Phi$ be a set of simple roots, then we have a torus invariant non-singular subvariety $X_u$ in $X$ of codimension $|u(\Pi)\cap\Phi^-|$ such that the associated cohomology classes $\{[X_u]\}_{u\in W}$ form a module basis of the integral singular cohomology $H^*(X)$.
The cohomology class $[X_u]$ is written as a monomial of torus invariant divisors $\invdiv{u\omega_i}$ of $X$ for all coweights $u\omega_i$ satisfying $u\alpha_i\in\Phi^-$ where $\{\omega_1,\cdots,\omega_{\rnk}\}$ is the set of fundamental coweights (see Section 2 and (\ref{def of Xu}) for details).
In this paper, we study the case for the root systems of classical type and of exceptional type $G_2$, and we give a combinatorial formula of the intersection numbers 
\begin{align*}
\intnum{[w_0X_{w_0 w}][X_u][X_v]}
\end{align*} 
of three subvarieties $X_u$, $X_v$ and $w_0X_{w_0 w}$ for $u,v,w\in W$
where $\mu_X$ is the fundamental homology class of $X$ and
$w_0$ is the longest element.
As an application, we will obtain a recursive formula for the structure constants $c_{u,v}^w$ in the expansion of the product
\begin{align*}
[X_u][X_v] = \sum_{w\in W} c_{u,v}^w [X_w]
\quad \text{where} \quad c_{u,v}^w \in\Z
\end{align*} 
as discussed in Section 4.

\vspace{10pt}
Let us state our formula for $\intnum{[w_0X_{w_0 w}][X_u][X_v]}$ in the case of the classical root system of type $A_{\rnk}$ (the results for the classical root systems of type $B_{\rnk}, C_{\rnk}$, and $D_{\rnk}$ are stated in Section 5). 
In this case, the Weyl group $W$ is the $(\rnk+1)$-th permutation group $\mathfrak{S}_{\rnk+1}$.
For each $u\in \mathfrak{S}_{\rnk+1}$, we let 
\begin{align}\label{def of D(u)}
&D(u):=\{ \{u(1),u(2),\cdots,u(i)\} \mid u(i)>u(i+1) \},  \\ \label{def of A(u)}
&A(u):=\{ \{u(1),u(2),\cdots,u(i)\} \mid u(i)<u(i+1) \}
\end{align}
where each $\{u(1),u(2),\cdots,u(i)\}$ is a subset of $[\rnk+1]$.
We define a Young diagram $\WY{u}{v}{w}$ as follows. Assume $d(u)+d(v)=d(w)$, and the collection $D(u)\coprod D(v) \coprod A(w)$ forms a nested chain of subsets of $[\rnk+1]$. In this case, $\WY{u}{v}{w}$ is defined to be the Young diagram consisting of the cardinalities of those chains ordered as a weakly decreasing sequence. Otherwise, $\WY{u}{v}{w}=\emptyset$.
For example, suppose $n=4$ and, let $u=12354$, $v=31254$, and $w=35421$. Then, we have that $D(u)=\{1235\}$, $D(v)=\{3, 3125\}=\{3, 1235\}$, and $A(w)=\{3\}$ where $1235$ denotes the set $\{1,2,3,5\}$ and we use the same notation for others. These sets forms a nested chain of subsets $3\subset 3\subset 1235\subset 1235$, and hence we obtain $\lambda_{12354, 31254}^{35421}=(4,4,1,1)$.

For a Young diagram $\lambda=(\lambda_1\geq\cdots\geq\lambda_{\rnk})$ with $\rnk$ rows (i.e. $\lambda_{\rnk}>0$) fitting into the $\rnk\times\rnk$ square, we define $I(\lambda)\in\Z$ to be the following integer.
Let $s$ be the number of lower-right corners of $\lambda$, i.e.,  $s=|\{ i\in[\rnk] \mid \lambda_{i}>\lambda_{i+1} \}|$ where $\lambda_{\rnk+1}:=0$.
Write
\begin{align*}
\{ i\in[\rnk] \mid \lambda_{i}>\lambda_{i+1} \} 
=\{ \coi_1,\cdots,\coi_{s} \}.
\end{align*}
We impose the condition $\coi_1<\coi_2<\cdots<\coi_{s}$ to determine them uniquely. 
Observe that $\coi_{s}=\rnk$.
For example, if $\rnk=4$ and $\lambda=(4,2,2,1)$, then $s=3$ and $\{\coi_1,\coi_2, \coi_3\}=\{1,3,4\}$.
For $r=1,\cdots,s$, define
\begin{align*}
\parasec_{r}:=\coi_{r}-\coi_{r-1}-1, \quad
\parathr_{r}:=\lambda_{i_{r}}-\lambda_{i_{r+1}}-1, \quad
\parafir_{r}:=\lambda_{i_{r}}+\coi_{r}-\rnk-1
\end{align*}
where we write $\coi_0=0$, and let
\begin{align*}
\parafor_r:=
\binom{ \parasec_{r} }{ \parafir_{r} }
\binom{ \parathr_{r} }{ \parafir_{r} }
\quad \text{for $r=1,\cdots,s$}.
\end{align*}
We use the convention $\binom{x}{y}=0$ unless $0\leq y\leq x$.
By shading each lower-right corner of $\lambda$, the pictorial meanings of $\parasec_r$, $\parathr_r$, and $\parafir_r$ become clear (as shown in Figure \ref{Intro picture}).
Namely, $\parasec_r$ is the number of boxes between the north side of the shaded box and the the upper-left corner placed above,
$\parathr_r$ is the similar number for the horizontal segment of the corner, and $\parafir_r$ is the number of boxes between the north side of the shaded box and the crossing point of the vertical segment and the anti-diagonal line where we count negatively if that part of the vertical segment is above the anti-diagonal. 
\vspace{-5pt}
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\caption{Three numbers $\parasec_r$, $\parathr_r$ and $\parafir_r$}
\label{Intro picture}
\end{figure}



\noindent
Now, let 
\begin{align*}
I(\lambda) := (-1)^{\rnk+s}\parafor_{1}\cdots \parafor_{s},
\end{align*}
and put $I(\emptyset)=0$.
The following is our main statement for type $A_{\rnk}$.
\begin{theorem}\label{intro main thm}
$\displaystyle{\intnum{[w_0X_{w_0w}][X_u][X_v]}=I(\WY{u}{v}{w})}$ for any $u,v,w\in\mathfrak{S}_{\rnk+1}$ where $\mu_X$ is the fundamental homology class of $X$.
\end{theorem}

We will prove Theorem \ref{intro main thm} by computing general intersection numbers of invariant divisors of $X$ in Section 3 and 4.
Section 5 is devoted to the classical root systems of type $B_{\rnk}$, $C_{\rnk}$ and $D_{\rnk}$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\Phi$ be a root system in the $\rnk$-dimensional Euclidean space $\Euc$ with its inner product. Let $\RL\subset \Euc$ be the root lattice of $\Phi$ and $\CL\subset \Euc^*$ be the coweight lattice of $\Phi$.  Then $\RL$ is the dual lattice of $\CL$ with respect to the natural pairing.

We choose a set of simple roots $\SimR=\{\alpha_1,\cdots,\alpha_{\rnk}\}\subset\Phi$, and let $\ddPhi:=\{\fcw_1,\cdots,\fcw_{\rnk}\}\subset E^*$ be the dual basis of $\SimR$ defined by $\langle \fcw_i, \alpha_j \rangle=\delta_{ij}$, i.e., $\fcw_1,\cdots,\fcw_{\rnk}$ are the \textit{fundamental coweights}.
For each $u\in W$, denote 
\begin{align*}
\sigma_{u}
:=\text{cone}(u\fcw_1,\cdots,u\fcw_{\rnk}) 
= \{ \textstyle{\sum_{i=1}^{\rnk}} \lambda_iu\fcw_i \mid \lambda_i\geq0 \}.
\end{align*} 
These cones $\{\sigma_{u}\}_{u\in W}$ form a non-singular complete fan $\Delta(\Phi)$ in $\Euc^*$ by including all their faces. 
The set of minimal generators of these cones are the set of \textit{coweights}:
\begin{align*}
 \dPhi=\bigcup_{v\in W} \{v\fcw_1,\cdots,v\fcw_{\rnk}\}.
\end{align*} 
For each element $u\in W$, 
the maximal cones containing a minimal generator $u\fcw_i$ are $\sigma_v$ for $v\in W$ such that $u\fcw_i=v\fcw_j$ for some $j$.
There is a cone of $\Delta(\Phi)$ generated by minimal generators $x_1,\cdots,x_k\in\dPhi$ if and only if there exists $u\in W$ such that each $x_i$ can be written as $u\fcw_{j}$ for some $j$.

We consider the toric manifold $X=X(\Phi)$ associated with the fan $\Delta(\Phi)$. 
For root systems $\Phi$ and $\Phi'$, it is easily verified that $X(\Phi)\cong X(\Phi')$ as toric varieties (in the sense of \cite{Cox-Little-Schenck} Sec. 3.3.) if and only if $\Phi\cong \Phi'$ as root systems (in the sense that their Cartan matrices are the same up to permuting the indexes).
We refer to \cite{Batyrev-Blume} and \cite{Klyachko} for general properties of $X$. 
Let $\invdiv{x}\subset X$ be the invariant divisor corresponding to the ray generated by $x\in\dPhi$. 
The Poincar$\acute{\text{e}}$ dual $\tau_{x}:=[\invdiv{x}]$ gives us a cohomology class of degree $2$ in the integral singular cohomology $H^*(X)$.
The cohomology ring $H^*(X)$ is isomorphic to the face ring of the underlying simplicial complex of the fan $\Delta(\Phi)$ modulo some linear relations (\cite{Fulton1}). More precisely, we have 
\begin{align*}
H^*(X) = \Z[\tau_x \mid x\in\dPhi]/I
\end{align*} 
where the ideal $I$ is generated by $\tau_{x_1}\cdots\tau_{x_k}$ for which $x_1,\cdots,x_k$ do not generate a face of  $\sigma_u$ for some $u\in W$ and $\sum_{x\in\dPhi} \langle x,\alpha \rangle \tau_{x}$ for any root $\alpha$.
Namely, we have the following equalities in $H^*(X)$ :
\begin{align}\label{prelim linear relation}
\sum_{x\in\dPhi} \langle x,\alpha \rangle \tau_{x}=0
\quad \text{for any root $\alpha$}.
\end{align} 

The above observation about rays of $\Delta(\Phi)$ implies that 
\begin{lemma}\label{prelim rays generating a cone}
We have $\tau_{x_1}\cdots\tau_{x_k} = 0$ unless there exists $u\in W$ such that each $x_i$ can be written as $u\fcw_{j}$ for some $j$.
\end{lemma}

Let $\mu_X$ be the fundamental homology class of $X$.
For subvarieties $Z_1,\cdots,Z_k\subset X$, we call $\intnum{[Z_1]\cdots[Z_k]}$ the \textit{intersection number} of $Z_1,\cdots,Z_k$ where $[Z_i]$ denotes the Poincar$\acute{\text{e}}$ dual of $Z_i$.
Note that the Weyl group $W$ acts on the fan $\Delta(\Phi)$, and hence acts on the toric manifold $X$.
We have $uX_{x}=X_{ux}$ for any $u\in W$ and $x\in\dPhi$ which means that $(u^{-1})^*\tau_{x} = \tau_{ux}$.
The next lemma says that intersection numbers for divisors $\invdiv{u\fcw_i}$ are invariant under the Weyl group action. 
\begin{lemma}\label{prelim Weyl inv}
Let $x_1,\cdots,x_{\rnk}\in\dPhi$. Then for any $u\in W$, we have
\begin{align*}
\intnum{\tau_{ux_1}\cdots\tau_{ux_{\rnk}}} = \intnum{\tau_{x_1}\cdots\tau_{x_{\rnk}}}.
\end{align*} 
\end{lemma}
\begin{proof}
Observe that $\tau_{ux_1}\cdots\tau_{ux_{\rnk}}=(u^{-1})^*(\tau_{x_1}\cdots\tau_{x_{\rnk}})$. Both of $\tau_{ux_1}\cdots\tau_{ux_{\rnk}}$ and $\tau_{x_1}\cdots\tau_{x_{\rnk}}$ can be written as the cohomology class $[p]$ of a point $p$ in $X$ multiplied by some integer, and these integers are the corresponding intersection numbers.
Since $u$ preserves the orientation of $X$, we have $(u^{-1})^*([p])=[u\cdot p]=[p]$ which proves the claim.
\end{proof}

\vspace{10pt}
For any $u\in W$, the product $\tau_{u\fcw_{1}} \cdots \tau_{u\fcw_{\rnk}}$ is the Poincar$\acute{\text{e}}$ dual of a point in $X$ since the invariant divisors $X_{u\fcw_1},\cdots,X_{u\fcw_{\rnk}}$ intersect transversally which means that
\begin{align}\label{general type I=1}
\intnum{\tau_{u\fcw_{1}} \cdots \tau_{u\fcw_{\rnk}}}=1.
\end{align}
We will compute the intersection number $\intnum{\tau_{x_1}\cdots\tau_{x_{\rnk}}}$ for arbitrary $x_1,\cdots,x_{\rnk}\in\dPhi$ for the root systems of classical type.
By Lemma \ref{prelim rays generating a cone}, we can assume that this number is of the form $\intnum{(\tau_{u\fcw_{i_1}})^{m_1} \cdots (\tau_{u\fcw_{i_s}})^{m_s}}$ for some $1\leq i_1< \cdots< i_s\leq\rnk$ and $1\leq m_{k}\leq\rnk \ (k=1,\cdots,s)$ satisfying $m_1+\cdots+m_s=n$ without loss of generality.
We call the number $m_{k}$ the \textit{multiplicity} of $\tau_{u\omega_{i_k}}$.
We compute this number by applying the linear relations (\ref{prelim linear relation}) to reduce the multiplicities $m_1,\cdots,m_s$. 
Although Lemma \ref{prelim Weyl inv} shows that this number is equal to $\intnum{(\tau_{\fcw_{i_1}})^{m_1} \cdots (\tau_{\fcw_{i_s}})^{m_s}}$, we will need Lemma \ref{prelim Weyl inv} again after applying the relations (\ref{prelim linear relation}).

In the next section, we will consider the classical root system of type $A_{\rnk}$, and compute the intersection numbers. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\texorpdfstring{Intersection numbers for Type $A_{\rnk}$}{Intersection numbers for Type An}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we compute the intersection numbers for the toric manifold $X$ of type $A_{\rnk}$. 
Let $E=\{x\in\R^{\rnk+1} \mid x_1+\cdots+x_{\rnk+1}=0\}$.
The roots are  $t_i-t_j \in E \ (1\leq i,j\leq \rnk+1)$ where $t_i\in\R^{\rnk+1}$ is the $i$-th standard vector.
We choose $\SimR=\{t_i-t_{i+1}\mid 1\leq i\leq \rnk\}$
as the set of simple roots, and write $\alpha_i=t_i-t_{i+1}$ for each $i$.
The Weyl group $W=\mathfrak{S}_{\rnk+1}$ is the $(\rnk+1)$-th permutation group acting on $E$ by $u(t_i-t_j)=t_{u(i)}-t_{u(j)}$ for each $u\in W$.
The minimal generators $\fcw_1,\cdots,\fcw_{\rnk}\in E^*$ of the fundamental Weyl chamber $\sigma_{\text{id}}$ are
\[
\fcw_i=(e_1+\cdots+e_i)-\frac{i}{\rnk+1}(e_1+\cdots+e_{\rnk+1}) 
\quad \text{for} \quad
i=1,\cdots,\rnk
\]
where $\{e_i\}_i\subset (\R^{\rnk+1})^*$ is the dual basis of $\{t_i\}_i\subset \R^{\rnk+1}$.

Denoting by $2^{[\rnk+1]}$ the set of all subsets of $[\rnk+1]=\{1,\cdots,\rnk+1\}$, we have a well-defined map $\dPhi \rightarrow 2^{[\rnk+1]}$ by sending $u\fcw_i \mapsto \{u(1),\cdots,u(i)\}$.
It is easy to see that this is an injection, and this establishes an identification
\begin{align}\label{type A identification}
\dPhi \quad \longleftrightarrow \quad \text{the set of non-empty proper subsets of $[\rnk+1]$}.
\end{align} 
In particular, the well-definedness implies that if $u\fcw_i=v\fcw_j$ then $i=j$.
Now, for each $\emptyset\subsetneq S\subsetneq[\rnk+1]$, we define
$\tau_S:=\tau_{u\omega_i}$ where $u\omega_i\in\dPhi$ corresponds to $S$ by this identification. 
Then, for $\emptyset\subsetneq S_1, \cdots, S_q\subsetneq[\rnk+1] \ (1\leq q\leq\rnk)$, it follows by Lemma \ref{prelim rays generating a cone} that $\tau_{S_1}\cdots\tau_{S_q}=0$ unless these sets form a nested chain of subsets, i.e. $S_1\subset\cdots\subset S_q$ up to reordering.

With Lemma \ref{prelim Weyl inv}, it is easy to show the following invariance property of intersection numbers which implies that $\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}}$ for $\emptyset\subsetneq S_1\subset \cdots \subset S_{\rnk} \subsetneq [\rnk+1]$ is determined by the set of integers $1\leq |S_1|\leq \cdots\leq |S_{\rnk}|\leq \rnk$.
\begin{lemma}\label{invariance for type A}
Let $\emptyset\subsetneq S_1\subset \cdots \subset S_{\rnk} \subsetneq [\rnk+1]$ and $\emptyset\subsetneq S'_1\subset \cdots \subset S'_{\rnk} \subsetneq [\rnk+1]$.
If $|S_i|=|S'_i|$ for all $i=1,\cdots,\rnk$, then $\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}} = \intnum{\tau_{S'_1}\cdots\tau_{S'_{\rnk}}}.$
\end{lemma}

Motivated by this property, we compute intersection numbers in terms of Young diagrams consisting of the cardinalities of the sets corresponding to the given invariant divisors. 
The linear relations (\ref{prelim linear relation}) are translated to 
\begin{align}\label{SR relation for type A}
\sum_{\substack{\emptyset \subsetneq S\subsetneq [\rnk+1] \\ k\in S, l\notin S}} \tau_{S} 
-
\sum_{\substack{\emptyset \subsetneq S\subsetneq [\rnk+1] \\ k\notin S, l\in S}} \tau_{S} 
=0
\qquad  \text{for each $k, l\in[\rnk+1]$}.
\end{align} 
In the following, we write $\tau_{\emptyset}=\tau_{[\rnk+1]}=1$.
This equality together with the above observation about $\tau_{S_1}\cdots\tau_{S_q}$ being $0$ implies the next lemma.
\begin{lemma}\label{type A separation}
Let $\emptyset\subset \Asetfir\subsetneq \Asetsec \subsetneq \Asetthr \subset [\rnk+1]$.
For any $b\in \Asetsec\backslash \Asetfir$ and $c\in \Asetthr\backslash \Asetsec$, we have
\begin{align*}
\tau_{\Asetfir} {\tau_{\Asetsec}}^2 \tau_{\Asetthr} = - \sum_{\substack{\Asetfir\subsetneq \Asetrun \subsetneq \Asetthr \\ \Asetrun\neq \Asetsec, b\in \Asetrun, c\notin \Asetrun}} \tau_{\Asetfir} \tau_{\Asetsec}\tau_{\Asetrun} \tau_{\Asetthr}.
\end{align*}
\end{lemma}




\vspace{10pt}
For a Young diagram fitting into the $\rnk\times\rnk$ square, we write the \textit{dotted anti-diagonal line} shifted down half the length of a single box from the standard anti-diagonal. 

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\end{picture}%
\caption{Young diagrams and the dotted anti-diagonal line}
\label{zigzag picture}
\end{figure}

Let $\emptyset\subsetneq S_1\subset \cdots \subset S_{\rnk} \subsetneq [\rnk+1]$, and denote by $\lambda$ the Young diagram consisting of $\lambda_i=|S_{\rnk+1-i}|$ for each $i$.
We write $\lambda_{\rnk+1}=0$.
Let $s$ be the number of the lower-right corners of $\lambda$, that is, 
\begin{align*}
s:=|\{ i\in[\rnk] \mid \lambda_{i}>\lambda_{i+1} \}|.
\end{align*}
\begin{proposition}\label{dotted vanishing type A}
\emph{(Vanishing property)}
$\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}} = 0$ unless each step of the zigzag line of the lower-right corners of $\lambda$ crosses the dotted anti-diagonal. 
\end{proposition}
\begin{proof}
We suppose that there is a step of the zigzag line of $\lambda$ which does not cross the dotted anti-diagonal, and show  $\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}} = 0$ by induction on $k:=\rnk-s$.
Since there is no such case for $k=0$, we consider the case $k=1$. In this case,
there is a unique vertical segment of length 2 in the zigzag line of $\lambda$.
If there is a (unique) horizontal segment of length 2, then the vertical and horizontal segments are not adjacent because of our assumption.
By applying Lemma \ref{type A separation} for the square corresponding to this vertical segment, it follows that the intersection number is zero since there is no summand.

For the general case, take a vertical segment of length $\geq2$.
Let us say that this vertical segment contains $S_{i}$ and $S_{i+1}$ (i.e. $S_{i}=S_{i+1}$). 
We separate this square in $\tau_{S_1}\cdots\tau_{S_{\rnk}}$ by Lemma \ref{type A separation}. Let $\lambda'$ be the Young diagram corresponding to a summand in the right-hand-side. 
Then the zigzag line of $\lambda'$ has a step which does not cross the dotted anti-diagonal. 
In fact, if the vertical segment does not cross the dotted anti-diagonal, then this segment survives as a non-crossing segment of length at least 1, and if it does then we can find another vertical segment which does not, and this segment is preserved for each $\lambda'$ in the summands.
Now the induction hypothesis shows that each term will vanish after taking the intersection number, and we get $\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}} = 0$.
\end{proof}









\vspace{20pt}
Let $\lambda=(\lambda_1\geq\cdots\geq\lambda_{\rnk})$ be a Young diagram with $\rnk$ rows (i.e. $\lambda_{\rnk}>0$) fitting into the $\rnk\times\rnk$ square.
Let $I(\lambda)\in\Z$ be the one defined in Section 1.
We here recall the definition for the convenience of the reader.
Let $s$ be the number of lower-right corners of $\lambda$, i.e.,  $s=|\{ i\in[\rnk] \mid \lambda_{i}>\lambda_{i+1} \}|$ where $\lambda_{\rnk+1}:=0$.
Write
\begin{align*}
\{ i\in[\rnk] \mid \lambda_{i}>\lambda_{i+1} \} 
=\{ \coi_1,\cdots,\coi_{s} \}.
\end{align*}
We impose the condition $\coi_1<\coi_2<\cdots<\coi_{s}$ to determine them uniquely. 
Observe that $\coi_{s}=\rnk$.
For $r=1,\cdots,s$, define
\begin{align}\label{definition of a b c}
\parasec_{r}:=\coi_{r}-\coi_{r-1}-1, \quad
\parathr_{r}:=\lambda_{i_{r}}-\lambda_{i_{r+1}}-1, \quad
\parafir_{r}:=\lambda_{i_{r}}+\coi_{r}-\rnk-1
\end{align}
where we write $\coi_0=0$, and let
\begin{align}\label{definition of y}
\parafor_r:=
\binom{ \parasec_{r} }{ \parafir_{r} }
\binom{ \parathr_{r} }{ \parafir_{r} }
\quad \text{for $r=1,\cdots,s$}.
\end{align}
See Figure \ref{Intro picture} for the pictorial meaning of these numbers.
We use the convention $\binom{x}{y}=0$ unless $0\leq y\leq x$.
Now, let 
\begin{align}\label{intro int num}
I(\lambda) := (-1)^{\rnk+s}\parafor_{1}\cdots \parafor_{s}.
\end{align}
The next is the main theorem of this section.


\begin{theorem}\label{intro formula of type A intersection}
If $\A_1,\cdots,\A_{\rnk}$ form a nested chain of subsets, 
then we have
\begin{align*}
\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}} 
= I(\lambda)
\end{align*}
where $\mu_X$ is the fundamental homology class and $\lambda$ is the Young diagram consisting of $|\A_1|,\cdots,|\A_{\rnk}|$ reordered as a weakly decreasing sequence. Otherwise, the intersection number is zero.
\end{theorem}\vspace{5pt}
\begin{proof}
Recall that $\lambda$ is the Young diagram defined by $\lambda_i=|S_{\rnk+1-i}|$ for $i=1,\cdots,\rnk$.
We denote $J(\lambda) := \intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}$, and we show that $J(\lambda)=I(\lambda)$.
Observe that the condition $0 \leq \parafir_{r} \leq \parathr_{r}$ for all $r=1,\cdots,s$ is equivalent to the condition that each step of the zigzag line of the corners of $\lambda$ crosses the dotted anti-diagonal.
If this condition is not satisfied, then both of $J(\lambda)$ and $I(\lambda)$ are zero. Hence, in the following, we can assume that this condition holds.

We prove the claim by induction on $k:=\rnk-s$.
For the case $k=0$, we have $\lambda_i=|S_{\rnk+1-i}|=\rnk+1-i$ for all $1\leq i\leq \rnk$. So we have $J(\lambda)=1$ by (\ref{general type I=1}).
Since $\parafor_1=\cdots=\parafor_{\rnk}=1$ in this case, we have $I(\lambda)=1$, and the claim follows.
For a general case, there is a lower-right corner (say $r$-th corner from the top) of $\lambda$ whose vertical line has length $\geq2$. 
Then, Lemma \ref{invariance for type A} and Proposition \ref{dotted vanishing type A} combined together show that 
\begin{align}\label{recurrence eq type A}
&J(\lambda)
=
\begin{cases}
-
\begin{pmatrix} \parathr_{r-1} \\ \parafir_{r-1} \end{pmatrix} 
J(\lambda') 
-
\begin{pmatrix} \parathr_{r} \\ \parafir_{r} \end{pmatrix} 
J(\lambda'') 
\qquad
&\text{(if $\lambda\neq\lambda',\lambda''$)}
\vspace{10pt} \\
-
\begin{pmatrix} \parathr_{r-1} \\ \parafir_{r-1} \end{pmatrix} 
J(\lambda') 
&\text{(if $\lambda\neq\lambda', \lambda=\lambda''$)}
\vspace{10pt} \\
-
\begin{pmatrix} \parathr_{r} \\ \parafir_{r} \end{pmatrix} 
J(\lambda'')
&\text{(if $\lambda=\lambda', \lambda\neq\lambda''$)}
\end{cases}
\end{align}
where $\parasec_{r}$, $\parathr_{r}$, and $\parafir_{r}$ are those for $\lambda$, and the Young diagrams $\lambda'$ and $\lambda''$ are given by
\begin{align*}
\lambda'_j = 
\begin{cases}
\rnk+1-j \ &\text{if $j=i_{r-1}+1$}, \\
\lambda_j  &\text{otherwise},
\end{cases}
\qquad 
\lambda''_j = 
\begin{cases}
\rnk+1-j \ &\text{if $j=i_r$}, \\
\lambda_j  &\text{otherwise}.
\end{cases}
\end{align*}
Note that there are no cases that $\lambda=\lambda'=\lambda''$ since our vertical line has length $\geq2$.
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% $\lambda'$
\put(34.8000,-18.1000){\makebox(0,0)[lb]{$\lambda'$}}%
% STR 2 0 3 0
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% $\lambda''$
\put(51.0000,-18.1000){\makebox(0,0)[lb]{$\lambda''$}}%
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% 
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% 2 4778 1376 4778 2520
% 
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\special{pa 4778 2520}%
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% LINE 2 0 3 0
% 2 4778 2520 4874 2520
% 
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% LINE 2 2 3 0
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% 
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% LINE 2 2 3 0
% 2 4778 2568 5970 1376
% 
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\special{pa 5970 1376}%
\special{dt 0.030}%
\end{picture}%
\caption{The Young diagrams $\lambda$, $\lambda'$ and $\lambda''$}
\label{lambda and lambda' and lambda''}
\end{figure}

\noindent
If $\lambda\neq\lambda'$, by the induction hypothesis, we have
\begin{align}\label{inductive comp 10}
J(\lambda') 
&= 
(-1)^{\rnk+(s+1)}
\parafor_1\cdots \parafor_{r-2} \cdot 
\binom{\parasec_{r-1}}{\parafir_{r-1}}
\cdot 1 \cdot
\binom{\parasec_{r}-1}{\parafir_{r}}
\binom{\parathr_{r}}{\parafir_{r}}
\cdot \parafor_{r+1}\cdots \parafor_{s}.
\end{align}
Similarly, if $\lambda\neq\lambda''$, we have
\begin{align}\label{inductive comp 20}
J(\lambda'')
&= 
(-1)^{\rnk+(s+1)}
\parafor_1\cdots \parafor_{r-1} \cdot 
\binom{\parasec_{r}-1}{\parafir_{r}-1}
\cdot 1 \cdot 
\parafor_{r+1}\cdots \parafor_{s}.
\end{align}
Observe that the right-hand-sides of (\ref{inductive comp 10}) and (\ref{inductive comp 20}) vanish when $\lambda=\lambda'$ and $\lambda=\lambda''$, respectively.
Hence, the right-hand-side of the equation (\ref{recurrence eq type A}) can be written as
\begin{align*}
J(\lambda)
&=
-
\binom{\parathr_{r-1}}{\parafir_{r-1}} \cdot
(-1)^{\rnk+s+1}\parafor_1\cdots \parafor_{r-2}
\binom{\parasec_{r-1}}{\parafir_{r-1}}
\binom{\parasec_{r}-1}{\parafir_{r}}
\binom{\parathr_{r}}{\parafir_{r}}
\parafor_{r+1}\cdots \parafor_{s}  \\
&\qquad\qquad
-
\binom{\parathr_{r}}{\parafir_{r}}
\cdot
(-1)^{\rnk+s+1}\parafor_1\cdots \parafor_{r-2}
\parafor_{r-1} \binom{\parasec_{r}-1}{\parafir_{r}-1} 
\parafor_{r+1}\cdots \parafor_{s} \\
&=(-1)^{\rnk+s}\parafor_1\cdots \parafor_{s}=I(\lambda).
\end{align*}
\end{proof}


%\vspace{10pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The ring structure of the cohomology}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The cohomology ring $H^*(X)$ of the toric manifold $X$ associated with the fan $\Delta(A_{\rnk})$ is given by the face ring of $\Delta(A_{\rnk})$ modulo the linear relations (\ref{prelim linear relation}) (\cite{Fulton1}). 
As an application of Theorem \ref{intro formula of type A intersection}, we describe the ring structure of the cohomology $H^*(X)$ 
in terms of an additive basis.

Recall that $\invdiv{u\fcw_i}$ for some $i\in[\rnk]$ and permutation $u\in \mathfrak{S}_{\rnk+1}$ is the invariant divisor of $X$ associated with the ray generated by $u\fcw_i\in\CL$. 
Let \vspace{-4pt}
\begin{align*}
X_{u} := \bigcap_{i} \invdiv{u\omega_i} 
\end{align*}

\vspace{-6pt}\noindent
for each permutation $u\in \mathfrak{S}_{\rnk+1}$ where 
$i$ runs over all descents in $u$. 
Here, a descent in $u$ is a number $i\in[\rnk]$ which satisfies $u(i)>u(i+1)$, and we denote by $d(u)$ the number of descents in $u$.
Denote by $[X_u]\in H^{2d(u)}(X)$ the Poincar$\acute{\text{e}}$ dual of $X_{u}$, then we have \vspace{-4pt}
\begin{align*}
[X_{u}]
 = \prod_{i} \tau_{\{u(1),\cdots,u(i)\}}
\end{align*}

\vspace{-6pt}\noindent
where $i$ runs over all descents in $u$ since invariant divisors of $X$ intersect transversely.
$\{[X_u]\}_{u\in \mathfrak{S}_{\rnk+1}}$ forms a module basis of $H^*(X)$ (See \cite{Klyachko} or \cite{Batyrev-Blume} for combinatorial proofs and \cite{De Mari-Procesi-Shayman} for a geometric proof). 
The class $[X_u]$ can be expressed by a Young diagram consisting of the descents in $u$ with the numbers in the nested chain of subsets in 
$D(u)$ (see (\ref{def of D(u)}) for the definition)
written above the diagram so that each column represents the written number above it. 
This expression effectively encodes the descents in $u$ and the information of the chain of subsets.
Denoting $Y^{u}:=w_0X_{w_0u}=\cap X_{u[i]}$ where the intersection runs over all ascents $i$ in $u$, the similar expression works for $[Y^u]$ and the chain of subsets in 
$A(u)$.
Here, $w_0$ is the longest element of $\mathfrak{S}_{\rnk+1}$.
\begin{figure}[h]
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% $3$
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\end{picture}%
\caption{Two examples for $\rnk=5$ in one-line notations}
\label{X_u and Young diagrams}
\end{figure}

For $u,v,w\in \mathfrak{S}_{\rnk+1}$, we have the Young diagram $\lambda_{uv}^w$ constructed in Section 1.
Recall that $\mu_X$ is the fundamental homology class of $X$.
The following corollary provides the combinatorial rule to compute the intersection number of $X_u$, $X_v$, and $Y^w$ in $X$.
\begin{corollary}\label{triple intersection}
$\displaystyle{\intnum{[Y^w][X_u][X_v]}=I(\WY{u}{v}{w})}$.
\end{corollary}
For example, for $\rnk=4$, we have 
\begin{align*}
\intnum{[Y^{35421}][X_{12354}][X_{31254}]} = 2.
\end{align*}
In Figure \ref{pic-typeA-2}, we left the numbers on the Young diagram so that we can see the nested chain of subsets appeared in the construction of $\WY{u}{v}{w}$.
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\caption{}
\label{pic-typeA-2}
\end{figure}

Since $\{[X_u]\}_{u\in \mathfrak{S}_{\rnk+1}}$ forms a module basis of $H^*(X)$, we can consider the expansion coefficients of the product
\begin{align}\label{expansion of product}
[X_{u}][X_{v}] = \sum_{w} \SC{u}{v}{w} [X_{w}].
\end{align}
For example, these coefficients for the product $[X_{s_i}][X_{s_j}]$ can be calculated directly if $|i-j|\geq1$ where $s_i$ is the simple reflection exchanging $i$ and $i+1$.
In fact, we have 
\begin{align}\label{direct computations}
[X_{s_i}][X_{s_j}] = 
\begin{cases}
[X_{s_is_j}](=[X_{s_js_i}]) \quad &\text{if $|i-j|\geq2$}, \\
0 &\text{if $|i-j|=1$}.
\end{cases}
\end{align}
since $\{1,\cdots,i-1,i+1\}$ and $\{1,\cdots,i,i+2\}$ do not form a chain of subsets.

Let us describe each structure constant $c_{u,v}^w$ in terms of intersection numbers computed above. Since a Weyl chamber $\sigma_{u}=\text{cone}(u\fcw_1,\cdots,u\fcw_{\rnk})$ is a maximal cone of the fan, $\sigma_{u}$ is identified with a fixed point of the canonical torus action on $X$ denoted by $p_u\in X$ where $p_u$ is the intersection $\cap_{i=1}^{\rnk}\invdiv{u\omega_i}$.
Then from the definition of $X_{w'}$, one can show that 
$p_u\in X_{w'}$ implies $u\geq w'$ (e.g. \cite{Bjorner-Brenti}; Theorem 2.6.3) where $>$ is the Bruhat order.
If $Y^{w}\cap X_{w'}\neq\emptyset$, then $Y^{w}\cap X_{w'}$ must contain a fixed point since it is an intersection of invariant divisors of $X$, and hence it follows that $w\geq w'$. 
From this observation, we see that $Y^{w}\cap X_{w'}=\emptyset$ unless $w\geq w'$. Also, it is easy to see that $Y^w$ and $X_{w'}$ intersect transversally when $w=w'$. 
Recalling that the class $[Y^{w}][X_{w'}]$ is supported on the intersection $Y^{w}\cap X_{w'}$, we obtain 
\begin{align}\label{intersection of up and down}
\intnum{[Y^{w}][X_{w'}]}=
\begin{cases}
0 \quad &\text{unless $w\geq w'$ and $d(w)=d(w')$} , \\
1 &\text{if } w=w'.
\end{cases} 
\end{align}
See \cite{De Mari-Procesi-Shayman} for a proof using a cellular decomposition of $X$.
Let $\imat$ be the matrix whose $(u,v)$-component is given by $\imatcomp{u}{v}=\intnum{[Y^u][X_v]}=I(\WY{v}{\hspace{1pt}\text{id}}{u})$ for all $u,v\in W$. This matrix $\imat$ is invertible over $\Z$ because of (\ref{intersection of up and down}). 
Now, each coefficient $\SC{u}{v}{w}$ in (\ref{expansion of product}) is a linear transform of the intersection numbers $I(\WY{u}{v}{w})$;
\begin{align}\label{formula for str const}
\SC{u}{v}{w}
= \sum_{w'} \imatcompinv{w}{w'} I(\WY{u}{v}{w'}).
\end{align}
We note that it suffices to take the sum for $w'$ satisfying $d(w)=d(w')$ and $w\geq w'$ since $\imatcompinv{w}{w'}$ is also upper-triangular in the sense of the right-hand-side of (\ref{intersection of up and down}). 

\noindent
So the formula (\ref{formula for str const}) exhibits the upper-triangularity of $\SC{u}{v}{w}$ in the sense that $\SC{u}{v}{w}=0$ unless $u,v\leq w$ since $I(\WY{u}{v}{w})$ satisfies the same property.

The transition formula (\ref{formula for str const}) together with (\ref{intersection of up and down}) provides us a recursive formula for the structure constants $\SC{u}{v}{w}$ which is manifestly integral;  
\begin{align*}
\SC{u}{v}{w} = I(\WY{u}{v}{w}) - \sum_{w>w'}\imatcomp{w}{w'}\SC{u}{v}{w'}.
\end{align*} 
Note again that it is enough to take the sum for all $w'$ satisfying $d(w)=d(w')$ and $w> w'$.
From this recursion, we recover (\ref{direct computations}), and we can compute the expansion of $[X_{s_i}]^2$. For example, if $\rnk=3$, we obtain
\begin{align*}
[X_{2134}]^2
&= [X_{2431}] - [X_{4213}] - [X_{3421}] - [X_{3241}] - [X_{3214}]. 
\end{align*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Other classical types}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that the argument in the previous section can be naturally generalized to arbitrary root systems by considering the non-singular subvariety
\begin{align}\label{def of Xu}
X_u = \bigcap_{i} \invdiv{u\fcw_i}
\end{align} 
for each $u\in W$ where $\invdiv{u\fcw_i}$ is the invariant divisor of $X$ corresponding to the ray generated by $u\fcw_i$ and $i$ runs over all $i$ satisfying $u(\alpha_i)\in\Phi^-$.
Here, $\Phi^-$ is the set of negative roots. It follows that the Poincar$\acute{\text{e}}$ duals $\{[X_u]\}_{u\in W}$ form an additive basis of the integral cohomlogy $H^*(X)$ (see \cite{De Mari-Procesi-Shayman}).
\begin{remark}
\emph{
The collections $\{c_{u, v}^w\}_{u,v,w\in W}$ and $\{\intnum{[Y^w][X_u][X_v]}\}_{u,v,w\in W}$ are independent on the choice of the simple roots $\Pi$.
}
\end{remark}

\subsection{\texorpdfstring{Intersection numbers for type $B_{\rnk}$}{Intersection numbers for type Bn}}\label{section int num type B}
For the classical root system of type $B_{\rnk}$, the roots are $\{t_i-t_j, \ \pm(t_i+t_j) , \ \pm t_{i} \in E \mid 1\leq i\neq j\leq \rnk\}$ where $E=\R^{\rnk}$. We choose $\SimR=\{t_i-t_{i+1}, \ t_{\rnk} \mid 1\leq i\leq \rnk-1\}$ as a set of simple roots, and write $\alpha_i=t_i-t_{i+1} (1\leq i\leq\rnk-1)$, $\alpha_{\rnk}=t_{\rnk}$. The Weyl group $\widetilde{\mathfrak{S}}_{\rnk}$ is the $\rnk$-th signed permutation group. Letting $t_{-i}:=-t_i$ for all $1\leq i\leq \rnk$, $u\in\widetilde{\mathfrak{S}}_{\rnk}$ acts on $E$ by $ut_i=t_{u(i)}$.
The minimal generators $\fcw_1,\cdots,\fcw_{\rnk}\in E^*$ of the fundamental Weyl chamber are $\fcw_i=e_1+\cdots+e_i$ for $i=1,\cdots,\rnk$.

Let $[\pm\rnk] = \{1,\cdots,\rnk, -1,\cdots,-\rnk\}$.
For $\A\in2^{[\pm\rnk]}$, consider a condition
\begin{align}\label{type B basics 100}\tag{$*$}
\text{ for any $i\in[\pm\rnk]$, if $i\in \A$ then $-i\notin \A$}.
\end{align} 
We have a well-defined map $\dPhi \rightarrow 2^{[\pm\rnk]}$ by $u\fcw_i \mapsto \{u(1),\cdots,u(i)\}$.
This leads us to an identification
\begin{align*}
\dPhi \quad \longleftrightarrow \quad \text{the set of non-empty subsets of $[\pm\rnk]$ satisfying (\ref{type B basics 100})}.
\end{align*} 
Now, for each $\emptyset\subsetneq S\subset[\pm\rnk]$ satisfying (\ref{type B basics 100}), we define
$\tau_{\A}:=\tau_{u\omega_i}$ where $u\omega_i\in\dPhi$ corresponds to $\A$ by this identification. 
For $\emptyset\subsetneq \A_1,\cdots, \A_{q}\subset[\pm\rnk] \ (1\leq q\leq\rnk)$ satisfying (\ref{type B basics 100}), we have that $\tau_{\A_1}\cdots\tau_{\A_{q}}=0$ unless these sets form a nested chain of subsets, as in the case for type $A_{\rnk}$.

For each $k\in[\pm\rnk]$, let $\Bsetsec\subset[\pm\rnk]$ satisfy (\ref{type B basics 100}), $k\in \Bsetsec$, and $|\Bsetsec|=\rnk$. From the linear relation (\ref{prelim linear relation}) for the root $\alpha=t_k$, we can deduce that
\begin{align*}
{\tau_{\Bsetsec}}^2
= -
\sum_{\substack{k\in \Bsetrun \\ \Bsetrun\subsetneq \Bsetsec}} \tau_{\Bsetrun} \tau_{\Bsetsec}
\end{align*}  
where the sum is taken over all $\emptyset \subsetneq \Bsetrun\subset [\pm\rnk]$ satisfying (\ref{type B basics 100}) with the prescribed conditions.
Similarly, for each $k,l\in[\pm\rnk]$, let $\Bsetsec\subset[\pm\rnk]$ satisfy (\ref{type B basics 100}), $k\in \Bsetsec$, and $\pm l\notin \Bsetsec$ (hence $1\leq |\Bsetsec|\leq\rnk-1$). 
Then from (\ref{prelim linear relation}) for the root $\alpha=t_k-t_l$, we obtain 
\begin{align*}
&{\tau_{\Bsetsec}}^2
=
-
\sum_{\substack{k\in \Bsetrun, \ \pm l\notin \Bsetrun \\ \Bsetrun\neq \Bsetsec}} \tau_{\Bsetrun} \tau_{\Bsetsec} 
-
\sum_{k,-l\in \Bsetrun} 2\tau_{\Bsetrun}\tau_{\Bsetsec} .
\end{align*} 
Observe that the second summand will vanish after multiplying $\tau_{\Bsetfir}$ and $\tau_{\Bsetthr}$ for $\Bsetfir\subset \Bsetsec\backslash{\{k\}}$ and $\Bsetsec\coprod\{l\}\subset \Bsetthr$ where we write $\tau_{\emptyset}=0$.
So these two equations can be used to prove the separation rule similar to Lemma \ref{type A separation}, and we obtain the same type of vanishing property as in Proposition \ref{dotted vanishing type A}. 
Now the argument in the proof of Theorem \ref{intro formula of type A intersection} also works for this case, and it follows that
\begin{theorem}
If $\emptyset\subsetneq \A_1,\cdots,\A_{\rnk}\subset[\pm\rnk]$ satisfying \emph{(\ref{type B basics 100})} form a nested chain of subsets, then we have
\begin{align*}
\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}
= 2^{\rnk-\lambda_1} I(\lambda)
\end{align*}
where 
$\mu_X$ is the fundamental homology class of $X$ and
$\lambda$ is the Young diagram consisting of $|\A_1|,\cdots,|\A_{\rnk}|$ reordered as a weakly decreasing sequence and $I$ is the function defined in \eqref{intro int num}. Otherwise, the intersection number is zero.
\end{theorem}

Let $\alpha_i:=t_{i}-t_{i+1}$ for $1\leq i\leq\rnk-1$ and $\alpha_{\rnk}:=t_{\rnk}$.
For each signed permutation $u\in \widetilde{\mathfrak{S}}_{\rnk}$, an element $i\in[\rnk]$ satisfies $u(\alpha_i)\in\Phi^-$ if and only if
\begin{itemize}
 \item[(D-1)] if $i\leq n-1$, then $u(i)>u(i+1)$ with the same sign or $u(i)<u(i+1)$ with different signs,
 \item[(D-2)] if $i=\rnk$, then $u(i)<0$.
\end{itemize}
Similarly, consider the conditions
\begin{itemize}
 \item[(A-1)] if $i\leq n-1$, then $u(i)<u(i+1)$ with the same sign or $u(i)>u(i+1)$ with different signs,
 \item[(A-2)] if $i=\rnk$, then $u(i)>0$.
\end{itemize}
Denoting 
\begin{align*}
D(u):=\{ u[i] \mid \text{$i$ satisfies (D)} \}  
\quad \text{and} \quad
A(u):=\{ u[i] \mid \text{$i$ satisfies (A)} \},
\end{align*}
we define a Young diagram $\lambda_{u,v}^w$ in the manner described in the last section.
Note that we put $I(\emptyset)=0$ as a convention.

Now, for signed permutations $u,v,w\in \widetilde{\mathfrak{S}}_{\rnk}$, the intersection number of $Y^w$, $X_u$, and $X_v$ in $X$ of type $B_{\rnk}$ is given by the following. 
\begin{corollary}\label{triple intersection for type B}
For signed permutations $u,v,w\in \widetilde{\mathfrak{S}}_{\rnk}$, we have
\[ \intnum{[Y^w][X_u][X_v]}=2^{\rnk-(\lambda_{u,v}^w)_1}I(\WY{u}{v}{w}) \]
where $I$ is the function defined in \eqref{intro int num}. 
\end{corollary}



For example, for $\rnk=4$ with the convention $\bar{k}=-k$, Corollary \ref{triple intersection for type B} computes
\begin{align*}
\intnum{[Y^{2\bar{3}\bar{1}\bar{4}}] [X_{2\bar{3}14}] [X_{2\bar{3}14}]}  = 4.
\end{align*}
\vspace{-20pt}
\begin{figure}[h]
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\caption{}
\label{EX for type B}
\end{figure}




\subsection{\texorpdfstring{Intersection numbers for type $C_{\rnk}$}{Intersection numbers for type Cn}}\label{section int num type C}
For the classical root system of type $C_{\rnk}$, the roots are $\{t_i-t_j, \ \pm(t_i+t_j), \ \pm2t_i \in E \mid 1\leq i\neq j\leq \rnk\}$ where $E=\R^{\rnk}$. We choose $\SimR=\{t_i-t_{i+1}, \ 2t_{\rnk} \mid 1\leq i\leq \rnk-1\}$ as a set of simple roots, and write $\alpha_i=t_i-t_{i+1} (1\leq i\leq\rnk-1)$, $\alpha_{\rnk}=2t_{\rnk}$. The Weyl group $\widetilde{\mathfrak{S}}_{\rnk}$ is the $\rnk$-th signed permutation group as above.
The minimal generators $\fcw_1,\cdots,\fcw_{\rnk}$ of the fundamental Weyl chamber are $\fcw_i=e_1+\cdots+e_i$ for $i=1,\cdots,\rnk-1$ and $\fcw_{\rnk}=\frac{1}{2}(e_1+\cdots+e_{\rnk})$.

We have a well-defined map $\dPhi \rightarrow 2^{[\pm\rnk]}$ by $v\fcw_i \mapsto \{v(1),\cdots,v(i)\}$, and obtain an identification $\dPhi$ and the set of non-empty subsets of $[\pm\rnk]$ satisfying (\ref{type B basics 100}).
For $\emptyset\subsetneq \A_1,\cdots, \A_{q}\subsetneq[\pm\rnk] \ (1\leq q\leq \rnk)$ satisfying (\ref{type B basics 100}), we have that $\tau_{\A_1}\cdots\tau_{\A_{q}}=0$ unless these sets form a nested chain of subsets where $\tau_{\A}$ is defined as in Section \ref{section int num type B}.

For each $k\in[\pm\rnk]$, let $\Csetsec\subset[\pm\rnk]$ satisfy (\ref{type B basics 100}), $k\in \Csetsec$, and $|\Csetsec|=\rnk$.
Then (\ref{prelim linear relation}) for the root $\alpha=2t_k$ shows that
\begin{align*}
{\tau_{\Csetsec}}^2
= -
\sum_{\substack{k\in \Csetrun \\ \Csetrun\subsetneq \Csetsec}} 2\tau_{\Csetrun} \tau_{\Csetsec}
\end{align*}  
where the sum is taken over all $\emptyset \subsetneq \Bsetrun\subset [\pm\rnk]$ satisfying (\ref{type B basics 100}) with the prescribed conditions.
For each $k,l\in[\pm\rnk]$, let $\Csetsec\subset[\pm\rnk]$ satisfy (\ref{type B basics 100}), $k\in \Csetsec$, and $\pm l\notin \Csetsec$ (hence $1\leq |\Csetsec|\leq\rnk-1$). 
Then from (\ref{prelim linear relation}) for the root $\alpha=t_k-t_l$, we obtain
\begin{align*}
&{\tau_{\Csetsec}}^2
=
-
\sum_{\substack{k\in \Csetrun, \ \pm l\notin \Csetrun \\ \Csetrun\neq \Csetsec}} \tau_{\Csetrun} \tau_{\Csetsec} 
-
\sum_{\substack{k\in \Csetrun, \ -l\in \Csetrun \\ |\Csetrun|\neq\rnk}} 2\tau_{\Csetrun}\tau_{\Csetsec} 
-
\sum_{\substack{k\in \Csetrun, \ -l\in \Csetrun \\ |\Csetrun|=\rnk}} \tau_{\Csetrun} \tau_{\Csetsec}.
\end{align*} 
With a similar observation made for type $B_{\rnk}$, we again have the same type of vanishing property as in Proposition \ref{dotted vanishing type A}. Hence, we obtain 
\begin{theorem}
If $\emptyset\subsetneq \A_1,\cdots,\A_{\rnk}\subsetneq[\pm\rnk]$ satisfying \emph{(\ref{type B basics 100})} form a nested chain of subsets, then we have
\begin{align*}
\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}
= 2^{\rnk-\lambda_1+m-1} I(\lambda)
\end{align*}
where $\mu_X$ is the fundamental homology class of $X$ and $\lambda$ is the Young diagram consisting of the numbers $|\A_1|,\cdots,|\A_{\rnk}|$ reordered as a weakly decreasing sequence and $I$ is the function defined in \eqref{intro int num} and $m$ is the number of rows of $\lambda$ of length $\rnk$. Otherwise, the intersection number is zero.
\end{theorem}


For signed permutations $u,v,w\in \widetilde{\mathfrak{S}}_{\rnk}$, let $\lambda_{u,v}^w$ be the Young diagram defined in Section \ref{section int num type B}. 
The intersection number of $Y^w$, $X_u$, and $X_v$ in $X$ of type $C_{\rnk}$ is given by the following.  
\begin{corollary}\label{triple intersection for type C}
For signed permutations $u,v,w\in \widetilde{\mathfrak{S}}_{\rnk}$, we have
\[ \intnum{[Y^w][X_u][X_v]}=2^{\rnk-(\lambda_{u,v}^w)_1+m-1}I(\WY{u}{v}{w}) \]
where $I$ is the function defined in \eqref{intro int num} and and $m$ is the number of rows of $\lambda_{u,v}^w$ of length $\rnk$.
\end{corollary}





\subsection{\texorpdfstring{Intersection numbers for type $D_{\rnk}$}{Intersection numbers for type Dn}}
For the classical root system of type $D_{\rnk}$, the roots are $\{t_i-t_j, \ \pm(t_i+t_j) \in E \mid 1\leq i\neq j\leq \rnk\}$ where $E=\R^{\rnk}$. We choose $\SimR=\{t_i-t_{i+1}, \ t_{\rnk-1}+t_{\rnk} \mid 1\leq i\leq \rnk-1\}$ as a set of simple roots, and write $\alpha_i=t_i-t_{i+1} (1\leq i\leq\rnk-2)$, $\alpha_{\rnk-1}=t_{\rnk-1}+t_{\rnk}$, $\alpha_{\rnk}=t_{\rnk-1}-t_{\rnk}$. The Weyl group $\widetilde{\mathfrak{S}}_{\rnk}^+$ is the $\rnk$-th \textit{even signed permutation group} defined by
\begin{align*}
\widetilde{\mathfrak{S}}_{\rnk}^+:=\{w\in\widetilde{\mathfrak{S}}_{\rnk} \mid \text{ the number of $i$ with $w(i)<0$ is even}\}
\end{align*}
where $\widetilde{\mathfrak{S}}_{\rnk}$ is the $\rnk$-th signed permutation group.
The minimal generators $\fcw_1,\cdots,\fcw_{\rnk}$ $\in E^*$ of the fundamental Weyl chamber are $\fcw_i=e_1+\cdots+e_i$ for $i=1,\cdots,\rnk-2$, \  $\fcw_{\rnk-1}=\frac{1}{2}(e_1+\cdots+e_{\rnk-1}+e_{\rnk})$ and $\fcw_{\rnk}=\frac{1}{2}(e_1+\cdots+e_{\rnk-1}-e_{\rnk})$.



For $\A\in2^{[\pm\rnk]}$, consider a condition
\begin{align}\label{type D basics 100}\tag{$**$}
\text{$|\A|\neq \rnk-1$, and if $i\in \A$ then $-i\notin \A$ for any $i\in[\pm\rnk]$ }.
\end{align} 
We have a well-defined map $\dPhi \rightarrow 2^{[\pm\rnk]}$ given by
\begin{align*}
&u\fcw_i \mapsto u[i]=\{u(1),\cdots,u(i)\} \quad \text{for } 1\leq i\leq \rnk-2, \\
&u\fcw_{\rnk-1} \mapsto u[n]_+=\{u(1),\cdots,u(\rnk-1), u(\rnk)\}, \\
&u\fcw_{\rnk} \mapsto u[n]_-=\{u(1),\cdots,u(\rnk-1),-u(\rnk)\}.
\end{align*} 
where $[\rnk]_+ = \{1,2,\cdots,n-1,n\}$ and $[\rnk]_- = \{1,2,\cdots,n-1,-n\}$.
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\caption{The Dynkin diagram and a maximal chain of subsets for type $D_{\rnk}$}
\label{type D chain}
\end{figure}
\vspace{-3pt}
\\ 
It follows that this map $\dPhi \rightarrow 2^{[\pm\rnk]}$ is an injection.
In fact, we cannot have 
\[\{u(1),\cdots,u(\rnk-1), u(\rnk)\}=\{v(1),\cdots,v(\rnk-1),-v(\rnk)\}\] 
for any $u,v\in\widetilde{\mathfrak{S}}_{\rnk}^+$ since the number of negative integers in the left hand side and the right-hand-side are different, and so
$u\fcw_{\rnk-1}$ and $v\fcw_{\rnk}$ are never mapped to the same element.
The other cases are left to the reader.
So we can make an identification
\begin{align}\label{type D identification}
\dPhi \quad \longleftrightarrow \quad \text{the set of non-empty subsets of $[\pm\rnk]$ satisfying (\ref{type D basics 100})}.
\end{align} 
Hence,  for each $\emptyset\subsetneq \A\subset[\pm\rnk]$ satisfying (\ref{type D basics 100}), we define
$\tau_{\A}:=\tau_{u\omega_i}$ where $u\omega_i\in\dPhi$ corresponds to $\A$ by this identification. 


Let us denote by $\mathcal{C}$ the set of chains of subsets $\{\DA_i\}_i$ of $[\pm n]$ of the following form:
there exists $u\in\widetilde{\mathfrak{S}}_{\rnk}^+$ such that $\DA_i=u[i]$ for $1\leq i\leq n-2$, $\DA_{n-1}=u[n]_+$, and $\DA_{n}=u[n]_-$.
Note that $\{\DA_i\}_i$ does not have a set of order $n-1$ and satisfies the same inclusion relation shown in Figure \ref{type D chain}.
A \textit{subchain} $\{\A_i\}_{i}$ of a chain $\{\DA_i\}_{i}$ in $\mathcal{C}$ is a sequence satisfying $\A_j\in\{\DA_i\}_{i}$ for $1\leq j\leq n$ and $S_{j}\subset S_{j'}$ for $1\leq j\leq j'\leq \rnk$ unless $|S_{j}|=|S_{j'}|=\rnk$.
For $\emptyset\subsetneq\A_1,\cdots,\A_{q}\subsetneq[\pm\rnk] \ (1\leq q\leq \rnk)$ satisfying (\ref{type D basics 100}), we have $\tau_{\A_1}\cdots\tau_{\A_{q}}=0$ unless the sequence forms a subchain of a chain in $\mathcal{C}$ up to reordering. 
Let $\{\A_i\}_i$ be a subchain of a chain in $\mathcal{C}$. 
For $\A_i$ satisfying $|\A_i|=\rnk$, we say that $\A_i$ is \textit{even} (resp. \textit{odd}) if the number of negative elements of $\A_i$ is even (resp. odd).
Recall that $\mu_X$ is the fundamental homology class of $X$.
The following is Lemma \ref{prelim Weyl inv} for type $D_{\rnk}$.
\begin{lemma}\label{invariance property for type D}
Let $\{\A_i\}_i$ and $\{\A'_i\}_i$ be subchains of some chains in $\mathcal{C}$. 
If $|\A_i|=|\A'_i|$ for $i=1,\cdots,\rnk$ and the number of even $\A_i$'s and the number of even $\A'_i$'s are the same, then $\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}} = \intnum{\tau_{\A'_1}\cdots\tau_{\A'_{\rnk}}}$.
\end{lemma}


Let $\{\A_i\}_i$ be a subchain of a chain in $\mathcal{C}$. We denote by $\lambda$ the \textit{signed Young diagram} consisting of $\lambda_i=|\A_{\rnk+1-i}|$ for $i=1,\cdots,\rnk$ where the label of $\lambda$ is defined as follows:
if we have $\lambda_i=\rnk$, then we label this row by $+$ (resp. $-$) if $\A_{\rnk+1-i}$ is even (resp. odd).
Recall from Section 3 that the \textit{dotted anti-diagonal line} drawn on the Young diagram is the dotted line shifted down half the length of a single box from the standard anti-diagonal. 
Our first aim is to prove the following.
\begin{proposition}\label{zigzag lemma type D}
\emph{(The vanishing property)}
$\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}=0$ unless each step of the zigzag line of the lower-right corners of $\lambda$ crosses the dotted anti-diagonal.
\end{proposition}
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\special{pa 1806 1832}%
\special{pa 1806 1942}%
\special{pa 1696 1942}%
\special{pa 1696 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 1696 1941 1805 2049
% 
\special{pn 8}%
\special{pa 1696 1942}%
\special{pa 1806 1942}%
\special{pa 1806 2050}%
\special{pa 1696 2050}%
\special{pa 1696 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 1696 2049 1805 2157
% 
\special{pn 8}%
\special{pa 1696 2050}%
\special{pa 1806 2050}%
\special{pa 1806 2158}%
\special{pa 1696 2158}%
\special{pa 1696 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 1696 2157 1805 2265
% 
\special{pn 8}%
\special{pa 1696 2158}%
\special{pa 1806 2158}%
\special{pa 1806 2266}%
\special{pa 1696 2266}%
\special{pa 1696 2158}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 1723 1913 1831
% 
\special{pn 8}%
\special{pa 1806 1724}%
\special{pa 1914 1724}%
\special{pa 1914 1832}%
\special{pa 1806 1832}%
\special{pa 1806 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 1831 1913 1941
% 
\special{pn 8}%
\special{pa 1806 1832}%
\special{pa 1914 1832}%
\special{pa 1914 1942}%
\special{pa 1806 1942}%
\special{pa 1806 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 1941 1913 2049
% 
\special{pn 8}%
\special{pa 1806 1942}%
\special{pa 1914 1942}%
\special{pa 1914 2050}%
\special{pa 1806 2050}%
\special{pa 1806 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 2049 1913 2157
% 
\special{pn 8}%
\special{pa 1806 2050}%
\special{pa 1914 2050}%
\special{pa 1914 2158}%
\special{pa 1806 2158}%
\special{pa 1806 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 2157 1913 2265
% 
\special{pn 8}%
\special{pa 1806 2158}%
\special{pa 1914 2158}%
\special{pa 1914 2266}%
\special{pa 1806 2266}%
\special{pa 1806 2158}%
\special{fp}%
% BOX 2 0 3 0
% 2 1913 1723 2022 1831
% 
\special{pn 8}%
\special{pa 1914 1724}%
\special{pa 2022 1724}%
\special{pa 2022 1832}%
\special{pa 1914 1832}%
\special{pa 1914 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 1913 1831 2022 1941
% 
\special{pn 8}%
\special{pa 1914 1832}%
\special{pa 2022 1832}%
\special{pa 2022 1942}%
\special{pa 1914 1942}%
\special{pa 1914 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 1913 1941 2022 2049
% 
\special{pn 8}%
\special{pa 1914 1942}%
\special{pa 2022 1942}%
\special{pa 2022 2050}%
\special{pa 1914 2050}%
\special{pa 1914 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 1913 2049 2022 2157
% 
\special{pn 8}%
\special{pa 1914 2050}%
\special{pa 2022 2050}%
\special{pa 2022 2158}%
\special{pa 1914 2158}%
\special{pa 1914 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 2022 1723 2130 1831
% 
\special{pn 8}%
\special{pa 2022 1724}%
\special{pa 2130 1724}%
\special{pa 2130 1832}%
\special{pa 2022 1832}%
\special{pa 2022 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 2022 1831 2130 1941
% 
\special{pn 8}%
\special{pa 2022 1832}%
\special{pa 2130 1832}%
\special{pa 2130 1942}%
\special{pa 2022 1942}%
\special{pa 2022 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 1480 1615 1588 1723
% 
\special{pn 8}%
\special{pa 1480 1616}%
\special{pa 1588 1616}%
\special{pa 1588 1724}%
\special{pa 1480 1724}%
\special{pa 1480 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 1588 1615 1696 1723
% 
\special{pn 8}%
\special{pa 1588 1616}%
\special{pa 1696 1616}%
\special{pa 1696 1724}%
\special{pa 1588 1724}%
\special{pa 1588 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 1696 1615 1805 1723
% 
\special{pn 8}%
\special{pa 1696 1616}%
\special{pa 1806 1616}%
\special{pa 1806 1724}%
\special{pa 1696 1724}%
\special{pa 1696 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 1615 1913 1723
% 
\special{pn 8}%
\special{pa 1806 1616}%
\special{pa 1914 1616}%
\special{pa 1914 1724}%
\special{pa 1806 1724}%
\special{pa 1806 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 1913 1615 2022 1723
% 
\special{pn 8}%
\special{pa 1914 1616}%
\special{pa 2022 1616}%
\special{pa 2022 1724}%
\special{pa 1914 1724}%
\special{pa 1914 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 2022 1615 2130 1723
% 
\special{pn 8}%
\special{pa 2022 1616}%
\special{pa 2130 1616}%
\special{pa 2130 1724}%
\special{pa 2022 1724}%
\special{pa 2022 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 2130 1615 2238 1723
% 
\special{pn 8}%
\special{pa 2130 1616}%
\special{pa 2238 1616}%
\special{pa 2238 1724}%
\special{pa 2130 1724}%
\special{pa 2130 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 1480 1507 1588 1615
% 
\special{pn 8}%
\special{pa 1480 1508}%
\special{pa 1588 1508}%
\special{pa 1588 1616}%
\special{pa 1480 1616}%
\special{pa 1480 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 1588 1507 1696 1615
% 
\special{pn 8}%
\special{pa 1588 1508}%
\special{pa 1696 1508}%
\special{pa 1696 1616}%
\special{pa 1588 1616}%
\special{pa 1588 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 1696 1507 1805 1615
% 
\special{pn 8}%
\special{pa 1696 1508}%
\special{pa 1806 1508}%
\special{pa 1806 1616}%
\special{pa 1696 1616}%
\special{pa 1696 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 1805 1507 1913 1615
% 
\special{pn 8}%
\special{pa 1806 1508}%
\special{pa 1914 1508}%
\special{pa 1914 1616}%
\special{pa 1806 1616}%
\special{pa 1806 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 1913 1507 2022 1615
% 
\special{pn 8}%
\special{pa 1914 1508}%
\special{pa 2022 1508}%
\special{pa 2022 1616}%
\special{pa 1914 1616}%
\special{pa 1914 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 2022 1507 2130 1615
% 
\special{pn 8}%
\special{pa 2022 1508}%
\special{pa 2130 1508}%
\special{pa 2130 1616}%
\special{pa 2022 1616}%
\special{pa 2022 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 2130 1507 2238 1615
% 
\special{pn 8}%
\special{pa 2130 1508}%
\special{pa 2238 1508}%
\special{pa 2238 1616}%
\special{pa 2130 1616}%
\special{pa 2130 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 2238 1507 2347 1615
% 
\special{pn 8}%
\special{pa 2238 1508}%
\special{pa 2348 1508}%
\special{pa 2348 1616}%
\special{pa 2238 1616}%
\special{pa 2238 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 2347 1507 2455 1615
% 
\special{pn 8}%
\special{pa 2348 1508}%
\special{pa 2456 1508}%
\special{pa 2456 1616}%
\special{pa 2348 1616}%
\special{pa 2348 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 2130 1723 2238 1831
% 
\special{pn 8}%
\special{pa 2130 1724}%
\special{pa 2238 1724}%
\special{pa 2238 1832}%
\special{pa 2130 1832}%
\special{pa 2130 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 2130 1831 2238 1941
% 
\special{pn 8}%
\special{pa 2130 1832}%
\special{pa 2238 1832}%
\special{pa 2238 1942}%
\special{pa 2130 1942}%
\special{pa 2130 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 2022 1941 2130 2049
% 
\special{pn 8}%
\special{pa 2022 1942}%
\special{pa 2130 1942}%
\special{pa 2130 2050}%
\special{pa 2022 2050}%
\special{pa 2022 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 2022 2049 2130 2157
% 
\special{pn 8}%
\special{pa 2022 2050}%
\special{pa 2130 2050}%
\special{pa 2130 2158}%
\special{pa 2022 2158}%
\special{pa 2022 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 1723 3600 1831
% 
\special{pn 8}%
\special{pa 3492 1724}%
\special{pa 3600 1724}%
\special{pa 3600 1832}%
\special{pa 3492 1832}%
\special{pa 3492 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 1831 3600 1941
% 
\special{pn 8}%
\special{pa 3492 1832}%
\special{pa 3600 1832}%
\special{pa 3600 1942}%
\special{pa 3492 1942}%
\special{pa 3492 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 1941 3600 2049
% 
\special{pn 8}%
\special{pa 3492 1942}%
\special{pa 3600 1942}%
\special{pa 3600 2050}%
\special{pa 3492 2050}%
\special{pa 3492 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 2049 3600 2157
% 
\special{pn 8}%
\special{pa 3492 2050}%
\special{pa 3600 2050}%
\special{pa 3600 2158}%
\special{pa 3492 2158}%
\special{pa 3492 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 2157 3600 2265
% 
\special{pn 8}%
\special{pa 3492 2158}%
\special{pa 3600 2158}%
\special{pa 3600 2266}%
\special{pa 3492 2266}%
\special{pa 3492 2158}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 2265 3600 2373
% 
\special{pn 8}%
\special{pa 3492 2266}%
\special{pa 3600 2266}%
\special{pa 3600 2374}%
\special{pa 3492 2374}%
\special{pa 3492 2266}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 1723 3708 1831
% 
\special{pn 8}%
\special{pa 3600 1724}%
\special{pa 3708 1724}%
\special{pa 3708 1832}%
\special{pa 3600 1832}%
\special{pa 3600 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 1831 3708 1941
% 
\special{pn 8}%
\special{pa 3600 1832}%
\special{pa 3708 1832}%
\special{pa 3708 1942}%
\special{pa 3600 1942}%
\special{pa 3600 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 1941 3708 2049
% 
\special{pn 8}%
\special{pa 3600 1942}%
\special{pa 3708 1942}%
\special{pa 3708 2050}%
\special{pa 3600 2050}%
\special{pa 3600 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 2049 3708 2157
% 
\special{pn 8}%
\special{pa 3600 2050}%
\special{pa 3708 2050}%
\special{pa 3708 2158}%
\special{pa 3600 2158}%
\special{pa 3600 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 2157 3708 2265
% 
\special{pn 8}%
\special{pa 3600 2158}%
\special{pa 3708 2158}%
\special{pa 3708 2266}%
\special{pa 3600 2266}%
\special{pa 3600 2158}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 1723 3816 1831
% 
\special{pn 8}%
\special{pa 3708 1724}%
\special{pa 3816 1724}%
\special{pa 3816 1832}%
\special{pa 3708 1832}%
\special{pa 3708 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 1831 3816 1941
% 
\special{pn 8}%
\special{pa 3708 1832}%
\special{pa 3816 1832}%
\special{pa 3816 1942}%
\special{pa 3708 1942}%
\special{pa 3708 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 1941 3816 2049
% 
\special{pn 8}%
\special{pa 3708 1942}%
\special{pa 3816 1942}%
\special{pa 3816 2050}%
\special{pa 3708 2050}%
\special{pa 3708 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 2049 3816 2157
% 
\special{pn 8}%
\special{pa 3708 2050}%
\special{pa 3816 2050}%
\special{pa 3816 2158}%
\special{pa 3708 2158}%
\special{pa 3708 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 2157 3816 2265
% 
\special{pn 8}%
\special{pa 3708 2158}%
\special{pa 3816 2158}%
\special{pa 3816 2266}%
\special{pa 3708 2266}%
\special{pa 3708 2158}%
\special{fp}%
% BOX 2 0 3 0
% 2 3816 1723 3925 1831
% 
\special{pn 8}%
\special{pa 3816 1724}%
\special{pa 3926 1724}%
\special{pa 3926 1832}%
\special{pa 3816 1832}%
\special{pa 3816 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 3816 1831 3925 1941
% 
\special{pn 8}%
\special{pa 3816 1832}%
\special{pa 3926 1832}%
\special{pa 3926 1942}%
\special{pa 3816 1942}%
\special{pa 3816 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 3816 1941 3925 2049
% 
\special{pn 8}%
\special{pa 3816 1942}%
\special{pa 3926 1942}%
\special{pa 3926 2050}%
\special{pa 3816 2050}%
\special{pa 3816 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 3816 2049 3925 2157
% 
\special{pn 8}%
\special{pa 3816 2050}%
\special{pa 3926 2050}%
\special{pa 3926 2158}%
\special{pa 3816 2158}%
\special{pa 3816 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 3925 1723 4033 1831
% 
\special{pn 8}%
\special{pa 3926 1724}%
\special{pa 4034 1724}%
\special{pa 4034 1832}%
\special{pa 3926 1832}%
\special{pa 3926 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 3925 1831 4033 1941
% 
\special{pn 8}%
\special{pa 3926 1832}%
\special{pa 4034 1832}%
\special{pa 4034 1942}%
\special{pa 3926 1942}%
\special{pa 3926 1832}%
\special{fp}%
% BOX 2 0 3 0
% 2 3925 1941 4033 2049
% 
\special{pn 8}%
\special{pa 3926 1942}%
\special{pa 4034 1942}%
\special{pa 4034 2050}%
\special{pa 3926 2050}%
\special{pa 3926 1942}%
\special{fp}%
% BOX 2 0 3 0
% 2 3925 2049 4033 2157
% 
\special{pn 8}%
\special{pa 3926 2050}%
\special{pa 4034 2050}%
\special{pa 4034 2158}%
\special{pa 3926 2158}%
\special{pa 3926 2050}%
\special{fp}%
% BOX 2 0 3 0
% 2 4033 1723 4141 1831
% 
\special{pn 8}%
\special{pa 4034 1724}%
\special{pa 4142 1724}%
\special{pa 4142 1832}%
\special{pa 4034 1832}%
\special{pa 4034 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 1615 3600 1723
% 
\special{pn 8}%
\special{pa 3492 1616}%
\special{pa 3600 1616}%
\special{pa 3600 1724}%
\special{pa 3492 1724}%
\special{pa 3492 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 1615 3708 1723
% 
\special{pn 8}%
\special{pa 3600 1616}%
\special{pa 3708 1616}%
\special{pa 3708 1724}%
\special{pa 3600 1724}%
\special{pa 3600 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 1615 3816 1723
% 
\special{pn 8}%
\special{pa 3708 1616}%
\special{pa 3816 1616}%
\special{pa 3816 1724}%
\special{pa 3708 1724}%
\special{pa 3708 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 3816 1615 3925 1723
% 
\special{pn 8}%
\special{pa 3816 1616}%
\special{pa 3926 1616}%
\special{pa 3926 1724}%
\special{pa 3816 1724}%
\special{pa 3816 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 3925 1615 4033 1723
% 
\special{pn 8}%
\special{pa 3926 1616}%
\special{pa 4034 1616}%
\special{pa 4034 1724}%
\special{pa 3926 1724}%
\special{pa 3926 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 4033 1615 4141 1723
% 
\special{pn 8}%
\special{pa 4034 1616}%
\special{pa 4142 1616}%
\special{pa 4142 1724}%
\special{pa 4034 1724}%
\special{pa 4034 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 4141 1615 4250 1723
% 
\special{pn 8}%
\special{pa 4142 1616}%
\special{pa 4250 1616}%
\special{pa 4250 1724}%
\special{pa 4142 1724}%
\special{pa 4142 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 3492 1507 3600 1615
% 
\special{pn 8}%
\special{pa 3492 1508}%
\special{pa 3600 1508}%
\special{pa 3600 1616}%
\special{pa 3492 1616}%
\special{pa 3492 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 3600 1507 3708 1615
% 
\special{pn 8}%
\special{pa 3600 1508}%
\special{pa 3708 1508}%
\special{pa 3708 1616}%
\special{pa 3600 1616}%
\special{pa 3600 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 3708 1507 3816 1615
% 
\special{pn 8}%
\special{pa 3708 1508}%
\special{pa 3816 1508}%
\special{pa 3816 1616}%
\special{pa 3708 1616}%
\special{pa 3708 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 3816 1507 3925 1615
% 
\special{pn 8}%
\special{pa 3816 1508}%
\special{pa 3926 1508}%
\special{pa 3926 1616}%
\special{pa 3816 1616}%
\special{pa 3816 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 3925 1507 4033 1615
% 
\special{pn 8}%
\special{pa 3926 1508}%
\special{pa 4034 1508}%
\special{pa 4034 1616}%
\special{pa 3926 1616}%
\special{pa 3926 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 4033 1507 4141 1615
% 
\special{pn 8}%
\special{pa 4034 1508}%
\special{pa 4142 1508}%
\special{pa 4142 1616}%
\special{pa 4034 1616}%
\special{pa 4034 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 4141 1507 4250 1615
% 
\special{pn 8}%
\special{pa 4142 1508}%
\special{pa 4250 1508}%
\special{pa 4250 1616}%
\special{pa 4142 1616}%
\special{pa 4142 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 4250 1615 4358 1723
% 
\special{pn 8}%
\special{pa 4250 1616}%
\special{pa 4358 1616}%
\special{pa 4358 1724}%
\special{pa 4250 1724}%
\special{pa 4250 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 4250 1507 4358 1615
% 
\special{pn 8}%
\special{pa 4250 1508}%
\special{pa 4358 1508}%
\special{pa 4358 1616}%
\special{pa 4250 1616}%
\special{pa 4250 1508}%
\special{fp}%
% BOX 2 0 3 0
% 2 4141 1723 4250 1831
% 
\special{pn 8}%
\special{pa 4142 1724}%
\special{pa 4250 1724}%
\special{pa 4250 1832}%
\special{pa 4142 1832}%
\special{pa 4142 1724}%
\special{fp}%
% STR 2 0 3 0
% 3 2470 1533 2470 1610 2 0
% $+$
\put(24.7000,-16.1000){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 4358 1615 4467 1723
% 
\special{pn 8}%
\special{pa 4358 1616}%
\special{pa 4468 1616}%
\special{pa 4468 1724}%
\special{pa 4358 1724}%
\special{pa 4358 1616}%
\special{fp}%
% BOX 2 0 3 0
% 2 4358 1507 4467 1615
% 
\special{pn 8}%
\special{pa 4358 1508}%
\special{pa 4468 1508}%
\special{pa 4468 1616}%
\special{pa 4358 1616}%
\special{pa 4358 1508}%
\special{fp}%
% STR 2 0 3 0
% 3 4480 1533 4480 1610 2 0
% $+$
\put(44.8000,-16.1000){\makebox(0,0)[lb]{$+$}}%
% STR 2 0 3 0
% 3 4480 1649 4480 1726 2 0
% $+$
\put(44.8000,-17.2600){\makebox(0,0)[lb]{$+$}}%
% LINE 2 2 3 0
% 2 3492 2536 4520 1507
% 
\special{pn 8}%
\special{pa 3492 2536}%
\special{pa 4520 1508}%
\special{dt 0.030}%
% BOX 2 0 3 0
% 2 4250 1723 4358 1831
% 
\special{pn 8}%
\special{pa 4250 1724}%
\special{pa 4358 1724}%
\special{pa 4358 1832}%
\special{pa 4250 1832}%
\special{pa 4250 1724}%
\special{fp}%
% BOX 2 0 3 0
% 2 4358 1723 4467 1831
% 
\special{pn 8}%
\special{pa 4358 1724}%
\special{pa 4468 1724}%
\special{pa 4468 1832}%
\special{pa 4358 1832}%
\special{pa 4358 1724}%
\special{fp}%
% STR 2 0 3 0
% 3 4480 1757 4480 1834 2 0
% $-$
\put(44.8000,-18.3400){\makebox(0,0)[lb]{$-$}}%
% BOX 2 0 3 0
% 2 3492 2373 3600 2482
% 
\special{pn 8}%
\special{pa 3492 2374}%
\special{pa 3600 2374}%
\special{pa 3600 2482}%
\special{pa 3492 2482}%
\special{pa 3492 2374}%
\special{fp}%
% BOX 2 0 3 0
% 2 1480 2373 1588 2482
% 
\special{pn 8}%
\special{pa 1480 2374}%
\special{pa 1588 2374}%
\special{pa 1588 2482}%
\special{pa 1480 2482}%
\special{pa 1480 2374}%
\special{fp}%
% LINE 2 2 3 0
% 2 1480 2536 2509 1507
% 
\special{pn 8}%
\special{pa 1480 2536}%
\special{pa 2510 1508}%
\special{dt 0.030}%
\end{picture}%
\caption{}
\label{zigzag for type D}
\end{figure}


For each $k,l\in[\pm\rnk]$, let $\Dsetsec\subset[\pm\rnk]$ satisfy (\ref{type D basics 100}), $k\in \Dsetsec$, and $\pm l\notin \Dsetsec$ (hence $|\Dsetsec|\leq\rnk-2$).
By the linear relation (\ref{prelim linear relation}) for the root $\alpha=t_k-t_l$, it follows that
\begin{align}\label{type D vanishing 130}
{\tau_{\Dsetsec}}^2 = 
%&
-
\sum_{\substack{k\in \Dsetrun, \ \pm l\notin \Dsetrun \\ \Dsetrun\neq \Dsetsec}} \tau_{\Dsetrun} \tau_{\Dsetsec}
-
\sum_{\substack{k\in \Dsetrun, \ -l\in \Dsetrun \\ |\Dsetrun|\neq\rnk}} 2\tau_{\Dsetrun} \tau_{\Dsetsec}
-
\sum_{\substack{k\in \Dsetrun, \ -l\in \Dsetrun \\ |\Dsetrun|=\rnk}} \tau_{\Dsetrun} \tau_{\Dsetsec}
\end{align}
where the sum is taken over all $\emptyset \subsetneq \Bsetrun\subset [\pm\rnk]$ satisfying (\ref{type D basics 100}) with the prescribed conditions.
If $\Dsetfir\subsetneq \Dsetsec$ with $k\notin \Dsetfir$, and if $\Dsetsec\subsetneq \Dsetthr$ with 
$l\in \Dsetthr$ then, 
\begin{align}\label{type D vanishing 140}
\tau_{\Dsetfir} {\tau_{\Dsetsec}}^2 \tau_{\Dsetthr} = 
-
\sum_{\substack{k\in \Dsetrun, \ \pm l\notin \Dsetrun \\ \Dsetfir\subsetneq \Dsetrun \subsetneq \Dsetthr, \ \Dsetrun\neq \Dsetsec}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec} \tau_{\Dsetthr}
\ \ - \ \ 
\delta_{|\Dsetthr|,\rnk}\tau_{\Dsetfir} \tau_\Dsetsec \tau_{(l,-l)\Dsetthr} \tau_{\Dsetthr}.
\end{align} 
where $\delta_{|\Dsetthr|,\rnk}$ is the Kronecker delta.
If $|\Dsetthr|=\rnk$, then after multiplying (\ref{type D vanishing 140}) by $\tau_{\overline{\Dsetthr}}$ where $\overline{\Dsetthr}=(p,-p)\Dsetthr$ for some $p\in \Dsetthr\backslash \{l\}$, we obtain 
\begin{align}\label{type D vanishing 141}
\tau_{\Dsetfir} {\tau_{\Dsetsec}}^2 \tau_{\Dsetthr}\tau_{\overline{\Dsetthr}} = 
-
\sum_{\substack{k\in \Dsetrun, \ \pm l, \pm p\notin \Dsetrun \\ \Dsetfir\subsetneq \Dsetrun \subsetneq \Dsetthr, \ \Dsetrun\neq \Dsetsec}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec} \tau_{\Dsetthr}\tau_{\overline{\Dsetthr}}.
\end{align} 


Let $\lambda$ as above. 
We denote 
\begin{align*}
&m_+(\lambda):= |\{i \mid \text{$\lambda_i=\rnk$ and the label of $\lambda_i$ is $+$} \}|, \\
&m_-(\lambda):= |\{i \mid \text{$\lambda_i=\rnk$ and the label of $\lambda_i$ is $-$} \}|.
\end{align*}

\begin{lemma}\label{type D vanishing 300}
Suppose that one of the following holds:
\begin{itemize}
 \item[(i)] $m_+(\lambda)=m_-(\lambda)=1$
 \item[(ii)] $(m_+(\lambda),m_-(\lambda))$ is equal to $(1,0)$ or $(0,1)$,
  \item[(iii)] $m_+(\lambda)=m_-(\lambda)=0$.
\end{itemize}
Then $\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}}=0$ unless each step of the zigzag line of the corners of $\lambda$ crosses the dotted anti-diagonal.
\end{lemma}
\begin{proof}
The claim for the case (i) can be proved by (\ref{type D vanishing 140}) and (\ref{type D vanishing 141}) as in the proof of Proposition \ref{dotted vanishing type A}.
For the case (ii), the same argument works together with (\ref{type D vanishing 140}), since we have already proved the claim for the case (i). 
Now, the case (iii) is shown again by the same proof used for Proposition \ref{dotted vanishing type A} together with (\ref{type D vanishing 130}), (\ref{type D vanishing 140}), and the case (ii).
\end{proof}





For each $k,l\in[\pm\rnk]$, let $\emptyset\subsetneq \Dsetsec\subsetneq[\pm\rnk]$ satisfy (\ref{type D basics 100}),  $k,l\in \Dsetsec$, and $|\Dsetsec|=\rnk$. 
If $\Dsetfir\subset \Dsetsec\backslash\{k,l\}$, then from the linear relation (\ref{prelim linear relation}) for the root $\alpha=t_k+t_l$, we obtain
\begin{align}\notag
\tau_{\Dsetfir} {\tau_{\Dsetsec}}^2 = 
&
-
\sum_{\substack{k\in \Dsetrun, \ \pm l\notin \Dsetrun}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec}
-
\sum_{\substack{\pm k\notin \Dsetrun, \ l\in \Dsetrun}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec}
\\ \label{type D vanishing 900}
&\quad\quad\quad\quad
-
\sum_{\substack{k, l\in \Dsetrun, \\ |\Dsetrun|\neq\rnk}} 2\tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec}
-
\sum_{\substack{k,l\in \Dsetrun, \ \Dsetrun\neq \Dsetsec, \\ |\Dsetrun|=\rnk}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec}
\end{align} 
where we denote $\tau_{\emptyset}=1$.
Especially if $\Dsetfir= \Dsetsec\backslash\{k,l\}$, then  $|\Dsetfir|=\rnk-2$ and we have
\begin{align}\label{type D vanishing 1000}
\tau_{\Dsetfir}{\tau_{\Dsetsec}}^2
&= 
-
\sum_{\substack{k,l\in \Dsetrun, \ \Dsetrun\neq \Dsetsec, \\ |\Dsetrun|=\rnk}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec} =0.
\end{align} 
The second equality follows since an element of $\Dsetrun$ which is neither $k$ nor $l$ has to be $-1$ times an element of $\Dsetsec$, which implies that $\Dsetfir\not\subset \Dsetrun$. 
On the other hand, letting 
$\overline{\Dsetsec}=(-k,k)\Dsetsec$, we obtain from (\ref{type D vanishing 900}) that 
\begin{align}\label{type D vanishing 1100}
\tau_{\Dsetfir} {\tau_\Dsetsec}^2 \tau_{\overline{\Dsetsec}} = 
&
-
\sum_{\substack{\pm k\notin \Dsetrun, \ l\in \Dsetrun \\ \Dsetfir\subsetneq \Dsetrun\subsetneq \Dsetsec}} \tau_{\Dsetfir} \tau_{\Dsetrun} \tau_{\Dsetsec} \tau_{\overline{\Dsetsec}}.
\end{align} 






\begin{lemma}\label{type D vanishing 1200}
Suppose that one of the following holds:
\begin{itemize}
 \item[(i)] $m_+(\lambda), m_-(\lambda)\geq1$,
 \item[(ii)] $m_+(\lambda)\geq1$ and $m_-(\lambda)=0$, 
 \item[(iii)] $m_+(\lambda)=0$ and $m_-(\lambda)\geq1$.
\end{itemize}
Then $\intnum{\tau_{S_1}\cdots\tau_{S_{\rnk}}}=0$ unless each step of the zigzag line of the lower-right corners of $\lambda$ crosses the dotted anti-diagonal.
\end{lemma}
\begin{proof}
The claim for the case (i) follows from (\ref{type D vanishing 1100}) and the case (i) of Lemma \ref{type D vanishing 300} by induction on $m_+(\lambda)+m_-(\lambda)$.
Let us consider the case (ii). 
We prove the claim by induction on $m_+(\lambda)$.
For the case $m_+(\lambda)=1$, the claim follows from the case (ii) of Lemma \ref{type D vanishing 300}.
For the general case, the induction hypothesis and the claim for the case (i) shows our claim by applying (\ref{type D vanishing 900}) to reduce the multiplicity for $\tau_{\A_{\rnk}}$ in $\tau_{\A_{1}}\cdots\tau_{\A_{\rnk}}$.
The claim for the case (iii) can be proved similarly.
\end{proof}

\vspace{20pt}
Now, Proposition \ref{zigzag lemma type D} follows from Lemma \ref{type D vanishing 1200} and the case (iii) of Lemma \ref{type D vanishing 300}.


\vspace{10pt}


For a signed Young diagram $\lambda$ with $\rnk$ rows fitting into the $\rnk\times\rnk$-square, let
\begin{align*}
&m:=|\{i \mid \text{$\lambda_i=\rnk$} \}|=m_+(\lambda) + m_-(\lambda) 
\end{align*}
be the number of rows of $\lambda$ of length $\rnk$.
Recall that the numbers $\parasec_{r}, \parathr_{r}, \parafir_{r}$, and $\parafor_{r}$ are defined in (\ref{definition of a b c}) and (\ref{definition of y}).
We now define
\begin{align*}
\widetilde{\parafor}_{1}
:=
\begin{cases}
\displaystyle{
2^{(\rnk-\lambda_1-1)(1-m)}
\binom{ \parasec_{1} }{ \parafir_{1} } 
\binom{ \parathr_{1} }{ \parafir_{1} } 
}
\qquad &\text{if $m\leq1$}, \vspace{3pt} \\ 
\displaystyle{
-\binom{\parathr_1-1}{\parafir_1-1}
}
&\text{if $m\geq2$ and $m_+(\lambda)m_-(\lambda)\neq0$}, \vspace{3pt} \\
\displaystyle{
\left(2^{\parasec_1}-\parasec_1-1\right)
 \binom{ \parathr_{1} }{ \parafir_{1} }
 +
 \binom{ \parathr_{1}-1 }{ \parafir_{1} } 
}
&\text{if $m\geq2$ and $m_+(\lambda)m_-(\lambda)=0$}
\end{cases}
\end{align*}


\begin{theorem}\label{main thm for type D}
If $\emptyset\subsetneq \A_1,\cdots,\A_{\rnk}\subsetneq[\pm\rnk]$ satisfying \emph{(\ref{type D basics 100})} form a subchain of a chain in $\mathcal{C}$, then we have
\begin{align*} 
\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}} = (-1)^{\rnk+s}\widetilde{y}_1y_2\cdots y_s.
\end{align*}
where $\mu_X$ is the fundamental homology class of $X$ and $\lambda$ is the signed Young diagram consisting of $|\A_1|,\cdots,|\A_{\rnk}|$ reordered as a weakly decreasing sequence and $m$ is the number of rows of $\lambda$ of length $\rnk$.
Otherwise, the intersection number is zero.
\end{theorem}

\begin{remark}
\emph{
In each case, the given number vanishes unless each step of the zigzag line of the lower-right corners of $\lambda$ crosses the dotted anti-diagonal. 
}
\end{remark}
\begin{proof}[Proof of Theorem~\ref{main thm for type D}]
We compute the intersection number 
\[J(\lambda):=(-1)^{\rnk-s}\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}\]
with sign where we can assume that each step of the zigzag line of the corners of $\lambda$ crosses the dotted anti-diagonal by Proposition \ref{zigzag lemma type D} and the remark above.
We first prove the case (i).
If $m_+(\lambda)=1$ and $m_-(\lambda)=0$ (or $m_+(\lambda)=1$ and $m_-(\lambda)=0$), then it follows that $J(\lambda) = \parafor_{1}\cdots \parafor_{s}$. This can be proved by induction similar to that used in the proof of Theorem \ref{intro formula of type A intersection} because of the separating properties (\ref{type D vanishing 140}).
So, let us consider the case of $m_+(\lambda)=m_-(\lambda)=0$.
In this case, Proposition \ref{zigzag lemma type D} shows that the separation rule (\ref{type D vanishing 130}) replaces the square $\tau_{\A_{\rnk}}^2$ in $\tau_{\A_1}\cdots\tau_{\A_{\rnk}}$ to
\begin{align*}
\sum_{\substack{k\in \A', \ \pm l\notin \A' \\ \A'\subsetneq \A_{\rnk}, \ |B'|\neq\rnk}} \tau_{\A'} \tau_{\A_{\rnk}}
+
\sum_{\substack{k\in \A', \ -l\in \A' \\ |\A'|=\rnk}} \tau_{\A'} \tau_{\A_{\rnk}}
\end{align*}
for some $k\in\A_{\rnk}$ and $\pm l\in\A_{\rnk}$
when we compute the intersection number $J(\lambda)$ with sign.
Namely, this replacement can be pictured as
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where we omit the coefficients in the picture.
Hence, with the claim for the previous case, we get $J(\lambda)=2^{\rnk-\lambda_1-1}\parafor_1\cdots \parafor_s$
as in the case of type $B_{\rnk}$. 




Let us consider the case (ii-a). 
We prove the claim by induction on the sum of the multiplicities for $\A_i$'s satisfying $|\A_i|\neq\rnk$.
The base case has 
$\lambda_i=\rnk+1-i$ for all $\lambda_i\neq\rnk$, so it is obvious that $-J(\lambda)=(-1)^{\rnk-s-1}\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}$ is equal to $1$ by iterating (\ref{type D vanishing 1100}). For the general case, we apply (\ref{type D vanishing 140}) and (\ref{type D vanishing 141}) 
to some square ${\tau_{\A_i}}^2$ with $|\A_i|\neq\rnk$ in $J(\lambda)$. 
If $\coi_{2}<i$, the computation with (\ref{type D vanishing 140}) works as in the proof of Theorem \ref{intro formula of type A intersection}.
If $\coi_{1}<i \leq \coi_{2}$, then the right-hand-side of (\ref{type D vanishing 141}) applied to $-J(\lambda)$ can be calculated as follows by the induction hypothesis:
\begin{align*}
\binom{\parathr_2}{\parafir_2}
\cdot
\binom{\parathr_1-1}{\parafir_1-1}
\binom{\parasec_2-1}{\parafir_2-1}
\parafor_{3}\cdots \parafor_{s} 
+
\binom{\parathr_1-1}{\parafir_1-1}
\cdot
\binom{\parasec_2-1}{\parafir_2}\binom{\parathr_2}{\parafir_2}
\parafor_{3}\cdots \parafor_{s} 
\end{align*}
which is equal to $\binom{\parathr_1-1}{\parafir_1-1}
\parafor_{2}\cdots \parafor_{s}$, and the claim follows.




Let us consider the case (ii-b). We can assume $m=m_+(\lambda)(=\parasec_1+1)$ without loss of generality.
We first consider the case that $\lambda_i=\rnk+1-i$ for all $\lambda_i\neq\rnk$. Note that 
$\binom{\parathr_1-1}{\parafir_1}=0$
in this case.
For this special case, we prove the claim by induction on $m=m_+(\lambda)$.
For the case $m=\parasec_1+1=2$, the intersection number is zero by (\ref{type D vanishing 1000}), and the claim follows since $2^{\parasec_1}-\parasec_1-1=0$.
For the general case, we have an inductive formula by (\ref{type D vanishing 900}), namely, 
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% BOX 2 0 3 0
% 2 1274 765 1381 873
% 
\special{pn 8}%
\special{pa 1274 766}%
\special{pa 1382 766}%
\special{pa 1382 874}%
\special{pa 1274 874}%
\special{pa 1274 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 1381 765 1488 873
% 
\special{pn 8}%
\special{pa 1382 766}%
\special{pa 1488 766}%
\special{pa 1488 874}%
\special{pa 1382 874}%
\special{pa 1382 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 954 765 1060 873
% 
\special{pn 8}%
\special{pa 954 766}%
\special{pa 1060 766}%
\special{pa 1060 874}%
\special{pa 954 874}%
\special{pa 954 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 1488 765 1594 873
% 
\special{pn 8}%
\special{pa 1488 766}%
\special{pa 1594 766}%
\special{pa 1594 874}%
\special{pa 1488 874}%
\special{pa 1488 766}%
\special{fp}%
% STR 2 0 3 0
% 3 1600 794 1600 870 2 0
% $+$
\put(16.0000,-8.7000){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 1166 765 1274 873
% 
\special{pn 8}%
\special{pa 1166 766}%
\special{pa 1274 766}%
\special{pa 1274 874}%
\special{pa 1166 874}%
\special{pa 1166 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 846 765 954 873
% 
\special{pn 8}%
\special{pa 846 766}%
\special{pa 954 766}%
\special{pa 954 874}%
\special{pa 846 874}%
\special{pa 846 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 873 846 979
% 
\special{pn 8}%
\special{pa 740 874}%
\special{pa 846 874}%
\special{pa 846 980}%
\special{pa 740 980}%
\special{pa 740 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 1193 846 1299
% 
\special{pn 8}%
\special{pa 740 1194}%
\special{pa 846 1194}%
\special{pa 846 1300}%
\special{pa 740 1300}%
\special{pa 740 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 1299 846 1405
% 
\special{pn 8}%
\special{pa 740 1300}%
\special{pa 846 1300}%
\special{pa 846 1406}%
\special{pa 740 1406}%
\special{pa 740 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 979 846 1086
% 
\special{pn 8}%
\special{pa 740 980}%
\special{pa 846 980}%
\special{pa 846 1086}%
\special{pa 740 1086}%
\special{pa 740 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 1086 846 1193
% 
\special{pn 8}%
\special{pa 740 1086}%
\special{pa 846 1086}%
\special{pa 846 1194}%
\special{pa 740 1194}%
\special{pa 740 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 1405 846 1513
% 
\special{pn 8}%
\special{pa 740 1406}%
\special{pa 846 1406}%
\special{pa 846 1514}%
\special{pa 740 1514}%
\special{pa 740 1406}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 1513 846 1619
% 
\special{pn 8}%
\special{pa 740 1514}%
\special{pa 846 1514}%
\special{pa 846 1620}%
\special{pa 740 1620}%
\special{pa 740 1514}%
\special{fp}%
% BOX 2 0 3 0
% 2 740 765 846 873
% 
\special{pn 8}%
\special{pa 740 766}%
\special{pa 846 766}%
\special{pa 846 874}%
\special{pa 740 874}%
\special{pa 740 766}%
\special{fp}%
% LINE 2 2 3 0
% 2 740 1672 1648 765
% 
\special{pn 8}%
\special{pa 740 1672}%
\special{pa 1648 766}%
\special{dt 0.030}%
% BOX 2 0 3 0
% 2 1060 1193 1166 1299
% 
\special{pn 8}%
\special{pa 1060 1194}%
\special{pa 1166 1194}%
\special{pa 1166 1300}%
\special{pa 1060 1300}%
\special{pa 1060 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 3071 873 3177 979
% 
\special{pn 8}%
\special{pa 3072 874}%
\special{pa 3178 874}%
\special{pa 3178 980}%
\special{pa 3072 980}%
\special{pa 3072 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 3284 873 3390 979
% 
\special{pn 8}%
\special{pa 3284 874}%
\special{pa 3390 874}%
\special{pa 3390 980}%
\special{pa 3284 980}%
\special{pa 3284 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 3390 873 3497 979
% 
\special{pn 8}%
\special{pa 3390 874}%
\special{pa 3498 874}%
\special{pa 3498 980}%
\special{pa 3390 980}%
\special{pa 3390 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 2964 873 3071 979
% 
\special{pn 8}%
\special{pa 2964 874}%
\special{pa 3072 874}%
\special{pa 3072 980}%
\special{pa 2964 980}%
\special{pa 2964 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 2964 1193 3071 1299
% 
\special{pn 8}%
\special{pa 2964 1194}%
\special{pa 3072 1194}%
\special{pa 3072 1300}%
\special{pa 2964 1300}%
\special{pa 2964 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 2964 1299 3071 1405
% 
\special{pn 8}%
\special{pa 2964 1300}%
\special{pa 3072 1300}%
\special{pa 3072 1406}%
\special{pa 2964 1406}%
\special{pa 2964 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 3071 979 3177 1086
% 
\special{pn 8}%
\special{pa 3072 980}%
\special{pa 3178 980}%
\special{pa 3178 1086}%
\special{pa 3072 1086}%
\special{pa 3072 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 3284 979 3390 1086
% 
\special{pn 8}%
\special{pa 3284 980}%
\special{pa 3390 980}%
\special{pa 3390 1086}%
\special{pa 3284 1086}%
\special{pa 3284 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 3390 979 3497 1086
% 
\special{pn 8}%
\special{pa 3390 980}%
\special{pa 3498 980}%
\special{pa 3498 1086}%
\special{pa 3390 1086}%
\special{pa 3390 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 2964 979 3071 1086
% 
\special{pn 8}%
\special{pa 2964 980}%
\special{pa 3072 980}%
\special{pa 3072 1086}%
\special{pa 2964 1086}%
\special{pa 2964 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 3071 1086 3177 1193
% 
\special{pn 8}%
\special{pa 3072 1086}%
\special{pa 3178 1086}%
\special{pa 3178 1194}%
\special{pa 3072 1194}%
\special{pa 3072 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 3284 1086 3390 1193
% 
\special{pn 8}%
\special{pa 3284 1086}%
\special{pa 3390 1086}%
\special{pa 3390 1194}%
\special{pa 3284 1194}%
\special{pa 3284 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 3390 1086 3497 1193
% 
\special{pn 8}%
\special{pa 3390 1086}%
\special{pa 3498 1086}%
\special{pa 3498 1194}%
\special{pa 3390 1194}%
\special{pa 3390 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 2964 1086 3071 1193
% 
\special{pn 8}%
\special{pa 2964 1086}%
\special{pa 3072 1086}%
\special{pa 3072 1194}%
\special{pa 2964 1194}%
\special{pa 2964 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 3497 979 3604 1086
% 
\special{pn 8}%
\special{pa 3498 980}%
\special{pa 3604 980}%
\special{pa 3604 1086}%
\special{pa 3498 1086}%
\special{pa 3498 980}%
\special{fp}%
% STR 2 0 3 0
% 3 3618 1007 3618 1083 2 0
% $+$
\put(36.1800,-10.8300){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 3497 1086 3604 1193
% 
\special{pn 8}%
\special{pa 3498 1086}%
\special{pa 3604 1086}%
\special{pa 3604 1194}%
\special{pa 3498 1194}%
\special{pa 3498 1086}%
\special{fp}%
% STR 2 0 3 0
% 3 3618 1114 3618 1190 2 0
% $-$
\put(36.1800,-11.9000){\makebox(0,0)[lb]{$-$}}%
% BOX 2 0 3 0
% 2 3497 873 3604 979
% 
\special{pn 8}%
\special{pa 3498 874}%
\special{pa 3604 874}%
\special{pa 3604 980}%
\special{pa 3498 980}%
\special{pa 3498 874}%
\special{fp}%
% STR 2 0 3 0
% 3 3618 900 3618 977 2 0
% $+$
\put(36.1800,-9.7700){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 3177 873 3284 979
% 
\special{pn 8}%
\special{pa 3178 874}%
\special{pa 3284 874}%
\special{pa 3284 980}%
\special{pa 3178 980}%
\special{pa 3178 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 3177 979 3284 1086
% 
\special{pn 8}%
\special{pa 3178 980}%
\special{pa 3284 980}%
\special{pa 3284 1086}%
\special{pa 3178 1086}%
\special{pa 3178 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 3177 1086 3284 1193
% 
\special{pn 8}%
\special{pa 3178 1086}%
\special{pa 3284 1086}%
\special{pa 3284 1194}%
\special{pa 3178 1194}%
\special{pa 3178 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 873 2964 979
% 
\special{pn 8}%
\special{pa 2856 874}%
\special{pa 2964 874}%
\special{pa 2964 980}%
\special{pa 2856 980}%
\special{pa 2856 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 1193 2964 1299
% 
\special{pn 8}%
\special{pa 2856 1194}%
\special{pa 2964 1194}%
\special{pa 2964 1300}%
\special{pa 2856 1300}%
\special{pa 2856 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 1299 2964 1405
% 
\special{pn 8}%
\special{pa 2856 1300}%
\special{pa 2964 1300}%
\special{pa 2964 1406}%
\special{pa 2856 1406}%
\special{pa 2856 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 979 2964 1086
% 
\special{pn 8}%
\special{pa 2856 980}%
\special{pa 2964 980}%
\special{pa 2964 1086}%
\special{pa 2856 1086}%
\special{pa 2856 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 1086 2964 1193
% 
\special{pn 8}%
\special{pa 2856 1086}%
\special{pa 2964 1086}%
\special{pa 2964 1194}%
\special{pa 2856 1194}%
\special{pa 2856 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 1405 2964 1513
% 
\special{pn 8}%
\special{pa 2856 1406}%
\special{pa 2964 1406}%
\special{pa 2964 1514}%
\special{pa 2856 1514}%
\special{pa 2856 1406}%
\special{fp}%
% BOX 2 0 3 0
% 2 3071 765 3177 873
% 
\special{pn 8}%
\special{pa 3072 766}%
\special{pa 3178 766}%
\special{pa 3178 874}%
\special{pa 3072 874}%
\special{pa 3072 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 3284 765 3390 873
% 
\special{pn 8}%
\special{pa 3284 766}%
\special{pa 3390 766}%
\special{pa 3390 874}%
\special{pa 3284 874}%
\special{pa 3284 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 3390 765 3497 873
% 
\special{pn 8}%
\special{pa 3390 766}%
\special{pa 3498 766}%
\special{pa 3498 874}%
\special{pa 3390 874}%
\special{pa 3390 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 2964 765 3071 873
% 
\special{pn 8}%
\special{pa 2964 766}%
\special{pa 3072 766}%
\special{pa 3072 874}%
\special{pa 2964 874}%
\special{pa 2964 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 3497 765 3604 873
% 
\special{pn 8}%
\special{pa 3498 766}%
\special{pa 3604 766}%
\special{pa 3604 874}%
\special{pa 3498 874}%
\special{pa 3498 766}%
\special{fp}%
% STR 2 0 3 0
% 3 3618 794 3618 870 2 0
% $+$
\put(36.1800,-8.7000){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 3177 765 3284 873
% 
\special{pn 8}%
\special{pa 3178 766}%
\special{pa 3284 766}%
\special{pa 3284 874}%
\special{pa 3178 874}%
\special{pa 3178 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 2856 765 2964 873
% 
\special{pn 8}%
\special{pa 2856 766}%
\special{pa 2964 766}%
\special{pa 2964 874}%
\special{pa 2856 874}%
\special{pa 2856 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 873 2856 979
% 
\special{pn 8}%
\special{pa 2750 874}%
\special{pa 2856 874}%
\special{pa 2856 980}%
\special{pa 2750 980}%
\special{pa 2750 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 1193 2856 1299
% 
\special{pn 8}%
\special{pa 2750 1194}%
\special{pa 2856 1194}%
\special{pa 2856 1300}%
\special{pa 2750 1300}%
\special{pa 2750 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 1299 2856 1405
% 
\special{pn 8}%
\special{pa 2750 1300}%
\special{pa 2856 1300}%
\special{pa 2856 1406}%
\special{pa 2750 1406}%
\special{pa 2750 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 979 2856 1086
% 
\special{pn 8}%
\special{pa 2750 980}%
\special{pa 2856 980}%
\special{pa 2856 1086}%
\special{pa 2750 1086}%
\special{pa 2750 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 1086 2856 1193
% 
\special{pn 8}%
\special{pa 2750 1086}%
\special{pa 2856 1086}%
\special{pa 2856 1194}%
\special{pa 2750 1194}%
\special{pa 2750 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 1405 2856 1513
% 
\special{pn 8}%
\special{pa 2750 1406}%
\special{pa 2856 1406}%
\special{pa 2856 1514}%
\special{pa 2750 1514}%
\special{pa 2750 1406}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 1513 2856 1619
% 
\special{pn 8}%
\special{pa 2750 1514}%
\special{pa 2856 1514}%
\special{pa 2856 1620}%
\special{pa 2750 1620}%
\special{pa 2750 1514}%
\special{fp}%
% BOX 2 0 3 0
% 2 2750 765 2856 873
% 
\special{pn 8}%
\special{pa 2750 766}%
\special{pa 2856 766}%
\special{pa 2856 874}%
\special{pa 2750 874}%
\special{pa 2750 766}%
\special{fp}%
% LINE 2 2 3 0
% 2 2750 1672 3657 765
% 
\special{pn 8}%
\special{pa 2750 1672}%
\special{pa 3658 766}%
\special{dt 0.030}%
% BOX 2 0 3 0
% 2 3071 1193 3177 1299
% 
\special{pn 8}%
\special{pa 3072 1194}%
\special{pa 3178 1194}%
\special{pa 3178 1300}%
\special{pa 3072 1300}%
\special{pa 3072 1194}%
\special{fp}%
% STR 2 0 3 0
% 3 1850 1101 1850 1177 2 0
% $=$
\put(18.5000,-11.7700){\makebox(0,0)[lb]{$=$}}%
% BOX 2 0 3 0
% 2 4665 873 4771 979
% 
\special{pn 8}%
\special{pa 4666 874}%
\special{pa 4772 874}%
\special{pa 4772 980}%
\special{pa 4666 980}%
\special{pa 4666 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4878 873 4986 979
% 
\special{pn 8}%
\special{pa 4878 874}%
\special{pa 4986 874}%
\special{pa 4986 980}%
\special{pa 4878 980}%
\special{pa 4878 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4986 873 5092 979
% 
\special{pn 8}%
\special{pa 4986 874}%
\special{pa 5092 874}%
\special{pa 5092 980}%
\special{pa 4986 980}%
\special{pa 4986 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4558 873 4665 979
% 
\special{pn 8}%
\special{pa 4558 874}%
\special{pa 4666 874}%
\special{pa 4666 980}%
\special{pa 4558 980}%
\special{pa 4558 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4558 1193 4665 1299
% 
\special{pn 8}%
\special{pa 4558 1194}%
\special{pa 4666 1194}%
\special{pa 4666 1300}%
\special{pa 4558 1300}%
\special{pa 4558 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 4558 1299 4665 1405
% 
\special{pn 8}%
\special{pa 4558 1300}%
\special{pa 4666 1300}%
\special{pa 4666 1406}%
\special{pa 4558 1406}%
\special{pa 4558 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 4665 979 4771 1086
% 
\special{pn 8}%
\special{pa 4666 980}%
\special{pa 4772 980}%
\special{pa 4772 1086}%
\special{pa 4666 1086}%
\special{pa 4666 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4878 979 4986 1086
% 
\special{pn 8}%
\special{pa 4878 980}%
\special{pa 4986 980}%
\special{pa 4986 1086}%
\special{pa 4878 1086}%
\special{pa 4878 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4986 979 5092 1086
% 
\special{pn 8}%
\special{pa 4986 980}%
\special{pa 5092 980}%
\special{pa 5092 1086}%
\special{pa 4986 1086}%
\special{pa 4986 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4558 979 4665 1086
% 
\special{pn 8}%
\special{pa 4558 980}%
\special{pa 4666 980}%
\special{pa 4666 1086}%
\special{pa 4558 1086}%
\special{pa 4558 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4665 1086 4771 1193
% 
\special{pn 8}%
\special{pa 4666 1086}%
\special{pa 4772 1086}%
\special{pa 4772 1194}%
\special{pa 4666 1194}%
\special{pa 4666 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 4558 1086 4665 1193
% 
\special{pn 8}%
\special{pa 4558 1086}%
\special{pa 4666 1086}%
\special{pa 4666 1194}%
\special{pa 4558 1194}%
\special{pa 4558 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 5092 979 5198 1086
% 
\special{pn 8}%
\special{pa 5092 980}%
\special{pa 5198 980}%
\special{pa 5198 1086}%
\special{pa 5092 1086}%
\special{pa 5092 980}%
\special{fp}%
% STR 2 0 3 0
% 3 5208 1007 5208 1083 2 0
% $+$
\put(52.0800,-10.8300){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 5092 873 5198 979
% 
\special{pn 8}%
\special{pa 5092 874}%
\special{pa 5198 874}%
\special{pa 5198 980}%
\special{pa 5092 980}%
\special{pa 5092 874}%
\special{fp}%
% STR 2 0 3 0
% 3 5208 900 5208 977 2 0
% $+$
\put(52.0800,-9.7700){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 4771 873 4878 979
% 
\special{pn 8}%
\special{pa 4772 874}%
\special{pa 4878 874}%
\special{pa 4878 980}%
\special{pa 4772 980}%
\special{pa 4772 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4771 979 4878 1086
% 
\special{pn 8}%
\special{pa 4772 980}%
\special{pa 4878 980}%
\special{pa 4878 1086}%
\special{pa 4772 1086}%
\special{pa 4772 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4771 1086 4878 1193
% 
\special{pn 8}%
\special{pa 4772 1086}%
\special{pa 4878 1086}%
\special{pa 4878 1194}%
\special{pa 4772 1194}%
\special{pa 4772 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 873 4558 979
% 
\special{pn 8}%
\special{pa 4452 874}%
\special{pa 4558 874}%
\special{pa 4558 980}%
\special{pa 4452 980}%
\special{pa 4452 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 1193 4558 1299
% 
\special{pn 8}%
\special{pa 4452 1194}%
\special{pa 4558 1194}%
\special{pa 4558 1300}%
\special{pa 4452 1300}%
\special{pa 4452 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 1299 4558 1405
% 
\special{pn 8}%
\special{pa 4452 1300}%
\special{pa 4558 1300}%
\special{pa 4558 1406}%
\special{pa 4452 1406}%
\special{pa 4452 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 979 4558 1086
% 
\special{pn 8}%
\special{pa 4452 980}%
\special{pa 4558 980}%
\special{pa 4558 1086}%
\special{pa 4452 1086}%
\special{pa 4452 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 1086 4558 1193
% 
\special{pn 8}%
\special{pa 4452 1086}%
\special{pa 4558 1086}%
\special{pa 4558 1194}%
\special{pa 4452 1194}%
\special{pa 4452 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 1405 4558 1513
% 
\special{pn 8}%
\special{pa 4452 1406}%
\special{pa 4558 1406}%
\special{pa 4558 1514}%
\special{pa 4452 1514}%
\special{pa 4452 1406}%
\special{fp}%
% BOX 2 0 3 0
% 2 4665 765 4771 873
% 
\special{pn 8}%
\special{pa 4666 766}%
\special{pa 4772 766}%
\special{pa 4772 874}%
\special{pa 4666 874}%
\special{pa 4666 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 4878 765 4986 873
% 
\special{pn 8}%
\special{pa 4878 766}%
\special{pa 4986 766}%
\special{pa 4986 874}%
\special{pa 4878 874}%
\special{pa 4878 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 4986 765 5092 873
% 
\special{pn 8}%
\special{pa 4986 766}%
\special{pa 5092 766}%
\special{pa 5092 874}%
\special{pa 4986 874}%
\special{pa 4986 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 4558 765 4665 873
% 
\special{pn 8}%
\special{pa 4558 766}%
\special{pa 4666 766}%
\special{pa 4666 874}%
\special{pa 4558 874}%
\special{pa 4558 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 5092 765 5198 873
% 
\special{pn 8}%
\special{pa 5092 766}%
\special{pa 5198 766}%
\special{pa 5198 874}%
\special{pa 5092 874}%
\special{pa 5092 766}%
\special{fp}%
% STR 2 0 3 0
% 3 5208 794 5208 870 2 0
% $+$
\put(52.0800,-8.7000){\makebox(0,0)[lb]{$+$}}%
% BOX 2 0 3 0
% 2 4771 765 4878 873
% 
\special{pn 8}%
\special{pa 4772 766}%
\special{pa 4878 766}%
\special{pa 4878 874}%
\special{pa 4772 874}%
\special{pa 4772 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 4451 765 4558 873
% 
\special{pn 8}%
\special{pa 4452 766}%
\special{pa 4558 766}%
\special{pa 4558 874}%
\special{pa 4452 874}%
\special{pa 4452 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 873 4451 979
% 
\special{pn 8}%
\special{pa 4346 874}%
\special{pa 4452 874}%
\special{pa 4452 980}%
\special{pa 4346 980}%
\special{pa 4346 874}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 1193 4451 1299
% 
\special{pn 8}%
\special{pa 4346 1194}%
\special{pa 4452 1194}%
\special{pa 4452 1300}%
\special{pa 4346 1300}%
\special{pa 4346 1194}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 1299 4451 1405
% 
\special{pn 8}%
\special{pa 4346 1300}%
\special{pa 4452 1300}%
\special{pa 4452 1406}%
\special{pa 4346 1406}%
\special{pa 4346 1300}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 979 4451 1086
% 
\special{pn 8}%
\special{pa 4346 980}%
\special{pa 4452 980}%
\special{pa 4452 1086}%
\special{pa 4346 1086}%
\special{pa 4346 980}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 1086 4451 1193
% 
\special{pn 8}%
\special{pa 4346 1086}%
\special{pa 4452 1086}%
\special{pa 4452 1194}%
\special{pa 4346 1194}%
\special{pa 4346 1086}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 1405 4451 1513
% 
\special{pn 8}%
\special{pa 4346 1406}%
\special{pa 4452 1406}%
\special{pa 4452 1514}%
\special{pa 4346 1514}%
\special{pa 4346 1406}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 1513 4451 1619
% 
\special{pn 8}%
\special{pa 4346 1514}%
\special{pa 4452 1514}%
\special{pa 4452 1620}%
\special{pa 4346 1620}%
\special{pa 4346 1514}%
\special{fp}%
% BOX 2 0 3 0
% 2 4345 765 4451 873
% 
\special{pn 8}%
\special{pa 4346 766}%
\special{pa 4452 766}%
\special{pa 4452 874}%
\special{pa 4346 874}%
\special{pa 4346 766}%
\special{fp}%
% BOX 2 0 3 0
% 2 4665 1193 4771 1299
% 
\special{pn 8}%
\special{pa 4666 1194}%
\special{pa 4772 1194}%
\special{pa 4772 1300}%
\special{pa 4666 1300}%
\special{pa 4666 1194}%
\special{fp}%
% STR 2 0 3 0
% 3 3880 1101 3880 1177 2 0
% $+$
\put(38.8000,-11.7700){\makebox(0,0)[lb]{$+$}}%
% STR 2 0 3 0
% 3 4109 1101 4109 1177 2 0
% $2$
\put(41.0900,-11.7700){\makebox(0,0)[lb]{$2$}}%
% STR 2 0 3 0
% 3 2110 1154 2110 1230 2 0
% $(\parasec_1-1)$
\put(21.1000,-12.3000){\makebox(0,0)[lb]{$(\parasec_1-1)$}}%
% LINE 2 2 3 0
% 2 4345 1672 5252 765
% 
\special{pn 8}%
\special{pa 4346 1672}%
\special{pa 5252 766}%
\special{dt 0.030}%
\end{picture}%
\]
where the first coefficient $\parasec_1-1$ comes from the choices of an element in $\Dsetsec'\backslash(\Dsetfir\cup\{k,l\})$ turned to be negative, and the second coefficient 2 comes from the two summands corresponding to $k\in \Dsetsec', \pm l\notin \Dsetsec'$ and $\pm k\notin \Dsetsec', l\in \Dsetsec'$.
Noticing that the intersection number for the first summand is equal to $1$ as we already saw, it follows that 
\begin{align*}
J(\lambda)
= \sum_{i=0}^{\parasec_1} 2^i (\parasec_1-1-i) 
= 2^{\parasec_1} - \parasec_1 -1.
\end{align*}


We now prove the claim (ii-b) by induction on the sum of the multiplicities for $\A_i$ satisfying $|\A_i|\neq\rnk$.
The base case $\lambda_i=\rnk+1-i$ for all $\lambda_i\neq\rnk$ is proved above.
For the general case, we apply (\ref{type D vanishing 140}). 
If $\coi_{2}<i$, the computation with (\ref{type D vanishing 140}) again works as in the proof of Theorem \ref{intro formula of type A intersection}.
If $\coi_{1}<i \leq \coi_{2}$, we also apply (\ref{type D vanishing 140}) to a square ${\A_i}^2$ in $-J(\lambda)$.
Namely, we have
\[
%WinTpicVersion3.08
\unitlength 0.1in
\begin{picture}( 49.7900, 10.1400)(  8.0000,-20.8400)
% LINE 2 0 3 0
% 2 800 1167 1678 1167
% 
\special{pn 8}%
\special{pa 800 1168}%
\special{pa 1678 1168}%
\special{fp}%
% LINE 2 0 3 0
% 2 1678 1167 1678 1398
% 
\special{pn 8}%
\special{pa 1678 1168}%
\special{pa 1678 1398}%
\special{fp}%
% LINE 2 0 3 0
% 2 1678 1398 1239 1398
% 
\special{pn 8}%
\special{pa 1678 1398}%
\special{pa 1240 1398}%
\special{fp}%
% LINE 2 0 3 0
% 2 1239 1398 1239 1947
% 
\special{pn 8}%
\special{pa 1240 1398}%
\special{pa 1240 1948}%
\special{fp}%
% LINE 2 0 3 0
% 2 1239 1947 800 1947
% 
\special{pn 8}%
\special{pa 1240 1948}%
\special{pa 800 1948}%
\special{fp}%
% STR 2 0 3 0
% 3 1700 1185 1700 1240 2 0
% $_+$
\put(17.0000,-12.4000){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 1700 1262 1700 1317 2 0
% $_+$
\put(17.0000,-13.1700){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 1700 1339 1700 1394 2 0
% $_+$
\put(17.0000,-13.9400){\makebox(0,0)[lb]{$_+$}}%
% LINE 2 0 3 0
% 2 2557 1398 2557 1870
% 
\special{pn 8}%
\special{pa 2558 1398}%
\special{pa 2558 1870}%
\special{fp}%
% LINE 2 0 3 0
% 2 2557 1870 2293 1870
% 
\special{pn 8}%
\special{pa 2558 1870}%
\special{pa 2294 1870}%
\special{fp}%
% LINE 2 0 3 0
% 2 2293 1870 2293 1947
% 
\special{pn 8}%
\special{pa 2294 1870}%
\special{pa 2294 1948}%
\special{fp}%
% LINE 2 0 3 0
% 2 2117 1947 2293 1947
% 
\special{pn 8}%
\special{pa 2118 1948}%
\special{pa 2294 1948}%
\special{fp}%
% LINE 2 0 3 0
% 2 2117 1167 2996 1167
% 
\special{pn 8}%
\special{pa 2118 1168}%
\special{pa 2996 1168}%
\special{fp}%
% LINE 2 0 3 0
% 2 2996 1167 2996 1398
% 
\special{pn 8}%
\special{pa 2996 1168}%
\special{pa 2996 1398}%
\special{fp}%
% LINE 2 0 3 0
% 2 2996 1398 2557 1398
% 
\special{pn 8}%
\special{pa 2996 1398}%
\special{pa 2558 1398}%
\special{fp}%
% STR 2 0 3 0
% 3 3010 1185 3010 1240 2 0
% $_+$
\put(30.1000,-12.4000){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 3010 1262 3010 1317 2 0
% $_+$
\put(30.1000,-13.1700){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 3010 1339 3010 1394 2 0
% $_+$
\put(30.1000,-13.9400){\makebox(0,0)[lb]{$_+$}}%
% LINE 2 2 3 0
% 2 3034 1167 2117 2084
% 
\special{pn 8}%
\special{pa 3034 1168}%
\special{pa 2118 2084}%
\special{dt 0.030}%
% LINE 2 0 3 0
% 2 3874 1475 3874 1947
% 
\special{pn 8}%
\special{pa 3874 1476}%
\special{pa 3874 1948}%
\special{fp}%
% LINE 2 0 3 0
% 2 3435 1947 3874 1947
% 
\special{pn 8}%
\special{pa 3436 1948}%
\special{pa 3874 1948}%
\special{fp}%
% LINE 2 0 3 0
% 2 3435 1167 4313 1167
% 
\special{pn 8}%
\special{pa 3436 1168}%
\special{pa 4314 1168}%
\special{fp}%
% LINE 2 0 3 0
% 2 4313 1167 4313 1398
% 
\special{pn 8}%
\special{pa 4314 1168}%
\special{pa 4314 1398}%
\special{fp}%
% LINE 2 0 3 0
% 2 4313 1398 4083 1398
% 
\special{pn 8}%
\special{pa 4314 1398}%
\special{pa 4084 1398}%
\special{fp}%
% STR 2 0 3 0
% 3 4330 1185 4330 1240 2 0
% $_+$
\put(43.3000,-12.4000){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 4330 1262 4330 1317 2 0
% $_+$
\put(43.3000,-13.1700){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 4330 1339 4330 1394 2 0
% $_+$
\put(43.3000,-13.9400){\makebox(0,0)[lb]{$_+$}}%
% LINE 2 0 3 0
% 2 3874 1475 4083 1475
% 
\special{pn 8}%
\special{pa 3874 1476}%
\special{pa 4084 1476}%
\special{fp}%
% LINE 2 0 3 0
% 2 4083 1475 4083 1398
% 
\special{pn 8}%
\special{pa 4084 1476}%
\special{pa 4084 1398}%
\special{fp}%
% LINE 2 2 3 0
% 2 1717 1167 800 2084
% 
\special{pn 8}%
\special{pa 1718 1168}%
\special{pa 800 2084}%
\special{dt 0.030}%
% LINE 2 2 3 0
% 2 4352 1167 3435 2084
% 
\special{pn 8}%
\special{pa 4352 1168}%
\special{pa 3436 2084}%
\special{dt 0.030}%
% LINE 2 0 3 0
% 2 4862 1167 5741 1167
% 
\special{pn 8}%
\special{pa 4862 1168}%
\special{pa 5742 1168}%
\special{fp}%
% LINE 2 0 3 0
% 2 5741 1167 5741 1475
% 
\special{pn 8}%
\special{pa 5742 1168}%
\special{pa 5742 1476}%
\special{fp}%
% LINE 2 0 3 0
% 2 5741 1475 5301 1475
% 
\special{pn 8}%
\special{pa 5742 1476}%
\special{pa 5302 1476}%
\special{fp}%
% LINE 2 0 3 0
% 2 5301 1475 5301 1947
% 
\special{pn 8}%
\special{pa 5302 1476}%
\special{pa 5302 1948}%
\special{fp}%
% LINE 2 0 3 0
% 2 5301 1947 4862 1947
% 
\special{pn 8}%
\special{pa 5302 1948}%
\special{pa 4862 1948}%
\special{fp}%
% STR 2 0 3 0
% 3 5760 1185 5760 1240 2 0
% $_+$
\put(57.6000,-12.4000){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 5760 1262 5760 1317 2 0
% $_+$
\put(57.6000,-13.1700){\makebox(0,0)[lb]{$_+$}}%
% STR 2 0 3 0
% 3 5760 1339 5760 1394 2 0
% $_+$
\put(57.6000,-13.9400){\makebox(0,0)[lb]{$_+$}}%
% LINE 2 2 3 0
% 2 5779 1167 4862 2084
% 
\special{pn 8}%
\special{pa 5780 1168}%
\special{pa 4862 2084}%
\special{dt 0.030}%
% STR 2 0 3 0
% 3 5760 1416 5760 1471 2 0
% $_-$
\put(57.6000,-14.7100){\makebox(0,0)[lb]{$_-$}}%
% STR 2 0 3 0
% 3 1820 1596 1820 1650 2 0
% $=$
\put(18.2000,-16.5000){\makebox(0,0)[lb]{$=$}}%
% STR 2 0 3 0
% 3 3130 1596 3130 1650 2 0
% $+$
\put(31.3000,-16.5000){\makebox(0,0)[lb]{$+$}}%
% STR 2 0 3 0
% 3 4560 1595 4560 1650 2 0
% $+$
\put(45.6000,-16.5000){\makebox(0,0)[lb]{$+$}}%
\end{picture}%
\]
with omitting the coefficients.
The right-hand-side can be calculated by the induction hypothesis and the claim for the case (ii-a), and $J(\lambda)=(-1)^{\rnk-s}\intnum{\tau_{\A_1}\cdots\tau_{\A_{\rnk}}}$ is 
\begin{align*}
&
\binom{\parathr_2}{\parafir_2} \cdot
\Bigg\{
(2^{\parasec_1}-\parasec_1-1)
\binom{\parathr_1}{\parafir_1}
\binom{\parasec_2-1}{\parafir_2-1}
\parafor_{3}\cdots \parafor_{s} 
+
\binom{\parathr_1-1}{\parafir_1}
\binom{\parasec_2-1}{\parafir_2-1}
\parafor_3\cdots \parafor_s \Bigg\}
\\
&\qquad\quad
+ 
\binom{\parathr_1}{\parafir_1} \cdot
\Bigg\{
(2^{\parasec_1}-\parasec_1-1)
\binom{\parasec_2-1}{\parafir_2}\binom{\parathr_2}{\parafir_2}
\parafor_{3}\cdots \parafor_{s} 
+ \ 
0 \ 
\Bigg\}
\\
&\qquad\quad
+
\binom{\parathr_1-1}{\parafir_1+1-1}
\binom{\parasec_2-1}{\parafir_2}\binom{\parathr_2}{\parafir_2}
\parafor_3\cdots \parafor_s \\
&\quad
=
(2^{\parasec_1}-\parasec_1-1)
\binom{\parathr_1}{\parafir_1}
\parafor_{2}
\cdots \parafor_{s} 
+
\binom{\parathr_1-1}{\parafir_1}
\parafor_2\cdots \parafor_s.
\end{align*}
\end{proof}


For example, for $\rnk=5$ with the convention $\bar{k}=-k$, we can calculate
\begin{align*}
\intnum{{ \tau_{\{\bar{1}\}} }^2 { \tau_{\{\bar{1},3,4,5,\bar{2}\}} }^3}  = -4
\end{align*}
by the case (ii-b) of Theorem \ref{main thm for type D} (See Figure \ref{EX for type D}).
\vspace{-6pt}
\begin{figure}[h]
\centering
%WinTpicVersion3.08
\unitlength 0.1in
\begin{picture}(  7.3000,  8.9000)( 24.6000,-19.8000)
% STR 2 0 3 0
% 3 2630 1160 2630 1260 2 0
% $3$
\put(26.3000,-12.6000){\makebox(0,0)[lb]{$3$}}%
% STR 2 0 3 0
% 3 2770 1160 2770 1260 2 0
% $4$
\put(27.7000,-12.6000){\makebox(0,0)[lb]{$4$}}%
% STR 2 0 3 0
% 3 2910 1160 2910 1260 2 0
% $5$
\put(29.1000,-12.6000){\makebox(0,0)[lb]{$5$}}%
% STR 2 0 3 0
% 3 3050 1160 3050 1260 2 0
% $\bar{2}$
\put(30.5000,-12.6000){\makebox(0,0)[lb]{$\bar{2}$}}%
% BOX 2 0 3 0
% 2 3160 1420 3020 1280
% 
\special{pn 8}%
\special{pa 3160 1420}%
\special{pa 3020 1420}%
\special{pa 3020 1280}%
\special{pa 3160 1280}%
\special{pa 3160 1420}%
\special{fp}%
% BOX 2 0 3 0
% 2 3160 1560 3020 1420
% 
\special{pn 8}%
\special{pa 3160 1560}%
\special{pa 3020 1560}%
\special{pa 3020 1420}%
\special{pa 3160 1420}%
\special{pa 3160 1560}%
\special{fp}%
% BOX 2 0 3 0
% 2 3020 1560 2880 1420
% 
\special{pn 8}%
\special{pa 3020 1560}%
\special{pa 2880 1560}%
\special{pa 2880 1420}%
\special{pa 3020 1420}%
\special{pa 3020 1560}%
\special{fp}%
% BOX 2 0 3 0
% 2 3020 1420 2880 1280
% 
\special{pn 8}%
\special{pa 3020 1420}%
\special{pa 2880 1420}%
\special{pa 2880 1280}%
\special{pa 3020 1280}%
\special{pa 3020 1420}%
\special{fp}%
% BOX 2 0 3 0
% 2 2880 1420 2740 1280
% 
\special{pn 8}%
\special{pa 2880 1420}%
\special{pa 2740 1420}%
\special{pa 2740 1280}%
\special{pa 2880 1280}%
\special{pa 2880 1420}%
\special{fp}%
% BOX 2 0 3 0
% 2 2880 1560 2740 1420
% 
\special{pn 8}%
\special{pa 2880 1560}%
\special{pa 2740 1560}%
\special{pa 2740 1420}%
\special{pa 2880 1420}%
\special{pa 2880 1560}%
\special{fp}%
% BOX 2 0 3 0
% 2 2740 1560 2600 1420
% 
\special{pn 8}%
\special{pa 2740 1560}%
\special{pa 2600 1560}%
\special{pa 2600 1420}%
\special{pa 2740 1420}%
\special{pa 2740 1560}%
\special{fp}%
% BOX 2 0 3 0
% 2 2740 1420 2600 1280
% 
\special{pn 8}%
\special{pa 2740 1420}%
\special{pa 2600 1420}%
\special{pa 2600 1280}%
\special{pa 2740 1280}%
\special{pa 2740 1420}%
\special{fp}%
% BOX 2 0 3 0
% 2 2600 1560 2460 1420
% 
\special{pn 8}%
\special{pa 2600 1560}%
\special{pa 2460 1560}%
\special{pa 2460 1420}%
\special{pa 2600 1420}%
\special{pa 2600 1560}%
\special{fp}%
% BOX 2 0 3 0
% 2 2600 1420 2460 1280
% 
\special{pn 8}%
\special{pa 2600 1420}%
\special{pa 2460 1420}%
\special{pa 2460 1280}%
\special{pa 2600 1280}%
\special{pa 2600 1420}%
\special{fp}%
% BOX 2 0 3 0
% 2 2600 1700 2460 1560
% 
\special{pn 8}%
\special{pa 2600 1700}%
\special{pa 2460 1700}%
\special{pa 2460 1560}%
\special{pa 2600 1560}%
\special{pa 2600 1700}%
\special{fp}%
% BOX 2 0 3 0
% 2 2600 1840 2460 1700
% 
\special{pn 8}%
\special{pa 2600 1840}%
\special{pa 2460 1840}%
\special{pa 2460 1700}%
\special{pa 2600 1700}%
\special{pa 2600 1840}%
\special{fp}%
% BOX 2 0 1 0
% 2 2600 1980 2460 1840
% 
\special{pn 8}%
\special{sh 0.300}%
\special{pa 2600 1980}%
\special{pa 2460 1980}%
\special{pa 2460 1840}%
\special{pa 2600 1840}%
\special{pa 2600 1980}%
\special{fp}%
% BOX 2 0 3 0
% 2 2740 1700 2600 1560
% 
\special{pn 8}%
\special{pa 2740 1700}%
\special{pa 2600 1700}%
\special{pa 2600 1560}%
\special{pa 2740 1560}%
\special{pa 2740 1700}%
\special{fp}%
% BOX 2 0 3 0
% 2 2880 1700 2740 1560
% 
\special{pn 8}%
\special{pa 2880 1700}%
\special{pa 2740 1700}%
\special{pa 2740 1560}%
\special{pa 2880 1560}%
\special{pa 2880 1700}%
\special{fp}%
% BOX 2 0 3 0
% 2 3020 1700 2880 1560
% 
\special{pn 8}%
\special{pa 3020 1700}%
\special{pa 2880 1700}%
\special{pa 2880 1560}%
\special{pa 3020 1560}%
\special{pa 3020 1700}%
\special{fp}%
% BOX 2 0 1 0
% 2 3160 1700 3020 1560
% 
\special{pn 8}%
\special{sh 0.300}%
\special{pa 3160 1700}%
\special{pa 3020 1700}%
\special{pa 3020 1560}%
\special{pa 3160 1560}%
\special{pa 3160 1700}%
\special{fp}%
% LINE 2 0 3 0
% 2 2460 1980 3160 1280
% 
\special{pn 8}%
\special{pa 2460 1980}%
\special{pa 3160 1280}%
\special{fp}%
% STR 2 0 3 0
% 3 2490 1160 2490 1260 2 0
% $\bar{1}$
\put(24.9000,-12.6000){\makebox(0,0)[lb]{$\bar{1}$}}%
% STR 2 0 3 0
% 3 3190 1300 3190 1400 2 0
% $+$
\put(31.9000,-14.0000){\makebox(0,0)[lb]{$+$}}%
% STR 2 0 3 0
% 3 3190 1440 3190 1540 2 0
% $+$
\put(31.9000,-15.4000){\makebox(0,0)[lb]{$+$}}%
% STR 2 0 3 0
% 3 3190 1580 3190 1680 2 0
% $+$
\put(31.9000,-16.8000){\makebox(0,0)[lb]{$+$}}%
\end{picture}%
\caption{The Young diagram corresponding to ${ \tau_{\{\bar{1}\}} }^2 { \tau_{\{\bar{1},3,4,5,\bar{2}\}} }^3$}
\label{EX for type D}
\end{figure}
\newpage
For each even signed permutation $u\in \widetilde{\mathfrak{S}}_{\rnk}^+$, 
an element $i\in[\rnk]$ satisfies $u(\alpha_i)\in\Phi^-$ if and only if
\begin{itemize}
 \item[(D-1)] if $i\leq\rnk-2$, then $u(i)>u(i+1)$ with the same sign, or $u(i)<u(i+1)$ with different signs,
 \item[(D-2)] if $i=\rnk-1$, then $u(\rnk-1), u(\rnk)<0$, or 
$u(\rnk-1)$ and $u(\rnk)$ have different signs and the absolute value of the negative one is less than the positive one,
 \item[(D-3)] if $i=\rnk$, then $u(\rnk-1)>u(\rnk)$ with the same sign, or $u(\rnk-1)<u(\rnk)$ with different signs.
\end{itemize}
Consider the similar condition 
\begin{itemize}
 \item[(A-1)] if $i\leq n-2$, then $u(i)<u(i+1)$ with the same sign or $u(i)>u(i+1)$ with different signs,
 \item[(A-2)] if $i=\rnk-1$, then $u(\rnk-1), u(\rnk)>0$, or 
$u(\rnk-1)$ and $u(\rnk)$ have different signs and the absolute value of the negative one is greater than the positive one,
 \item[(A-3)] if $i\neq n$, then $u(\rnk-1)<u(\rnk)$ with the same sign or $u(\rnk-1)>u(\rnk)$ with different signs,
\end{itemize}
Denote
\begin{align*}
D(u):=&\{ u[i] \mid \text{$i\leq\rnk-2$ and $i$ satisfies (D)} \} \\
&\qquad
\cup \{ u[\rnk]_+ \mid \text{$i=\rnk-1$ satisfies (D)}\}
\cup \{ u[\rnk]_- \mid \text{$i=\rnk$ satisfies (D)}\} \\
A(u):=&\{ u[i] \mid \text{$i$ satisfies (A)} \} \\
&\qquad
\cup \{ u[\rnk]_+ \mid \text{$i=\rnk-1$ satisfies (A)}\}
\cup \{ u[\rnk]_- \mid \text{$i=\rnk$ satisfies (A)}\}
\end{align*}
where 
\begin{align*}
[\rnk]_+ = \{1,2,\cdots,\rnk-1,\rnk\} \quad \text{and} \quad
[\rnk]_- = \{1,2,\cdots,\rnk-1,-\rnk\}
\end{align*}
(cf. Figure \ref{type D chain}).
We define a signed Young diagram $\lambda_{u,v}^w$ for $u,v,w\in \widetilde{\mathfrak{S}}_{\rnk}^+$ in the manner described in  Section 4.
Note that we put $I(\emptyset)=0$ as a convention.

Now, the intersection number of $Y^w$, $X_u$ and $X_v$ in $X$ of type $D_{\rnk}$ is given by the following. 
\begin{corollary}\label{triple intersection for type D}
For even signed permutations $u,v,w\in \widetilde{\mathfrak{S}}_{\rnk}^+$, we have
\[ \intnum{[Y^w][X_u][X_v]}=\widetilde{I}(\WY{u}{v}{w}) \]
where $\widetilde{I}=(-1)^{\rnk+s}\widetilde{y}_1y_2\cdots y_s$ is the function described in \emph{Theorem \ref{main thm for type D}}.
\end{corollary}

\vspace{10pt}
For example, for $\rnk=5$ with the convention $\bar{k}=-k$, the Young diagram $\lambda_{\bar{1}345\bar{2}, \bar{1}345\bar{2}}^{\bar{1}\bar{2}543}$ is the one in Figure \ref{EX for type D}, and hence we obtain
\begin{align*}
\intnum{[Y^{\bar{1}\bar{2}543}][X_{\bar{1}345\bar{2}}][X_{\bar{1}345\bar{2}}]}  = -4.
\end{align*}




















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{On exceptional types}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we include the computation of intersection numbers of invariant divisors of the toric manifold $X$ for the root system of exceptional type $G_2$.
For other exceptional types $F_4, E_6, E_7$, and $E_8$, 
it would be interesting to find combinatorial objects which effectively compute the intersection numbers of invariant divisors.\\

Let $E=\{x\in\R^{3} \mid x_1+x_2+x_3=0\}$.
The roots are 
\begin{align*}
 &\pm(t_1-t_2), \ \pm(t_1-t_3), \ \pm(t_2-t_3), \\
 &\pm(2t_1-t_2-t_3), \ \pm(2t_2-t_1-t_3), \ \pm(2t_3-t_1-t_2)
\end{align*} 
where $t_i\in\R^{3}$ is the $i$-th standard vector.
We choose $\SimR=\{t_1-t_2, -2t_1+t_2+t_3\}$
as the set of simple roots, and write $\alpha_1=t_1-t_2$ and $\alpha_2=-2t_1+t_2+t_3$.
The Weyl group $W$ is the dihedral group of order 12 which is identified with the subgroup
\[
\WG:=
\{ u\in \widetilde{\mathfrak{S}}_3 \mid \text{$u(1)$, $u(2)$, and $u(3)$ have the same sign} \}
\]
of the 3rd signed permutation group.
Under this identification, the action of the Weyl group on $E$ is written as the natural action of $\WG$ on the indexes $i$ of $t_i$; $u\cdot t=t_{u(1)}$ for $u\in \WG$ where $t_{-i}:=-t_i$ ($1\leq i\leq 3$). 
This action of $W_{G_2}$ preserves $\Phi$.
The minimal generators $\fcw_1, \fcw_2\in E^*$ of the fundamental Weyl chamber $\sigma_{\text{id}}$ are
\[
\fcw_1=e_3-e_2, \ \fcw_2=\frac{1}{3}(2e_3-e_1-e_2)
\]
where $\{e_i\}_i\subset (\R^3)^*$ is the dual basis of $\{t_i\}_i\subset \R^3$.

Denoting by $2^{[\pm3]}$ the set of all subsets of $[\pm3]=\{1,2,3,-1,-2,-3\}$, we have a well-defined map $\dPhi \rightarrow 2^{[\rnk+1]}$ by sending 
\[
e_{u(3)}-e_{u(2)} \mapsto \{u(3), -u(2)\}, \quad
\frac{1}{3}(2e_{u(3)}-e_{u(1)}-e_{u(2)}) \mapsto \{u(3)\}
\]
for $u\in \WG$.
This is an injection, and hence we can identify $\Phi^*$ with the following subset of $2^{[\pm3]}$; 
\begin{align*}
\mathcal{S}:=\{ 3\bar{2}, \bar{3}2, 3\bar{1}, \bar{3}1, 2\bar{1}, \bar{2}1, 3,\bar{3}, 2, \bar{2}, 1, \bar{1}\}
\end{align*} 
where $\bar{k}=-k$ for $1\leq k\leq3$ and each sequence $ab$ in $\mathcal{S}$ is the set $\{a,b\}$, i.e. $3\bar{2}=\{3,-2\}$ for example.
Now, for each $S\in \mathcal{S}$, we have 
$\tau_S:=\tau_{u\omega_i}\in H^2(X)$ where $u\omega_i\in\dPhi$ corresponds to $S$ by this identification. 
Then, for $S_1,S_2\in\mathcal{S}$, it follows by Lemma \ref{prelim rays generating a cone} that $\tau_{S_1}\tau_{S_2}=0$ unless these sets form a nested chain of subsets, i.e. $S_1\subset S_2$ or $S_1\supset S_2$.

The linear relations (\ref{prelim linear relation}) for $\alpha=\alpha_1,\alpha_2$ are translated to 
\begin{align*}
&\tau_{3\bar{2}} 
+ \tau_{\bar{3}1} 
+ 2\tau_{\bar{2}1}  
+ \tau_{\bar{2}}  
+ \tau_{1}
=
\tau_{\bar{3}2}
+\tau_{3\bar{1}} 
+2\tau_{2\bar{1}} 
+\tau_{2}  
+ \tau_{\bar{1}}, \\
&3\tau_{3\bar{1}} 
+ 3\tau_{2\bar{1}}  
+ \tau_{3} 
+ \tau_{2} 
+ 2\tau_{\bar{1}} 
=
3\tau_{\bar{3}1}
+3\tau_{\bar{2}1} 
+\tau_{\bar{3}}
+\tau_{\bar{2}}   
+2\tau_{1},
\end{align*}
respectively.
From these relations together with the above observation about the vanishing of  $\tau_{S_1}\tau_{S_2}$, we see that
\begin{align*}
\tau_{3\bar{2}}\tau_{3}=1, \ 
\tau_{3\bar{2}}\tau_{3\bar{2}}=-1, \ 
\tau_{3}\tau_{3}=-3.
\end{align*}
Now, let 
\[
I_{G_2}(2,1):=1, \ I_{G_2}(1,1):=-3, \ I_{G_2}(2,2):=-1
\]
where $(2,1)$, $(1,1)$, and $(2,2)$ are Young diagrams with 2 rows.
Now the next claim follows from Lemma \ref{prelim Weyl inv}; 
if $S_1,S_2\in\mathcal{S}$ form a nested chain of subsets, 
then we have
\begin{align}\label{type G2 int num}
\intnum{\tau_{\A_1}\tau_{\A_2}} 
= I_{G_2}(\lambda)
\end{align}
where $\mu_X$ is the fundamental homology class and $\lambda$ is the Young diagram consisting of $|\A_1|$ and $|\A_2|$ reordered as a weakly decreasing sequence. Otherwise, the intersection number is zero.



Finally, we list the presentations of $[X_u]$ as monomials of $\tau_{\A}$ for all $u\in \WG$ in one-line notations;
\begin{align*}
&
[X_{123}]=1, \hspace{21pt}
[X_{213}]=\tau_{3\bar{1}}, 
[X_{132}]=\tau_{2\bar{3}}, 
[X_{231}]=\tau_{1\bar{3}}, 
[X_{312}]=\tau_{2}, \hspace{7pt}
[X_{321}]=\tau_{1}, \\
&
[X_{\bar{1}\bar{2}\bar{3}}]=\tau_{\bar{3}2}\tau_{\bar{3}}, \ 
[X_{\bar{2}\bar{1}\bar{3}}]=\tau_{\bar{3}}, \ 
[X_{\bar{1}\bar{3}\bar{2}}]=\tau_{\bar{2}}, \ \hspace{1pt}
[X_{\bar{2}\bar{3}\bar{1}}]=\tau_{\bar{1}}, \ 
[X_{\bar{3}\bar{1}\bar{2}}]=\tau_{\bar{2}1}, \ 
[X_{\bar{3}\bar{2}\bar{1}}]=\tau_{\bar{1}2}.
\end{align*}
Since we have $[Y^u]=(w_0^{-1})^*[X_{w_0 u}]=w_0^*[X_{w_0 u}]$ where $w_0=\bar{1}\bar{2}\bar{3}$ is the longest permutation, we obtain the list of  $[Y^u]$;
\begin{align*}
&
[Y^{\bar{1}\bar{2}\bar{3}}]=1, \hspace{22pt}
[Y^{\bar{2}\bar{1}\bar{3}}]=\tau_{\bar{3}1}, 
[Y^{\bar{1}\bar{3}\bar{2}}]=\tau_{\bar{2}3}, 
[Y^{\bar{2}\bar{3}\bar{1}}]=\tau_{\bar{1}3}, 
[Y^{\bar{3}\bar{1}\bar{2}}]=\tau_{\bar{2}}, \hspace{7pt}
[Y^{\bar{3}\bar{2}\bar{1}}]=\tau_{\bar{1}}, \\
&
[Y^{123}]=\tau_{3\bar{2}}\tau_{3}, \ 
[Y^{213}]=\tau_{3}, \ \hspace{1pt}
[Y^{132}]=\tau_{2}, \ 
[Y^{231}]=\tau_{1}, \ 
[Y^{312}]=\tau_{2\bar{1}}, \ 
[Y^{321}]=\tau_{1\bar{2}}.
\end{align*}
With these lists, we can compute intersection numbers $\intnum{[Y^w][X_u][X_v]}$ for all $u,v,w\in \WG$ by (\ref{type G2 int num}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
The author would like to thank Tatsuya Horiguchi, Hiroaki Ishida, Ivan Limonchenko, and Tomoo Matsumura for valuable comments. He also thanks Miho Hatanaka for reading of the first version of this paper carefully.

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