Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\mathbf \ell = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\mathbf \ell$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\mathbf \ell$. In recent work, the author and Harada described precisely when the Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\mathbf \ell$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\mathbf \ell$ comes from a weight of $G$. In the present paper, we interpret untwistedness of Grossberg-Karshon twisted cubes associated to any word $\mathbf i$ and any integer sequence $\mathbf \ell$ using the combinatorics of $\mathbf i$ and $\mathbf \ell$. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\mathbf \ell$-walk-avoiding.


Introduction
Let G be a complex semisimple algebraic group of rank r and B a Borel subgroup. Formulating a combinatorial model for a basis of a representation provides a fruitful connection between representation theory and algebraic geometry as exhibited by the theory of crystal bases and string polytopes. Kaveh [Kav15] show that the string polytopes can be obtained as Newton-Okounkov bodies of the flag variety G/B, and this association is extended to the generalized string polytopes and Bott-Samelson varieties by Fujita [Fuj18]. Furthermore, using the result of Anderson [And13], for this case there is a toric degeneration of Bott-Samelson variety to a toric variety whose Newton polytope is the generalized string polytope.
On the other hand, Grossberg and Karshon [GK94] also constructed one-parameter family of complex structures on Bott-Samelson varieties which makes the Bott-Samelson varieties into toric varieties, called Bott manifolds, and consequently obtained a Demazure-type character formula which can be interpreted combinatorially in terms of twisted cubes. This degeneration of complex structures can be interpreted as the toric degeneration of Bott-Samelson variety to a Bott manifold by Pasquier [Pas10]. Indeed, there is a flat family X over C such that X(t) is isomorphic to the Bott-Samelson variety for all t ∈ C \ {0} and X(0) is a Bott manifold. This connection is generalized to flag Bott-Samelson varieties and flag Bott manifolds in [FLS].
These twisted cubes are combinatorially much simpler than generalized string polytopes but they are not actual polytopes in the sense that they may not be convex nor closed and the intersection of faces may not be a face (cf. [GK94,§2.5 and Figure 1 therein] and Figure 1.1). More precisely, a Grossberg-Karshon twisted cube is a pair (C = C(c, ℓ), ρ), where C is a subset of R n and ρ is a density function whose support is C, taking values in {±1}. The defining parameters c = {c jk } 1≤j<k≤n and ℓ = (ℓ 1 , . . . , ℓ n ) are fixed constants with c jk ∈ Z and ℓ j ∈ R.
The main result of this paper concerns twisted cubes obtained from representationtheoretic data. Indeed, we consider a (not necessarily reduced) word decomposition i = (i 1 , . . . , i n ) ∈ [r] n of an element s i1 · · · s in in the Weyl group W of G and nonnegative integers m = (m 1 , . . . , m n ). Here, [r] := {1, . . . , r}. In this situation, the sequence i and non-negative integers m define a Bott-Samelson variety Z i and the line bundle L i,m on it. Moreover, the associated Grossberg-Karshon twisted cube (C(c(i), ℓ(i, m)), ρ) encodes the character of B-representation space H 0 (Z i , L i,m ) of holomorphic sections. Here, the integers c(i) and ℓ(i, m) are determined by i and m (see Section 2 for more details).
In this paper, we present a necessary and sufficient conditions on i and ℓ such that the associated Grossberg-Karshon twisted cube is untwisted (see Definition 1.3), i.e., C(c, ℓ) is a closed convex polytope and the density function is equal to 1 on C(c, ℓ), so that the Grossberg-Karshon character formula is a purely combinatorial positive formula. In other words, there is no minus sign in the formula.
In order to introduce our result, we prepare some terminology (see Section 2 for precise definitions). We say a word i = (i 1 , . . . , i n ) is a jumping walk if for each 1 ≤ j ≤ n, the set {i 1 , . . . , i j−1 } and an element i j are adjacent in the Dynkin diagram, i.e., the distance d({i 1 , . . . , i j−1 }, i j ) = min{d(i k , i j ) | k = 1, . . . , j − 1} is one. Here, d(a, b) is the distance of two nodes a and b on the Dynkin diagram. Therefore, to make a jumping walk, one can jump for the next step, but cannot away far. For example, in type A 5 1 2 3 4 5 the word i = (1, 3, 2, 4) is not a jumping walk since d(1, 3) = 2, but i = (3, 2, 1, 4, 5) is a jumping walk. In the above diagram, one can see the jumping walk (3, 2, 1, 4, 5). The word i = (i 1 , i 2 , . . . , i n ) is a hesitant jumping ℓ-walk if i 1 = i 2 , the subword (i 2 , . . . , i n ) is a jumping walk, and the integers satisfies an inequality ℓ 1 − ℓ 2 < ℓ 2 + · · · + ℓ n . Finally, we say that i is hesitant-jumping-ℓ-walk-avoiding if there is no subword which is a hesitant jumping ℓ-walk. Now we state our main theorem.
We note that if we consider the situation when the line bundle L i,m comes from the line bundle L λ of G/B, the untwistedness of the corresponding Grossberg-Karshon twisted cube can be detected using hesitant λ-walk avoidance by Harada and the author [HL15]. The Picard number of the Bott-Samelson variety Z i is n, the length of the sequence i, but on the other hand, that of the flag variety G/B is r, the rank of the group G. Accordingly, not every line bundle over the Bott-Samelson variety comes from the line bundle L λ of G/B usually. Since our main result can be applied to any line bundles over Z i , this result is more powerful than the previous one [HL15] (see Remark 1.8 and Corollary 2.9).
Additionally, for given i ∈ [r] n and m, with an appropriate choice of a valuation ν on the function field C(Z i ), Harada and Yang [HY16] construct a Newton-Okounkov body ∆ = ∆(Z i , L i,m , ν) of the Bott-Samelson variety Z i . They proved that when the twisted cube (C(c(i), ℓ(i, m)), ρ) is untwisted, then the Newton-Okounkov body ∆ and the twisted cube C(c(i), ℓ(i, m)) are the same (up to certain coordinate changes). Our result presents a sufficient condition on i and m so that the Newton-Okounkov body ∆ coincides with the twisted cube.
This paper is organized as follows. In Section 1, we recall the necessary definitions and establish terminology and notation. In Section 2, we introduce the notions of jumping walks, hesitant jumping walks, and hesitant-jumping-ℓ-walk-avoidance. Using this terminology we then make the statement of our main result, which is that untwistedness is equivalent to hesitant-jumping-ℓ-walk-avoidance. The proof of the main result occupies Section 3.

Background on Grossberg-Karshon twisted cubes
We begin by recalling the definition of twisted cubes introduced by Grossberg and Karshon [GK94, §2.5]. Let n be a fixed positive integer. A twisted cube is defined to be a pair (C(c, ℓ), ρ) where C(c, ℓ) is a subset of R n and ρ : R n → R is a density function with support equals to C(c, ℓ). Here, c = {c jk } 1≤j<k≤n and ℓ = (ℓ 1 , ℓ 2 , . . . , ℓ n ) are fixed integers. In order to simplify the notation in what follows, we define the following functions on R n : We also define a function sgn : R → {±1} by sgn(x) = 1 for x < 0 and sgn(x) = −1 for x ≥ 0.
We now give the definition of twisted cubes.
Definition 1.1. Let n, c, ℓ and A j be as above. Let C(c, ℓ) denote the following subset of R n : Moreover we define a density function ρ : R n → R by Obviously, the support supp(ρ) of the density function ρ is C(c, ℓ). We call the pair (C(c, ℓ), ρ) the twisted cube associated to c and ℓ.
A twisted cube may not be combinatorially equivalent to a cube [0, 1] n in the standard sense. In particular, the set C(c, ℓ) may be neither convex nor closed, as the following example shows. See also the discussion in [GK94, §2.5].
As mentioned in the introduction, the primary goal of this manuscript is to give necessary and sufficient conditions for the untwistedness of the twisted cube, in terms of the combinatorics of the defining parameters. The following makes this notion precise. Definition 1.3 (cf. [HY15, Definition 2.2]). We say that Grossberg-Karshon twisted cube (C = C(c, ℓ), ρ) is untwisted if C is a closed convex polytope, and the density function ρ is constant and equal to 1 on C and 0 elsewhere.
The main result of [HY15] characterizes the untwistedness of the Grossberg-Karshon twisted cube in terms of the basepoint-freeness of a certain toric divisor on a toric variety constructed from the data of c and ℓ. In particular, their result can be stated in terms of the Cartier data {m σ } associated to the divisor on the toric variety. Before reviewing the relevant result from [HY15] we introduce some terminology.
Let Σ(c) be the fan consisting of maximal cones generated by {e σj j | 1 ≤ j ≤ n} for each σ = (σ 1 , . . . , σ n ) ∈ {+, −} n . Then the toric variety X(Σ(c)) constructed by the fan Σ(c) is called a Bott manifold. For the torus-invariant divisor the Cartier data for D(c, ℓ) is same as {m σ }. Here, D e − j be the toric-invariant divisor corresponding to the ray spanned by e − j for 1 ≤ j ≤ n. Recall that we will study the case when the defining parameters for the Grossberg-Karshon twisted cube arise from certain representation-theoretic data. We now briefly describe how to derive the c and ℓ in this case.
Following the setting in [GK94], let G be a complex semisimple linear algebraic group of rank r over C. Choose a Cartan subgroup H ⊂ G, and let g = h ⊕ α g α be the decomposition into root spaces. We choose a set ∆ + of positive roots, and let B be the Borel subgroup whose Lie algebra is h ⊕ α∈∆ + g −α . Let {α 1 , . . . , α r } denote the simple roots, {α ∨ 1 , . . . , α ∨ r } the coroots, and {̟ 1 , . . . , ̟ r } the fundamental weights. Note that fundamental weights are characterized by the relation ̟ i , α ∨ j = δ ij . Let W be the Weyl group of G and s α ∈ W denote the simple reflection in W corresponding to the root α. For simplicity, we denote s i for the reflection s αi corresponding to the simple root α i .
Let i = (i 1 , . . . , i n ) be a sequence of elements in [r] and m = (m 1 , . . . , m n ) ∈ Z n ≥0 . Then i corresponds to a decomposition of an element w = s i1 s i2 · · · s in in W which is not necessarily reduced. For such i and m, we define constants c(i) = {c jk } 1≤j<k≤n and ℓ(i, m) = (ℓ 1 , . . . , ℓ n ) by the formulas in [GK94, §3.7] Note that the constants c jk are Cartan integers of G. The following example illustrates these definitions.
Example 1.6. Consider G = SL(3, C) with simple roots {α 1 , α 2 }. Let i = (1, 2, 1) and m = (1, 1, 1). Then we have Example 1.7. Consider G = SO(8) with simple roots {α 1 , . . . , α 4 }. See Table 1 for the numbering on simple roots. Let i = (1, 2, 3, 2, 4) and m = (2, 1, 3, 1, 1). Then the integers c jk and ℓ are given by Geometrically, a word i = (i 1 , . . . , i n ) and the integer vector m defines a Bott-Samelson variety Z i and a line bundle L i,m on it. More precisely, the Bott-Samelson variety Z i is defined to be the quotient where P i is the parabolic subgroup associated with the simple roots α i , i.e., its Lie algebra is g αi ⊕ b, and the right action of B n on P i1 × · · · × P in is given by where the right action of B n on P i1 × · · · × P in × C is defined by . In this setting, the Bott-Samelson variety Z i determined by the sequence i has a toric degeneration to the toric variety X(Σ(c)) (see [Pas10,§2] and [GK94]). Moreover, the line bundle L i,m over the Bott-Samelson variety Z i degenerates into the line bundle over the Bott manifold X(Σ(c)). In particular, the degeneration of the line bundle L i,m over X(Σ(c)) is given by the divisor D(c, ℓ).
The set H 0 (Z i , L i,m ) of holomorphic sections possesses a B-representation structure and is called a generalized Demazure module. Moreover, the ordinary Demazure module can be obtained in this way. Indeed, suppose that i = (i 1 , . . . , i n ) ∈ [r] n is a reduced decomposition of an element w in the Weyl group W , i.e., w = s i1 · · · s in and ℓ(w) = n. Then a dominant integral weight λ = r i=1 λ i ̟ i 1 gives a line bundle L λ on the flag variety G/B, so that on the Schubert variety X(w) for 1 ≤ j ≤ n. Then, we have that and the morphism µ induces an isomorphism of B-modules where λ ′ = i∈[n]\{i1,...,in} λ i ̟ i . We note that the relation (1.8) and the second isomorphism of B-modules in (1.9) holds even when i is not reduced. See [Fuj18, Section 2] and reference therein for more details on generalized Demazure modules.
As a consequence of equations (1.6) and (1.7), we obtain the following result which will be used later: Suppose that m is given by (1.7). Then the constant ℓ(i, m) = (ℓ 1 , . . . , ℓ n ) is given by the formula Proof. From (1.6) and (1.7), we get 1.7)).
This proves the lemma.
As mentioned in the introduction, Grossberg and Karshon derive a Demazuretype character formula for the B-representation H 0 (Z i , L i,m ) corresponding to i and m, expressed as a sum over the lattice points Z n ∩C(c, ℓ) in the Grossberg-Karshon twisted cube (C(c, ℓ), ρ) (see [GK94, Theorems 5 and 6]). The lattice points appear with a plus or minus sign according the density function ρ. Accordingly, their formula is a positive formula if ρ is constant and equal to 1 on all of C(c, ℓ). From the point of view of representation theory it is therefore of interest to determine conditions on the integer vector ℓ = (ℓ 1 , . . . , ℓ n ) and the word decomposition i = (i 1 , . . . , i n ) such that the associated Grossberg-Karshon twisted cube is in fact untwisted.
2. Diagram jumping walks, hesitant jumping walk avoidance, and the statement of the main theorem In order to state our main theorem, it is useful to introduce some terminology. From now on, we assume that the group G is simply-laced, i.e., the Dynkin diagram of G only contains simple links, so that G is of type A, D, or E. In what follows, we fix an ordering on the simple roots as in Table 1; our conventions agree with that in the standard textbook of Humphreys [Hum78].
In order to simplify the notation, we define distance d(A, B) of two subsets A, B ⊂ [r] to be Here, d(a, b) for a, b ∈ [r] is the minimal distance of elements a, b in the corresponding Dynkin diagram. For example, suppose that G is of type A 5 . Then we have the following enumerations.    In what follows, we also find it useful to consider words which are 'almost' jumping walks, except that the word begins with a repetition (thus disqualifying it from being a walk), i.e. the initial index appears twice.
• the subword (i 1 , . . . , i n ) is a jumping walk. In other words, except for the 'hesitation' at the first step, the remainder of the word is a jumping walk. We refer to the subword (i 1 , . . . , i n ) as the jumping component of the hesitant jumping walk.
We now present a corollary of Theorem 2.7 which is already observed by Harada and the author [HL15]. In order to state the corollary, it is useful to introduce some terminology in the paper [HL15]. We call a word i = (i 1 , . . . , i n ) is a diagram walk if d(i j , i j+1 ) = 1 for all 1 ≤ j < n. For a dominant weight λ = r i=1 λ r ̟ r , we say i = (i 0 , i 1 , . . . , i n ) is a hesitant λ-walk if i 0 = i 1 , the subword (i 1 , . . . , i n ) is a diagram walk, and λ in > 0. Lastly, we say i is hesitant-λ-walk-avoiding if there is no subword which is a hesitant λ-walk. With these expressions, we present the following corollary. Proof. By Theorem 2.7, it is enough to show that the word i is hesitant-jumpingℓ-walk-avoiding if and only if it is hesitant-λ-walk-avoiding. We prove the contrapositive of the claim, that is, we will prove that i has a subword which is a hesitant jumping ℓ-walk if and only if it has a subword which is a hesitant λ-walk.
On the other hand, suppose that i has a subword j = (i j0 , i j1 , . . . , i js ) which is a hesitant jumping ℓ-walk. Then, the condition i j0 = i j1 provides ℓ j0 = ℓ j1 so that we get the inequality 0 = ℓ j0 − ℓ j1 < ℓ j1 + · · · + ℓ js . Because ℓ j ≥ 0 for all j, there exists t ∈ [s] such that ℓ jt > 0. By the definition of jumping walk, one can always find a subword of j which is a hesitant λ-walk starting with i j0 , i j1 and ending at i js . This proves that i has a subword which is a hesitant λ-walk, so the result follows.
up to a certain coordinate change (see [HY16,Theorem 3.4] for more details). As a consequence, we get the following corollary which is related to the question (2) in [HL15, Section 5].
Corollary 2.10. Let G be a complex simply-connected semisimple algebraic group of rank r. Let i = (i 1 , . . . , i n ) ∈ [r] n be a word and let m = (m 1 , . . . , m n ) ∈ Z n ≥0 . Let c = {c jk } and ℓ ∈ Z n be determined from i and m as in (1.5) and (1.6). If the word i is hesitant-jumping-ℓ-walk-avoiding, then the twisted cube C(c, ℓ) coincides with the Newton-Okounkov body ∆(Z i , L i,m , ν) up to a coordinate change, where ν is the valuation constructed in [HY16].

Proof of the main theorem
In this section, we will present a proof of the main theorem. We start with a proposition which will be used in the proof. One can see that the subsequent proposition holds only when simply-laced cases.
Lemma 3.2. Suppose that G is simply-laced and (i 1 , . . . , i n−1 ) is a jumping walk. For i n ∈ [r], we have the following.
Here, c j,k are Cartan integers: Proof. Note that since (i 1 , . . . , i n−1 ) is a jumping walk, the set I := {i 1 , . . . , i n−1 } forms an interval on the Dynkin diagram, i.e., if j ∈ [n] satisfies min I < j < max I, then j ∈ I. We consider the first case. Since the distance between i n and the set I is zero, i n ∈ I. Suppose that i n sits in the right-most/left-most position among I, i.e., i n = min I or i n = max I. Then, c in,i1 + · · · + c in,in−1 = 2 − 1 = 1 ≥ 0, which is the desired inequality. Suppose that i n is neither maximum nor minimum of I. Then, we have that c in,i1 + · · · + c in,in−1 = 2 − 1 − 1 = 0 ≥ 0. Consequently, we prove the claim for (1). For the second and the third cases, by the definition of Cartan integers, we get the required equalities.
Suppose that we have an increasing sequence (j 0 < j 1 < · · · < j s ) satisfies conditions. Consider the subword j = (i j0 , i j1 , . . . , i js ). The first condition c j0,j1 = 2 implies that i j0 = i j1 . Accordingly, the word j hesitates at the first. When t = 2, the second condition becomes c j1,j2 = −1. Then by (3.1), we have that d(i j2 , i j1 ) = 1, so that (i j0 , i j1 , i j2 ) is a hesitant jumping walk. Using an induction on s and Lemma 3.2, we prove the "only if" part of the proposition.
3.1. Necessity. We first prove that if i has a subword which is a hesitant jumping ℓ-walk, then the corresponding twisted cube is twisted. Suppose that j = (i j0 , i j1 , . . . , i js ) is a subword of i which is a hesitant jumping walk. Also suppose that (3.2) ℓ j0 − ℓ j1 < ℓ j1 + · · · + ℓ js .
Consequently, we prove the equation (3.6), so the necessity of the theorem follows.
3.2. Sufficiency. We now prove that twistedness implies the existence of a subword which is a hesitant jumping ℓ-walk. To give a proof, we prepare one lemma.
By Theorem 1.4, there exists an element σ of {+, −} n and an index k such that m σ,k < 0. For such a choice of σ, we may assume without loss of generality that k is chosen to be the maximal such index, i.e., m σ,k < 0 and m σ,s ≥ 0 for s > k. Recall that m σ,k = ℓ k − s>k c ks m σ,s .
By assumption m σ,k < 0, we have that s>k c ks m σ,s > ℓ k ≥ 0.
Since m σ,s ≥ 0 for s > k, this implies that there exists some p > k with c k,p > 0 and m σ,p > 0. Choose j 1 to be the minimal such index. Consequently, c k,s ≤ 0 or m σ,s = 0 for all k < s < j 1 , so we have that ℓ k < s>k c ks m σ,s ≤ c k,j1 m σ,j1 + s>j1 c k,s m σ,s (3.7) By definition, we have that c k,j1 = α ij 1 , α ∨ i k > 0 if and only if i k = i j1 . Furthermore, in this case we get c k,j1 = 2 and c j1,s = c k,s for all s. From these observations, we get: Combining (3.7) and (3.8), we have that (3.9) ℓ k − ℓ j1 < ℓ j1 − s>j1 c j1,s m σ,s = m σ,j1 .
First suppose − s>j1 c j1,s m σ,s ≤ 0. In this case, we have that Accordingly, the sequence (j 0 = k < j 1 ) satisfies the three required conditions of hesitant jumping ℓ-walk, so we are done.