\documentclass[12pt]{article} \usepackage{e-jc} \specs{P1.40}{25(1)}{2018} %\usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} %%%%%%%%%%%%%%% dodalem JW \usepackage{color} %%%%%%%%%%%% \usepackage[centertags]{amsmath} % \usepackage[margin=1in]{geometry} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{caption} \usepackage{newlfont} \usepackage{verbatim} \usepackage{amscd} \usepackage{amsgen} \usepackage{comment} %\usepackage{youngtab} \usepackage{xspace} \usepackage{amssymb} \usepackage{stmaryrd} \usepackage{amssymb,amsmath} \usepackage{amscd} \usepackage{enumitem} \usepackage{color} \usepackage{xy} \usepackage{graphicx} % THEOREMS --------------------------------------------------------------- \theoremstyle{plain} \newtheorem{thm}{Theorem}%[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{con}[thm]{Conjecture} \newtheorem{lema}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{obs}[thm]{Observation} \newtheorem{comp}[thm]{Computation} %\newtheorem*{ques}{Question} \newtheorem{ques}[thm]{Question} %\newtheorem*{tthm}{Theorem} %\newtheorem{pr}{Program} %\newtheorem{pro}{Problem} \theoremstyle{definition} \newtheorem{df}[thm]{Definition} \newtheorem{exm}[thm]{Example} %\newtheorem*{exmm}{Example} \newtheorem{rem}[thm]{Remark} \newtheorem{tab}{table}[section] \newtheorem{fact}[thm]{Fact} \newtheorem{pr}{Algorithm} % \newtheorem{constr}{Construction}[section] %\numberwithin{equation}{section} %\renewcommand{\theequation}{\thesection.\arabic{equation}} %\renewcommand{\baselinestretch}{1.2} %%% ---------------------------------------------------------------------- \def\p{\mathbb{P}} \def\P{\mathbb{P}} \def\RR{\mathbb{R}} \def\z{\mathbb{Z}} \def\n{\mathbb{N}} \def\Z{\mathbb{Z}} \def\Q{\mathbb{Q}} \def\N{\mathbb{N}} \def\c{\mathbb{C}} \def\C{\mathbb{C}} \def\o{\mathcal{O}} \def\Id{Id} \def\Eval{Eval} \DeclareMathOperator{\inte}{int} \DeclareMathOperator{\rk}{Rk}\DeclareMathOperator{\Proj}{Proj} \DeclareMathOperator{\diag}{diag}\DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\sing}{Sing} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\divi}{div} \DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\rad}{Rad} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\h}{ht} \DeclareMathOperator{\Spec}{Spec} \def\p{\mathbb{P}} \def\a{\mathbb{A}} \def\ob{\begin{obs}} \def\kob{\end{obs}} \def\dow{\begin{proof}} \def\kdow{\end{proof}} \def\kwadrat{\hfill$\square$} \def\tw{\begin{thm}} \def\ktw{\end{thm}} \def\hip{\begin{con}} \def\khip{\end{con}} \def\lem{\begin{lema}} \def\klem{\end{lema}} \def\ex{\begin{exm}} \def\prog{\begin{pr}} \def\kprog{\end{pr}} \def\wn{\begin{cor}} \def\kwn{\end{cor}} \def\uwa{\begin{rem}} \def\kuwa{\end{rem}} \def\kex{\end{exm}} \def\dfi{\begin{df}} \def\kdfi{\end{df}} \def\W{\mathcal W} \setcounter{section}{0} \def\cA{{\mathcal A}} \def\cT{{\mathcal T}} \def\flo{{\mathcal G}} \def\soc{{\mathcal S}} \def\g{g} \def\n{n^-} \def\pp{p} \def\b{b} \definecolor{zielony}{rgb}{0.5, 0.9, 0.1} \definecolor{czerwony}{rgb}{0.9, 0.2, 0.1} \definecolor{niebieski}{rgb}{0.3, 0.1, 0.9} \newcommand{\red}[1]{\color{czerwony} #1 \color{black}} \newcommand{\green}[1]{\color{zielony} #1 \color{black}} \newcommand{\blue}[1]{\color{niebieski} #1 \color{black}} \newcommand{\barvinok}{\textsc{barvinok}\xspace} \newcommand{\iscc}{\textsc{iscc}\xspace} \newcommand{\isl}{\textsc{isl}\xspace} \xyoption{line} \def\fa{\begin{fact}} \def\kfa{\end{fact}} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\vol}{Vol} \usepackage{url} \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} \renewcommand{\comment}[1]{{$\star$\sf\textbf{#1}$\star$}} \title{\bf Obstructions to combinatorial formulas for plethysm} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Thomas Kahle\\ \small Fakult\"at f\"ur Mathematik\\[-0.8ex] \small Otto-von-Guericke Universit\"at\\[-0.8ex] \small 39106 Magdeburg, Germany\\ \small\tt thomas.kahle@ovgu.de\\ \and Mateusz Micha\l ek \\ \small Max Planck Institut for Mathematics in the Sciences\\[-0.8ex] \small Leipzig, Germany\\[-0.8ex] \small and\\[-0.8ex] \small Mathematical Institute of the Polish Academy of Sciences\\[-0.8ex] \small Warsaw, Poland\\ \small\tt wajcha2@poczta.onet.pl } \date{\dateline{Nov 10, 2016}{Jan 16, 2018}{Mar 2, 2018}\\ \small Mathematics Subject Classifications: 20G05, 11P21, 11H06, 05A16, 52B20, 52B55} % Primary: % 20G05 Representation theory % 11P21 Lattice points in specified regions % Secondary: % 11H06 Lattices and convex bodies % 05A16 Asymptotic enumeration % 52B20 Lattice polytopes % 52B55 Computational aspects related to convexity % 20C15 Ordinary representations and characters \begin{document} \maketitle \begin{abstract} Motivated by questions of Mulmuley and Stanley we investigate quasi-poly\-no\-mials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients. \end{abstract} Many problems in representation theory have combinatorial solutions. A~well-known, important example is the Littlewood--Richardson rule which gives the multiplicities of isotypic components in the tensor product of two irreducible $GL(n)$ representations \cite{fulton91:_repres_theor}. The solution is combinatorial in the sense that the answer is given by the number of lattice points in explicit rational convex polyhedra, i.e.~the multiplicities are equal to values of an Ehrhart quasi-polynomial. Berenstein and Zelevinsky provided another interpretation of the Littlewood--Richardson rule~\cite{BZ}, isomorphic, however, on the level of polytopes \cite{pak2005combinatorics}. The study of different polyhedral structures turned out to be very useful. Knutson and Tao \cite{knutson1999honeycomb} showed that the honeycomb polytopes (in spirit similar to \cite{BZ}) are nonempty if and only if they contain a lattice point. This was the crucial last step in the solution to the Horn problem~\cite{horn1962eigenvalues} which goes back to Weyl's work on eigenvalues of partial differential equations~\cite{weyl1912asymptotische}. In this paper we study the more complicated plethysm coefficients $m^{d,k}_\lambda$ defined~by \[ S^d(S^k W)=\bigoplus_{\lambda} m^{d,k}_\lambda S^\lambda W, \] where the sum is over partitions $\lambda$ of~$dk$. The computation of the multiplicity functions in plethysms can be seen as an operation on Schur polynomials~\cite{macdonald1998symmetric}. Viewed like this, it is surprising that when the plethysm is written in terms of other Schur polynomials, the coefficients are always nonnegative. Of course this must be true, because the coefficients are multiplicities of irreducible representations, but it would be desirable to have a combinatorial explanation for nonnegativity. % In \cite[Problem~9]{Stanley} Richard Stanley asked for a \emph{positive} combinatorial method to compute plethysm coefficients. A connection between plethysm and lattice point counting was shown at least in~\cite{KMplethysm,colmenarejo2015stability,christandl2012computing}. These connections are not direct in the sense that plethysm coefficients are not seen to equal counts, but always involve some opaque arithmetics. The functions $f(s)=m^{d,sk}_{s\lambda}$ and $g(s)=m^{sd,k}_{s\lambda}$ share many properties with Ehrhart functions of rational polytopes: \begin{itemize} \item Both $f$ and $g$ are nonnegative and $f(0)=g(0)=1$. \item Both $f$ and $g$ are quasi-polynomials. This deep fact follows from~\cite[Corollary~2.12]{meinrenken1999singular}. See~\cite[Remark~3.12]{KMplethysm}. \item For regular $\lambda$, the leading term of $f$ is proportional to the Ehrhart function of a rational polytope corresponding to the Littlewood--Richardson rule~\cite[Section~4]{KMplethysm}. \end{itemize} \begin{rem} The scaling factor between the leading coefficient of $f$ and the Ehrhart function in the third bullet is the Plancherel measure on Young diagrams. It is conjectured that the same is true for any~$\lambda$, however we are not aware of a proof. \end{rem} \begin{exm} The multiplicity of $\lambda=s\cdot(2k-1,1)$ inside $S^2(S^{sk})$ equals $0$ when $s$ is odd and $1$ when $s$ even. In particular, it is equal to the number of integral points in the (zero-dimensional) polytope $\{s/2\}$. The multiplicity of $s\cdot(2k,k)$ inside $S^3(S^{sk})$ is equal to the number of integral points in the (one-dimensional) polytope $s\cdot[1/3,1/2]$ or $s\cdot [1/2,2/3]$. \end{exm} Ketan Mulmuley posed several conjectures concerning the behavior of plethysm and Kronecker coefficients in relation to geometric complexity theory \cite{mulmuley2001geometric,mulmuleyGCT6long}. \begin{ques}\label{q:combinatorial} Fix positive integers $k,d$, and $\lambda$ a partition of $kd$. Are the quasi-polynomials $s\mapsto m^{d,sk}_{s\lambda}$ and $s\mapsto m^{sd,k}_{s\lambda}$ Ehrhart functions of rational polytopes? \end{ques} The question about $m^{sd,k}_{s\lambda}$ comes directly from \cite[Hypothesis~1.6.4]{mulmuleyGCT6long}. We give a negative answer for both functions in Remark~\ref{r:ehrhart-proof}. Our main goal, however, is a generalized version of these questions for which we need some additional terminology. Following \cite[Section~5.1]{mulmuleyGCT6short} we define a \emph{shifted} or \emph{inhomogeneous rational polytope} as a system of inequalities \[ P (A,b,c) = \{x\in\mathbb{Q}^m : Ax \le b+c\}, \] where $b,c$ are arbitrary rational vectors and $A$ is a rational matrix. Splitting the right hand side as $b + c$ is motivated by the definition of the dilations of $P$ as \[ P(A,sb,c) = \{x\in\mathbb{Q}^m : Ax \le sb + c\}. \] An \emph{asymptotic Ehrhart quasi-polynomial} is a counting function of the form \[ s\mapsto \#(P(A,sb,c)\cap\Z^m). \] The \emph{dimension} of a shifted rational polytope $P$ is by definition the dimension of $P(A,sb,c)$ for large~$s$ (for small $s$ the polytope can be empty). Contrary to the case of Ehrhart quasi-polynomials, an asymptotic Ehrhart quasi-polynomial does not need to be a quasi-polynomial, although it is for large arguments. Moreover, a quasi-polynomial may be an asymptotic Ehrhart quasi-polynomial but not an Ehrhart quasi-polynomial. Asymptotic Ehrhart quasi-polynomials do not have to satisfy Ehrhart reciprocity. % See the discussion at the end of this paper for an example. Further, the dimension of a shifted rational polytope can be strictly greater than the degree of the associated asymptotic Ehrhart quasi-polynomial. See~\cite{stanley1982linear} and~\cite[Chapter~I]{stanley96:_combin_commut_algeb} for structure theory of asymptotic Ehrhart quasi-polynomials and the relation to linear diophantine equations. In \cite[Hypothesis~5.3]{mulmuleyGCT6short} it is conjectured that the multiplicity of $s\cdot\pi$ in $S^{s\cdot\lambda}(S^\mu)$ is an asymptotic Ehrhart quasi-polynomial with additional complexity-theoretic properties. The asymptotic part of the conjecture is motivated by the failure of Ehrhart type explanations already for Kronecker coefficients~\cite{briand2009reduced,King} which are often considered less complicated than plethysm coefficients. In the present paper we make progress towards a negative answer of the following simplified version. %, motivated by the results of~\cite{King}. % which discards all the complexity theory. % of~\cite[Hypothesis~5.3]{mulmuleyGCT6short}. \begin{ques}\label{q:mulmuley2} Is $m^{sd,k}_{s\lambda}$ an asymptotic Ehrhart quasi-polynomial? \end{ques} Our main result, Theorem~\ref{t:main}, implies a negative answer to both cases in Question~\ref{q:combinatorial} and strong restrictions on a positive solution to Question~\ref{q:mulmuley2}. We show that $m^{sd,k}_{s\lambda}$ need not be an asymptotic Ehrhart quasi-polynomial of a polytope of dimension equal to the degree of its growth. At the moment we are not able either to exclude or confirm that $m^{sd,k}_{s\lambda}$ is an asymptotic Ehrhart quasi-polynomial for an inhomogeneous polytope of dimension strictly larger than its degree. % As the main result of our % work, we show that both of the above mentioned multiplicity functions % for plethysm do not have to come from any shifted rational polytope of % dimension equal to the degree of the quasi-polynomial. As for % (homogeneous) rational polytopes the degree of the quasi-polynomial % equals the dimension of the polytope, this is a stronger statement % than the one presented in Question \ref{q:combinatorial}. Still, In previous work the authors gave a formula for plethysm coefficients, which is a sum of Ehrhart functions of various polytopes with (positive and negative) coefficients~\cite{KMplethysm}. This allows to gather experimental data on the questions for many rays. A-posteriori, the specific plethysm in Theorem~\ref{t:main} can also be confirmed using well-known formulas for $S^3(S^k)$. General formulas for $S^k(S^3)$ are unknown, though. Our methods are inspired by \cite{King, briand2009reduced}. \begin{thm}\label{t:main} The multiplicity functions of $S^{s\cdot(7,5,0)}$ in $S^3(S^{4s})$ and $S^{4s}(S^3)$ equal \begin{align*} \phi: s & \mapsto \frac{s+r(s)}{3} % % % It also equals (s+1)/3 rounded up or down depending on the parity of s. % % % % \frac{1}{3} \left( s + 6 \left\lfloor \frac{s}{2} % \right\rfloor + 2 + % \left\lfloor \frac{s}{3} \right\rfloor - % \left\lfloor \frac{s-1}{3} \right\rfloor - % \left\lfloor \frac{s-2}{3} \right\rfloor + % \left\lfloor \frac{s-3}{3} \right\rfloor \right) \\ % &\,\, = \frac{1}{3} s + 2 \left\lfloor \frac{s}{2} % \right\rfloor + \frac{2}{3} + % \begin{cases} % 1/3 & \text { if $s\equiv 0 \mod 3$ }\\ % 0 & \text { if $s\equiv 1 \mod 3$ }\\ % -1/3 & \text { if $s\equiv 2 \mod 3$ }\\ % \end{cases} \end{align*} where $r(s)$ has period $6$ and takes the values $3,-1,1,0,2,-2$ on respectively the integers $0,\dots,5$. There do not exist rational vectors $a,b,c \in\mathbb{Q}^n$ (of arbitrary length $n$) such that $\phi$ equals the counting function of a one-dimensional inhomogeneous polytope $P(a,b,c)$. % \[ % \{x\in\mathbb{Q} : ax \le b + c\}. % \] % In particular, $\phi$ is not an Ehrhart function of any rational polyhedron. % No % positive combination of Ehrhart functions of rational polyhedra % equals~$\phi$. \end{thm} \begin{rem}\label{r:ehrhart-proof} Before giving a general proof, it is easy to see that $\phi$ cannot be an Ehrhart function: it violates Ehrhart--Macdonald reciprocity~\cite{ehrhart1967I,macdonald1971polynomials}. The value $f(-s)$ of an Ehrhart quasi-polynomial $f$ of a rational polytope $P$ at a negative integer argument counts (up to a global sign) the number of interior lattice points in~$s\cdot P$. In particular, $|f(-s)|\leq f(s)$. However, \[ -\phi(-1)=1>0=\phi(1). \] % The first 7 values of the function $\phi$ in the theorem are % $1,0,1,1,2,1,3$. One can without much trouble observe that this can % not be an Ehrhart function using the jump down from $1$ to $0$ and % then the $2,1,3$ pattern. Interestingly, the jumps in values that make Ehrhart reciprocity fail are also the crucial ingredient for our proof of non-representability by a one-dimensional inhomogeneous polytope. %This statement gives a counterexample to Mulmuley's original question %from the arXiv version [exact reference from referee reply]. \end{rem} % There are also several ways to exclude $\phi$ from being an Ehrhart % function. The easiest is to note that the polytope would have to be % of dimension $1$ and volume~$4/3$. Since $\phi(6)=9$, six times the % polytope is a lattice polytope. This reduces the problem to % checking a finite number of intervals as candidates for the % polytope, none of which has $\phi$ as its Ehrhart function. \begin{proof} The equality of the multiplicities of the given ray in both plethysms is a consequence of Hermite reciprocity \cite{hermite1854}, \cite[Exercise~6.18]{fulton91:_repres_theor}. There are now various ways to determine the formula from the interpretation as a multiplicity in $S^3(S^{4s})$. One is to simply evaluate the explicit formula from~\cite{KMplethysm} along a ray. Another way is to observe that the function must exhibit linear growth and that its period is at most six~\cite{CGR83, manivel2014secants}. With this information and some values the function can be interpolated. Let $P$ be an inhomogeneous line segment as in the statement. Then the $s$-th dilation of $P$ can be written as \[ P (s) = \{x\in\Q: \max_i (sb_i + c_i) \le x \le \min_i (sb'_i + c'_i)\}. \] Asymptotically, $P(s)$ becomes an interval of length~$\frac{s}{3}$. This means that there exists an $s_0$ such that for all $s > s_0$ \begin{equation}\label{eq:main_bound} P(s) = \{x\in\Q: s b + c \le x \le s(b+\textstyle\frac{1}{3}) + c')\} =: Q(s), \end{equation} for some $b,c,c'\in\Q$. Making $s_0$ even larger it can be assumed that $s_0b$ are integers and that $s_0$ is divisible by~6 (the period of~$\phi$). Since $\#Q(s)$ and $\#P(s)$ have the same linear term and constants that are $s_0$-periodic (by the divisibility assumptions) they agree for all~$s$. The proof of nonexistence of the family $Q(s)$ such that $\phi(s) = \#Q(s)$ is by examination of constraints on $b$, coming from the known values of the counting function~$\phi$. Without loss of generality we have \begin{equation} \label{eq:shiftP2} 0 \le b < 1. % By shifting $d$ by integers \end{equation} Indeed, changing $b$ by an integer also only shifts $Q$ by an integral value. As $\phi(5)+2=\phi(6)$ there must be at least two integers $x$ such that $5(b+\frac{1}{3})+c'1$. In particular, for any $s$ there is always at least one integer $x$ satisfying $s(b+\frac{1}{3})+c'1$ must contain an integer). On the other hand, by \eqref{eq:shiftP2} there is at most one integer $x$ satisfying $sb+c'0=\phi(1)$. \end{proof} \begin{rem} We have confirmed that all examples with two rows and fewer boxes are in fact Ehrhart functions. In this sense, our counter-example is minimal. \end{rem} \begin{rem} A different way to describe an inhomogeneous polytope is by linear integral equations and nonnegativity. In this setting Stanley gave a general reciprocity theorem which relates the Hilbert series (expanded at $\infty$) of the module of positive integral solutions of linear Diophantine equations to the Hilbert series of negative solutions to the same equations. In \cite[Theorem~4.2]{stanley1982linear} there are, however, combinatorially defined correction terms. \end{rem} % In the remainder of this note we discuss some ramifications of the % theorem. % \begin{rem} While this is not visible from the representation in~\cite{KMplethysm}, a general theorem of Meinrenken and Sjamaar \cite{meinrenken1999singular} implies that the chambers of plethysm quasi-polynomials are polyhedral cones (see \cite[Remark~3.12]{KMplethysm}, \cite{paradan2015witten}). One may thus ask the stronger question if they are Ehrhart quasi-polynomials in general. Of course, Theorem~\ref{t:main} gives a negative answer to this much stronger property too. \end{rem} Lattice point counting in polytopes and Ehrhart functions are arguably the most natural combinatorial explanations one may hope for here, but they are not the only ones. Our theorem also excludes other possibilities. \begin{cor} The plethysm coefficients are not positive combinations of counting functions of lattice points in integral scalings of any convex regions. \end{cor} \begin{proof} One-dimensional convex sets are polyhedra and the counting function in Theorem~\ref{t:main} would need to specialize to a positive combination of Ehrhart functions. \end{proof} % One could ask next if the plethysm quasi-polynomials are counting % functions of any other, necessarily nonconvex, shapes. In principle % there could be a nonconvex cone that projects to the weight space and % the counting function of lattice points in the fibers coincide with % the plethysm multiplicities. % The plethysm counting function in Theorem~\ref{t:main} cannot be written as a positive combination of Ehrhart quasi-polynomials as these also have to satisfy Ehrhart reciprocity. However, it can be written as a sum of an asymptotic Ehrhart quasi-polynomial and an honest Ehrhart quasi-polynomial as in the following proposition (which is very easy to confirm). \begin{prop}\label{p:sum} Fix a small rational $\varepsilon > 0$. Let $P(s) = [\varepsilon,s\cdot\frac{1}{3}+\varepsilon]$ and $Q(s) = s\cdot \{\frac{1}{2}\}$. Then $\phi(s) = \#(P(s)\cap\Z) + \#(Q(s) \cap \Z)$. \end{prop} \begin{rem} An interesting open question is the following. Suppose we are given a piecewise quasi-polynomial whose chambers are cones. Is there a finite computational test if the quasi-polynomial is an (asymptotic) Ehrhart function? Assume it is not an Ehrhart function globally, but is an Ehrhart function on each individual ray. Is there a finite number of rays such that finding polytopes for each of the finitely many rays proves the Ehrhart nature for all rays? Is it some multi-dimensional generalization of Ap\'ery sets known from numerical semigroups? \end{rem} \begin{rem} According to \cite[Remark~3.8]{KMplethysm} the function $\phi$ in Theorem~\ref{t:main} also counts, for example, the multiplicity of $\lambda = s\cdot (7+2t,5+2t,2t)$ in $S^3(S^{s\cdot (5+2t)})$ for any positive~$t$. Therefore counterexamples also exist for strictly interior rays. \end{rem} Recently, there has been a lot of interest in \emph{Kronecker coefficients}~\cite{manivel2014asymptotics2, briand2009reduced,sam2015stembridge,baldoni2015multiplicity}. These are different from plethysm, but also hard to compute and important in geometric complexity theory. It is known that on rays they are given by quasi-polynomials \cite[Theorem 1]{manivel2014asymptotics} that do not have to be Ehrhart functions~\cite{King}. However, the problem is open for specific rays that are important in GCT. Emmanuel Briand pointed us towards following observation. Two row partitions are useful because Hermite reciprocity applies to them. Further, by the work of Vallejo \cite{Vallejo}, cf.~\cite{macdonald1998symmetric}, the plethysm coefficients in Theorem~\ref{t:main} are equal to special Kronecker coefficients: \[ \phi(s)=K(3^{4s},3^{3s},(7s,5s))=K((4s,4s,4s),(4s,4s,4s),(7s,5s)), \] where $K$ depends on three partitions and is the Kronecker coefficient. Indeed, let $p_n(a,b)$ denote the number of Young diagrams with $n$ boxes contained inside an $a\times b$ rectangle. Both the multiplicity of $\lambda$ with two rows in the plethysm $S^a(S^b)$ and the Kronecker coefficient $K(a^{b},a^{b},\lambda)$ can be expressed as the difference \[ p_{\lambda_2}(a,b)-p_{\lambda_2-1}(a,b). \] For example by transposition, this immediately implies Hermite reciprocity. \begin{exm} The multiplicity of $(7,5)$ inside $S^4(S^3)$ equals zero. Here we have $4$ diagrams with $5$ boxes inside a $4\times 3$ rectangle: $(4,1),(3,2),(3,1,1),(2,2,1)$. We also have $4$ such diagrams with $4$ boxes: $(4,0),(3,1),(2,2),(2,1,1)$, hence the difference is indeed zero. \end{exm} The rectangular Kronecker coefficients $K(a^b,a^b,\lambda)$ are of particular interest in GCT (see e.g.~\cite{ChristianandGretta}). Moreover, they exactly govern unimodality of $q$-binomial coefficients, observed already in 1878 by Sylvester (see \cite[p.~2]{PakPanova}). In a different direction, the following interesting question was recently brought to our attention by Mich\`ele Vergne. \begin{ques} Are the plethysm coefficients Ehrhart polynomials after rescaling? Specifically, are $\tilde f(s):=f(ks)$ or $\tilde g(s):=g(ks)$ Ehrhart polynomials for some $k$? \end{ques} One can prove that the answer is positive when $f$ or $g$ are of degree one, thus to find a counterexample one needs to consider partitions with more than one row. \subsection*{Acknowledgments} The project was started at the Simons Institute at UC Berkeley during the ``Algorithms and Complexity in Algebraic Geometry'' semester. Micha{\l}ek was realizing the project as a part of the PRIME DAAD program. Micha{\l}ek would like to thank Klaus Altmann and Bernd Sturmfels for the great working environment. Kahle was supported by the research focus dynamical systems of the state Saxony Anhalt. % \bibliographystyle{amsalpha} % \bibliography{plethysm} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{CDW12} \bibitem[BOR09]{briand2009reduced} Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, \emph{Reduced {K}ronecker coefficients and counter--examples to {M}ulmuley's strong saturation conjecture {SH}}, Computational Complexity \textbf{18} (2009), no.~4, 577--600. \bibitem[BV15]{baldoni2015multiplicity} Velleda Baldoni and Mich\`ele Vergne, \emph{Multiplicity of compact group representations and applications to {K}ronecker coefficients}, preprint, \arxiv{1506.02472} (2015). \bibitem[BZ92]{BZ} Arkady~D Berenstein and Andrei~V Zelevinsky, \emph{Triple multiplicities for {$sl(r+1)$} and the spectrum of the exterior algebra of the adjoint representation}, Journal of Algebraic Combinatorics \textbf{1} (1992), no.~1, 7--22. \bibitem[CDW12]{christandl2012computing} Matthias Christandl, Brent Doran, and Michael Walter, \emph{Computing multiplicities of {L}ie group representations}, IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), IEEE, 2012, pp.~639--648. \bibitem[CGR84]{CGR83} Y.~M. 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