Horn inequalities for nonzero Kronecker coefficients

TL;DR

This paper extends Horn inequalities, originally for Littlewood-Richardson coefficients, to the setting of nonzero Kronecker coefficients, providing new linear inequalities for these tensor product multiplicities.

Contribution

It introduces a set of Horn inequalities applicable to triples of partitions with nonzero Kronecker coefficients, expanding the classical theory.

Findings

Extended Horn inequalities to Kronecker coefficients.

Established linear inequalities for nonzero Kronecker coefficients.

Bridged the gap between Littlewood-Richardson and Kronecker coefficient theories.

Abstract

The Kronecker coefficients and the Littlewood-Richardson coefficients are nonnegative integers depending on three partitions. By definition, these coefficients are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of a Littlewood-Richardson coefficient implies linear inequalities on the triple of partitions, called Horn inequalities. In thispaper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.

Equations

$$[\alpha ]\otimes [\beta ]=\sum _{\gamma }{g}_{\alpha \beta \gamma }[\gamma ].$$

$$S^{\alpha }V\otimes S^{\beta }V=\sum _{\gamma }{c}_{\alpha \beta }^{\gamma }{S}^{\gamma }V.$$

$$(n-{\alpha }_1)+(n-{\beta }_1)\ge n-{\gamma }_1.$$

$$g_{\alpha \beta \gamma }=c_{\bar{\alpha }\bar{\beta }}^{\bar{\gamma }}.$$

$${\bar{\gamma }}_{e+j-1}\le {\bar{\beta }}_{j-1},$$

$$l(\alpha )\le e+1,l(\beta )\le f+1,\text{mbox}andl(\gamma )\le e+f+1.$$

$$n+{\gamma }_1+{\gamma }_{e+j}\le {\alpha }_1+{\beta }_1+{\beta }_j$$

$$g(\alpha ,\beta ,\gamma )=\sum _{l(x)\le 2e,l(y)\le 2}c(x,{\bar{\beta }}_J;{\bar{\gamma }}_K)\cdot c({\gamma }_1\ge {\gamma }_j,y;{\beta }_1\ge {\beta }_j)\cdot g(\bar{\alpha },x,y).$$

$${\tau }^I=(d-r+1-i_1\ge d-r+2-i_2\ge \cdots \ge d-i_r).$$

$$n+|{\bar{\alpha }}_I|-{\alpha }_1+|{\bar{\beta }}_J|-{\beta }_1\ge |{\bar{\gamma }}_K|-{\gamma }_1,$$

$$c_{{\tau }^I{\tau }^J}^{{\tau }^K}=1.$$

$$|{\bar{\alpha }}_I|+|{\bar{\beta }}_J|\ge |{\bar{\gamma }}_K|,$$

$$n+|{\bar{\alpha }}_I|-{\alpha }_1+|{\bar{\beta }}_J|-{\beta }_1=|{\bar{\gamma }}_K|-{\gamma }_1.$$

$$\begin{array}[]{ll}\sum_{a,b,x,y,u,v}&c(\bar{\alpha}_{I_{-}},\bar{\beta}_{J_{-}};y)\cdot c(x,y;\bar{\gamma}_{K_{+}})\cdot c(u,v;\bar{\gamma}_{K_{-}})\cdot\\ &c(a,u;\bar{\alpha}_{I})\cdot c(b,v;\bar{\beta}_{J})\cdot g(a,b,x),\end{array}$$

$$\displaystyle\begin{array}[]{lll}l(x)\leq(e-r)(f-s),&l(a)\leq e-r,&l(u)\leq e-r,\\[-3.0pt] l(y)\leq r+s,&l(b)\leq f-s,&l(v)\leq f-s.\end{array}$$

$$2n+2{\gamma }_1+{\gamma }_j\ge 2{\alpha }_1+2{\beta }_1+{\beta }_j.$$

$$c_{\nu \hat{\nu }}=\dim (V_{\nu }^{\ast }\otimes V_{\hat{\nu }}{)}^G.$$

$$V_{\hat{\nu }}=\sum _{\nu \in X(T{)}^{+}}{c}_{\nu \hat{\nu }}{V}_{\nu },$$

$$\mathrm{LR}(G,\hat{G})=\{(\nu ,\hat{\nu })\in X(T{)}^{+}\times X(\hat{T}{)}^{+}:c_{\nu \hat{\nu }}\ne 0\}.$$

$$\mathrm{LR}(G,X)=\{\mathcal{L}\in {\mathrm{Pic}}^G(X):H^0(X,\mathcal{L}{)}^G\ne \{0\}\}.$$

$$\displaystyle S^{\gamma}(E\otimes F)=\sum_{\begin{array}[]{c}\mbox{partitions }\alpha,\,\beta\mbox{ s.t.}\\ l(\alpha)\leq e,\,l(\beta)\leq f\end{array}}g_{\alpha\,\beta\,\gamma}\;S^{\alpha}E\otimes S^{\beta}F.$$

$$\mathrm{Kron}(e,f,ef)=\mathrm{LR}(\mathrm{GL}(E)\times \mathrm{GL}(F),\mathrm{GL}(E\otimes F))\cap ({\mathbb{Z}}^e\times {\mathbb{Z}}^f\times ({\mathbb{Z}}_{\ge 0}{)}^{ef}).$$

$$\mathrm{LR}(e,e,e)=\mathrm{LR}({\mathrm{GL}}_e,{\mathrm{GL}}_e\times {\mathrm{GL}}_e)\cap (({\mathbb{Z}}_{\ge 0}{)}^e{)}^3.$$

$$\displaystyle S^{\gamma}(E\oplus F)=\sum_{\begin{array}[]{c}\mbox{partitions }\alpha,\,\beta\mbox{ s.t.}\\ l(\alpha)\leq e,\,l(\beta)\leq f\end{array}}c_{\alpha\beta}^{\gamma}S^{\alpha}E\otimes S^{\beta}F.$$

$$\mathrm{LR}(e,f,e+f)=\mathrm{LR}({\mathrm{GL}}_e\times {\mathrm{GL}}_f,{\mathrm{GL}}_{e+f})\cap ({\mathbb{Z}}^e\times {\mathbb{Z}}^f\times ({\mathbb{Z}}_{\ge 0}{)}^{e+f}).$$

$$X=\mathcal{F}l(E)\times \mathcal{F}l(F)\times \mathcal{F}l(1,\cdots ,e+f+1;E\otimes F)$$

$$H^0(X,\mathcal{L})\simeq S^{\alpha }E\otimes S^{\beta }F\otimes S^{\gamma }(E\otimes F{)}^{\ast },$$

$$g_{\alpha \beta \gamma }=\dim (H^0(X,\mathcal{L}{)}^G).$$

$$\mathrm{Kron}(e+1,f+1,e+f+1)=({\mathbb{Z}}_{\ge 0}{)}^{2e+2f+3}\cap \mathrm{LR}(G,X).$$

$$X^{\mathrm{ss}}(\mathcal{L})=\{x\in X:\exists k>0\text{mbox}and\sigma \in H^0(X,{\mathcal{L}}^{\otimes k}{)}^G\sigma (x)\ne 0\}.$$

Full text

The nonvanishing of the Littlewood-Richardson coefficient $c_{\alpha\beta}^{\gamma}$ implies that $(\alpha,\beta,\gamma)$ satisfies some linear inequalities called Horn inequalities. In this paper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.

Along the way, we describe the set of tripless $(\alpha,\beta,\gamma)$ of partitions such that $c_{\alpha\beta}^{\gamma}\neq 0$ and $l(\alpha)\leq e$, $l(\beta)\leq f$ and $l(\gamma)\leq e+f$, for some given positive integers $e$ and $f$. This set is the natural analogue of the classical Horn semigroup when one thinks about $c_{\alpha\beta}^{\gamma}$ as the branching multiplicities for the subgroup $\operatorname{GL}_{e}\times\operatorname{GL}_{f}$ of $\operatorname{GL}_{e+f}$.

1 Introduction

If $\alpha=(\alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{e}\geq 0)$ is a partition, we set $|\alpha|=\sum_{i}\alpha_{i}$ in such a way $\alpha$ is a partition of $|\alpha|$. Consider the symmetric group $S_{n}$ on $n$ letters. The irreducible representations of $S_{n}$ are parametrized by the partitions of $n$, see e.g. [Mac95, I. 7] . Let $[\alpha]$ denote the representation of $S_{|\alpha|}$ corresponding to $\alpha$. The Kronecker coefficients $g_{\alpha\,\beta\,\gamma}$, depending on three partitions $\alpha,\,\beta$, and $\gamma$ of the same integer $n$, are defined by

[TABLE]

The length $l(\alpha)$ of the partition $\alpha$ is the number of nonzero parts $\alpha_{i}$. Let $V$ be a complex vector space of dimension $d$. If $l(\alpha)\leq d$ then $S^{\alpha}V$ denotes the Schur power (see e.g. [FH91]): it is an irreducible polynomial representation of the linear group $\operatorname{GL}(V)$. Let $\beta$ be a second partition such that $l(\beta)\leq d$. Then the Littlewood-Richardson coefficients $c_{\alpha\,\beta}^{\gamma}$ are defined by

[TABLE]

The partition obtained by suppressing the first part of $\alpha$ is denoted by $\bar{\alpha}=(\alpha_{2}\geq\alpha_{3}\dots)$. Observe that $\bar{\alpha}_{1}=\alpha_{2}$. We state a classical result due to Littlewood and Murnaghan (see for example [JK81]).

Proposition 1

Let $\alpha$, $\beta$ and $\gamma$ be three partitons of the same integer $n$.

1. (i)

If $g_{\alpha\,\beta\,\gamma}\neq 0$ then

[TABLE] 2. (ii)

If $(n-\alpha_{1})+(n-\beta_{1})=n-\gamma_{1}$ then

[TABLE]

In this paper, we prove many other inequalities similar to the identity (3), that are consequences of the nonvanishing of $g_{\alpha\,\beta\,\gamma}$. For the partitions $(\alpha,\beta,\gamma)$ satisfying equality in such an inequality, we prove a reduction rule for $g_{\alpha\,\beta\,\gamma}$ similar to the identity (4).

Observe that the formula (4) shows that the Kronecker coefficients extend the Littlewood-Richardson ones. Indeed, given $\bar{\alpha}$, $\bar{\beta}$ and $\bar{\gamma}$, one can find $\alpha=(\alpha_{1},\bar{\alpha}),\beta=(\beta_{1},\bar{\beta})$ and $\gamma=(\gamma_{1},\bar{\gamma})$ such that $|\alpha|=|\beta|=|\gamma|=:n$, $(n-\alpha_{1})+(n-\beta_{1})=n-\gamma_{1}$. Then $c_{\bar{\alpha}\,\bar{\beta}}^{\bar{\gamma}}=g_{\alpha\,\beta\,\gamma}$ is a Kronecker coefficient. If $c_{\bar{\alpha}\,\bar{\beta}}^{\bar{\gamma}}\neq 0$ then $(\bar{\alpha},\bar{\beta},\bar{\gamma})$ satisfy the Horn inequalities (see e.g. [Ful00] or below for details). If $g_{\alpha\,\beta\,\gamma}\neq 0$, our inequalities for $(\alpha,\beta,\,\gamma)$ extend some Horn inequalities. Fix such an inequality $\varphi(\bar{\alpha},\bar{\beta},\bar{\gamma})\geq 0$. We want to find an inequality $\tilde{\varphi}(\alpha,\beta,\gamma)\geq 0$ such that

1. (i)

If $g_{\alpha\,\beta\,\gamma}\neq 0$ then $\tilde{\varphi}(\alpha,\beta,\gamma)\geq 0$; 2. (ii)

If $(n-\alpha_{1})+(n-\beta_{1})=n-\gamma_{1}$ then $\tilde{\varphi}(\alpha,\beta,\gamma)=\varphi(\bar{\alpha},\bar{\beta},\bar{\gamma})$.

For example, a Weyl’s theorem [Wey12] asserts that if $c_{\bar{\alpha}\,\bar{\beta}}^{\bar{\gamma}}\neq 0$ then

[TABLE]

whenever $l(\bar{\alpha})\leq e$ and $j\geq 2$.

Before stating our extension of Weyl’s theorem, we introduce some notation. Let ${\mathcal{S}}(r,d)$ denote the set of subsets of $\{1,\cdots,d\}$ with $r$ elements. Given $I=\{i_{1}<\cdots<i_{r}\}\in{\mathcal{S}}(r,d)$ and $\alpha=(\alpha_{1}\geq\cdots\geq\alpha_{d})$ a partition of length at most $d$, we set $\alpha_{I}=(\alpha_{i_{1}}\geq\cdots\geq\alpha_{i_{r}})$. Observe that $\bar{\alpha}_{I}=(\alpha_{i_{1}+1}\geq\cdots\geq\alpha_{i_{r}+1})$.

Theorem 1

Let $e$ and $f$ be two positive integers. Let $\alpha$, $\beta$, and $\gamma$ be three partitions of the same integer $n$ such that

[TABLE]

Let $j\in\{2,\dots,f+1\}$.

1. (i)

If $g_{\alpha\,\beta\,\gamma}\neq 0$ then

[TABLE] 2. (ii)

Set $J=\{1,\dots,f\}-\{j-1\}$ and $K=\{1,\dots,e+f\}-\{e+j-1\}$. If $n+\gamma_{1}+\gamma_{e+j}=\alpha_{1}+\beta_{1}+\beta_{j}$ then

[TABLE]

Remark. In the statement of Theorem 1 (and sometimes below) we denote $c_{\alpha\,\beta}^{\gamma}$ and $g_{\alpha\,\beta\,\gamma}$ respectively by $c(\alpha,\beta;\gamma)$ and $g(\alpha,\beta,\gamma)$.

Theorem 1 extends Weyl’s theorem in the sense that if $(n-\alpha_{1})+(n-\beta_{1})=n-\gamma_{1}$ then the inequality (7) is equivalent to $\gamma_{e+j}\leq\beta_{j}$, that is to the inequality (5).

For $I\in{\mathcal{S}}(r,d)$, consider the partition

[TABLE]

Set $|\alpha_{I}|:=\sum_{i\in I}\alpha_{i}$. Observe that $|\bar{\alpha}_{I}|:=\sum_{i\in I}\alpha_{i+1}$. We can now state our main result.

Theorem 2

Let $\alpha$, $\beta$, and $\gamma$ be three partitions of the same integer $n$ satisfying the conditions (6).

Assume that $g_{\alpha\,\beta\,\gamma}\neq 0$. Then

[TABLE]

for any $0<r<e$, $0<s<f$, $I\in{\mathcal{S}}(r,e)$, $J\in{\mathcal{S}}(s,f)$ and $K\in{\mathcal{S}}(r+s,e+f)$ such that

[TABLE]

If $(n-\alpha_{1})+(n-\beta_{1})=n-\gamma_{1}$ then the inequality (8) is equivalent to

[TABLE]

which is a Horn inequality (see [Ful00] or Section 4).

Remark. Since inequalities (3), (7) and (8) are linear in $(\alpha,\beta,\gamma)$, the condition $g_{\alpha\,\beta\,\gamma}\neq 0$ in Proposition 1 and Theorem 1 and 2 can be replaced by the weaker condition $g_{k\alpha\,k\beta\,k\gamma}\neq 0$ for some positive $k$.

We get a reduction formula for the coefficients $g_{\alpha\,\beta\,\gamma}$ if the inequality (8) is saturated. If $I\in{\mathcal{S}}(r,d)$, we denote by $I_{-}\in{\mathcal{S}}(d-r,d)$ the complement of $I$ in $\{1,\dots,d\}$. By symmetry we also set $I_{+}=I$.

Theorem 3

Let $\alpha$, $\beta$, and $\gamma$ be three partitions of the same integer $n$ satisfying the conditions (6).

Let $(I,J,K)$ be a triple that appears in Theorem 2 (in particular satisfying the condition (9)). We assume that

[TABLE]

Then $g(\alpha,\beta,\gamma)$ is equal to

[TABLE]

where the sum runs over the partitions $a,b,x,y,u,v$ satisfying

[TABLE]

Note that in Theorem 3, we needn’t assume that $g_{\alpha\beta\gamma}\neq 0$.

Let $\operatorname{Kron}(e+1,f+1,e+f+1)$ denote the set of triples $(\alpha,\,\beta,\,\gamma)$ of partitions such that $|\alpha|=|\beta|=|\gamma|$, $g_{\alpha\beta\gamma}\neq 0$ and $l(\alpha)\leq e+1,\,l(\beta)\leq f+1,\,l(\gamma)\leq e+f+1$. Then $\operatorname{Kron}(e+1,f+1,e+f+1)$ is a finitely generated semigroup in ${{\mathbb{Z}}_{\geq 0}}^{2e+2f+3}$. In particular, the cone ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ generated by $\operatorname{Kron}(e+1,f+1,e+f+1)$ is a closed convex polyhedral cone.

Theorem 4

The inequalities (7) in Theorem 1 and the inequalities (8) in Theorem 2 are essential, that is correspond to codimension one faces of ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$.

One can guess to describe the complete minimal list ${\mathcal{L}}$ of inequalities characterizing ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$. Such a list is known for the Littlewood-Richardson coefficients (see Theorem 7 below for details). In principle, [Res10] gives ${\mathcal{L}}$. Nevertheless, it is known to be untractable to make this description very explicit. Indeed, one first need to describe the so-called adapted one-parameter subgroups by describing the collection of hyperplanes spanned by subsets of a given set: a tricky combinatorial problem. And secondly one need to understand an unknown Schubert problem. In this paper we describe a natural subset of ${\mathcal{L}}$ related with the Horn cone.

Inequality (3) defines a codimension one face ${\mathcal{F}}_{LM}$ of ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$. Here “$LM$” stands for Littlewood-Murnaghan. Each Horn inequality (10) or Weyl inequality (5) define a face ${\mathcal{F}}$ of codimension two contained in ${\mathcal{F}}_{LM}$. By convex geometry ${\mathcal{F}}$ has to be contained in a second codimension one face ${\mathcal{F}}^{\prime}$ of ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$. Basically, Theorem 1 and 2 describe this face ${\mathcal{F}}^{\prime}$.

Comparaison between Theorems 1 and 2. With $I,J$ and $K$ respectively equal to $\{1,\dots,e\}$, $\{1,\dots,f\}-\{j-1\}$, and $\{1,\dots,e+f\}-\{e+j-1\}$ (where $j\in\{2,\dots,f+1\}$), we have $c_{\tau^{I}\,\tau^{J}}^{\tau^{K}}=1$. The inequality (8) gives

[TABLE]

This inequality is satisfied if $g_{\alpha\,\beta\,\gamma}\neq 0$. But the corresponding face has codimension 2 in ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$. Hence the inequality (16) is not essential. More precisely, it is a consequence of inequalities (3) and (7).

In Section 2, we define and compare several semigroups. In Section 3, we recall some results from [Res10] that allows to describe some cones generated by these semigroups. In Section 4, we describe the support of the LR-coefficients $c_{\alpha\beta}^{\gamma}$ for partitions satisfying $l(\alpha)\leq e$, $l(\beta)\leq f$ and $l(\gamma)\leq e+f$, for fixed positive integers $e$ and $f$. Note that these assumptions are natural if one thinks about the LR-coefficients as multiplicities for the branching from $\operatorname{GL}_{e}\times\operatorname{GL}_{f}$ to $\operatorname{GL}_{e+f}$. It is a variation of the classical Horn problem. The next sections contain the proofs of the statements of the introduction.

Acknowledgements. The author is partially supported by the French National Agency (Project GeoLie ANR-15-CE40-0012) and the Institut Universitaire de France (IUF).

2 Semigroups

2.1 Definitions

2.1.1 Kronecker semigroups

We extend the definition of $g_{\alpha\beta\gamma}$ to any triple $(\alpha,\beta,\gamma)$ of partitions by setting $g_{\alpha\beta\gamma}=0$ if the condition $|\alpha|=|\beta|=|\gamma|$ does not hold. Let $e$, $f$, and $g$ be three positive integers. We define $\operatorname{Kron}(e,f,g)$ to be the set of triples $(\alpha,\,\beta,\,\gamma)$ of partitions such that $g_{\alpha\beta\gamma}\neq 0$ and $l(\alpha)\leq e,\,l(\beta)\leq f,\,l(\gamma)\leq g$. It is well known that $\operatorname{Kron}(e,f,g)$ is a finitely generated semigroup of ${{\mathbb{Z}}_{\geq 0}}^{e+f+g}$.

2.1.2 Littlewood-Richardson semigroups

We define $\operatorname{LR}(e,f,g)$ to be the set of triples $(\alpha,\,\beta,\,\gamma)$ of partitions such that $c_{\alpha\beta}^{\gamma}\neq 0$ and $l(\alpha)\leq e,\,l(\beta)\leq f,\,l(\gamma)\leq g$. It is well known that $\operatorname{LR}(e,f,g)$ is a finitely generated semigroup of ${{\mathbb{Z}}_{\geq 0}}^{e+f+g}$.

2.1.3 Branching semigroups

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. Fix maximal tori $T\subset{\hat{T}}$ and Borel subgroups $B\supset T$ and ${\hat{B}}\supset{\hat{T}}$ of $G$ and ${\hat{G}}$. Let $X(T)$ denote the group of characters of $T$ and let $X(T)^{+}$ denote the set of dominant characters. The irreducible representation of highest weight $\nu\in X(T)^{+}$ is denoted by $V_{\nu}$. Similarly, we use the notation $X(\hat{T})$, $X(\hat{T})^{+}$, $V_{\hat{\nu}}$ relatively to $\hat{G}$. The subspace of $G$-fixed vectors of the $G$-module $V$ is denoted by $V^{G}$. Set

[TABLE]

The branching problem is equivalent to the knowledge of these coefficients since

[TABLE]

as a $G$-module. Consider the set

[TABLE]

By a result of Brion and Knop (see [É92]), $\operatorname{LR}(G,\hat{G})$ is a finitely generated semigroup.

2.1.4 GIT semigroups

Let $G$ be a complex reductive group acting on an irreducible projective variety $X$. Let $\operatorname{Pic}^{G}(X)$ denote the group of $G$-linearized line bundles on $X$. The space $\operatorname{H}^{0}(X,{\mathcal{L}})$ of regular sections of ${\mathcal{L}}$ is a $G$-module. Consider the set

[TABLE]

Since $X$ is irreducible, the product of two nonzero $G$-invariant sections is a nonzero $G$-invariant section and $\operatorname{LR}(G,X)$ is a semigroup.

2.2 Relations between these semigroups

2.2.1 Kronecker semigroups as branching semigroups.

Let $E$ and $F$ be two complex vector spaces of dimension $e$ and $f$. Consider the group $G=\operatorname{GL}(E)\times\operatorname{GL}(F)$. Using Schur-Weyl duality, the Kronecker coefficient $g_{\alpha\,\beta\,\gamma}$ can be interpreted in terms of representations of $G$. Namely (see for example [Mac95, FH91]) $g_{\alpha\,\beta\,\gamma}$ is the multiplicity of $S^{\alpha}E\otimes S^{\beta}F$ in $S^{\gamma}(E\otimes F)$. More precisely, let $\gamma$ be a partition such that $l(\gamma)\leq ef$. Then the simple $\operatorname{GL}(E\otimes F)$-module $S^{\gamma}(E\otimes F)$ decomposes as a sum of simple $G$-modules as follows

[TABLE]

As a consequence

[TABLE]

2.2.2 Littlewood-Richardson semigroups as branching

semigroups

Since the Littlewood-Richardson coefficients are multiplicities for the tensor product decomposition of $\operatorname{GL}_{n}$, we have

[TABLE]

The Littlewood-Richardson coefficients have another interpretation in terms of representations of linear groups. Consider the embedding of $\operatorname{GL}(E)\times\operatorname{GL}(F)$ in $\operatorname{GL}(E\oplus F)$ as a Levi subgroup by its natural action on $E\oplus F$. Then (see[Mac95, Chapter I, 5.9])

[TABLE]

In particular

[TABLE]

2.2.3 Branching semigroups as GIT

semigroups

We use notation of Section 2.1.3 and we assume that $G$ and ${\hat{G}}$ are semisimple simply connected. Consider the diagonal action of $G$ on $X=G/B\times{\hat{G}}/{\hat{B}}$. Note that $\operatorname{Pic}^{G}(X)$ identifies with $X(T)\times X({\hat{T}})$. Then Borel-Weyl’s theorem implies that $\operatorname{LR}(G,{\hat{G}})=\operatorname{LR}(G,X)$.

2.2.4 Kronecker semigroups as GIT semigroups

If $V$ is a complex finite dimensional vector space, let ${\mathcal{F}l}(V)$ denote the variety of complete flags of $V$. Given integers $a_{i}$ such that $1\leq a_{1}<\cdots<a_{s}\leq\dim(V)-1$, we denote by ${\mathcal{F}l}(a_{1},\cdots,a_{s};\,V)$ the variety of flags $V_{1}\subset\cdots\subset V_{s}\subset V$ such that $\dim(V_{i})=a_{i}$ for any $i$. If $\alpha$ is a partition with at most dim$(V)$ parts then ${\mathcal{L}}_{\alpha}$ (resp. ${\mathcal{L}}^{\alpha}$) denotes the $\operatorname{GL}(V)$-linearized line bundle on ${\mathcal{F}l}(V)$ such that the space $\operatorname{H}^{0}({\mathcal{F}l}(V),{\mathcal{L}}_{\alpha})$ (resp. $\operatorname{H}^{0}({\mathcal{F}l}(V),{\mathcal{L}}^{\alpha})$) is isomorphic to $S^{\alpha}V^{*}$ (resp. $S^{\alpha}V$) as a $\operatorname{GL}(V)$-module.

Assume that $E$ and $F$ are two linear spaces of dimension $e+1$ and $f+1$. Set $G=\operatorname{GL}(E)\times\operatorname{GL}(F)$. Consider the variety

[TABLE]

endowed with its natural $G$-action. Let $\alpha$, $\beta$, and $\gamma$ be three partitions such that $l(\alpha)\leq e+1$, $l(\beta)\leq f+1$, and $l(\gamma)\leq e+f+1$. Consider the $\operatorname{GL}(E)$-linearized line bundle ${\mathcal{L}}^{\alpha}$ on ${\mathcal{F}l}(E)$, and respectively ${\mathcal{L}}^{\beta}$ on ${\mathcal{F}l}(F)$. Since $l(\gamma)\leq e+f+1$, the line bundle ${\mathcal{L}}_{\gamma}$ on ${\mathcal{F}l}(E\otimes F)$ is the pullback of a line bundle (still denoted by ${\mathcal{L}}_{\gamma}$) on ${\mathcal{F}l}(1,\cdots,e+f+1;E\otimes F)$. Consider the line bundle ${\mathcal{L}}={\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma}$ on $X$ endowed with its natural $G$-action. Then

[TABLE]

and, by the formula (22),

[TABLE]

The map $(\alpha,\beta,\gamma)\mapsto{\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma}$ extends to a linear isomorphism from ${\mathbb{Z}}^{2e+2f+3}$ onto $\operatorname{Pic}^{G}(X)$. This isomorphism allows to identify $\operatorname{LR}(G,X)$ with a subset of ${\mathbb{Z}}^{2e+2f+3}$. The equality (29) implies that

[TABLE]

3 Descriptions of branching and GIT cones

3.1 GIT cones

Assume that the connected reductive group $G$ acts on the smooth projective variety $X$ and that $\operatorname{Pic}^{G}(X)$ has finite rank. Consider the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G,X)$ generated in $\operatorname{Pic}^{G}(X)\otimes{\mathbb{Q}}$ by the points of $\operatorname{LR}(G,X)$. The $G$-linearized ample line bundles on $X$ generated an open convex cone $\operatorname{Pic}^{G}(X)^{+}_{\mathbb{Q}}$ in $\operatorname{Pic}^{G}(X)\otimes{\mathbb{Q}}$. In this section, we recall from [Res10] a description of the faces of ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G,X)$ that intersect $\operatorname{Pic}^{G}(X)^{+}_{\mathbb{Q}}$.

Let ${\mathcal{L}}$ be a $G$-linearized line bundle on $X$. Consider the associated set of semistable points

[TABLE]

Assume that $X^{\rm ss}({\mathcal{L}})$ is nonempty. Then the projective variety ${\rm Proj}(\bigoplus_{k\geq 0}\operatorname{H}^{0}(X,{\mathcal{L}}^{\otimes k})^{G})$ is denoted by $X^{\rm ss}({\mathcal{L}})//G$. For later use, observe that $\dim(\operatorname{H}^{0}(X,{\mathcal{L}}^{\otimes k})^{G})$ is $O(k^{\dim(X^{\rm ss}({\mathcal{L}})//G)})$. If moreover ${\mathcal{L}}$ is ample, $X^{\rm ss}({\mathcal{L}})//G$ is the categorical quotient of $X^{\rm ss}({\mathcal{L}})$ by $G$. In general, there is a canonical $G$-invariant regular map

[TABLE]

Let $\lambda$ be a one parameter subgroup of $G$. The set

[TABLE]

is a parabolic subgroup of $G$. Consider an irreducible component $C$ of the fixed point set $X^{\lambda}$ of $\lambda$ in $X$. Set

[TABLE]

By Białynicki-Birula’s theorem, $C^{+}$ is an irreducible smooth locally closed subvariety of $X$. Moreover it is stable by the action of $P(\lambda)$. Consider on $G\times C^{+}$ the following action of the group $P(\lambda)$:

[TABLE]

There exists a quotient variety denoted by $G\times_{P(\lambda)}C^{+}$. We denote by $[g:x]$, the class of $(g,x)\in G\times C^{+}$. The following formula

[TABLE]

endows $G\times_{P(\lambda)}C^{+}$ with a $G$-action. Consider the $G$-equivariant morphism

[TABLE]

The pair $(C,\lambda)$ is said to be well covering if there exists a $P(\lambda)$-stable open subset $\Omega$ of $C^{+}$ such that

1. (i)

the restriction of $\eta$ to $G\times_{P(\lambda)}\Omega$ is an open immersion; 2. (ii)

$\Omega$ intersects $C$.

For any ${\mathcal{L}}\in\operatorname{Pic}^{G}(X)$, there exists an integer $\mu^{\mathcal{L}}(C,\lambda)$ such that

[TABLE]

for any $t\in{\mathbb{C}}^{*}$, $z\in C$ and $\tilde{z}$ in the fiber ${\mathcal{L}}_{z}$ over $z$ in ${\mathcal{L}}$.

Theorem 5

(see [Res10])

1. (i)

For any well covering pair $(C,\lambda)$ and any ${\mathcal{L}}\in\operatorname{LR}(G,X)$, we have $\mu^{\mathcal{L}}(C,\lambda)\leq 0$. 2. (ii)

For any face ${\mathcal{F}}$ of ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G,X)$ intersecting $\operatorname{Pic}^{G}(X)^{+}_{\mathbb{Q}}$ there exists a well covering pair $(C,\lambda)$ such that $({\mathcal{L}}\otimes 1)\in{\mathcal{F}}$ if and only if $\mu^{\mathcal{L}}(C,\lambda)=0$, for any ample ${\mathcal{L}}$ in ${{\mathbb{Q}}_{\geq 0}\operatorname{LR}}(G,X)$. 3. (iii)

Let $(C,\lambda)$ be a well covering pair and ${\mathcal{L}}$ be ample in $\operatorname{LR}(G,X)$. Then $\mu^{\mathcal{L}}(C,\lambda)=0$ if and only if $X^{\rm ss}({\mathcal{L}})\cap C$ is not empty.

3.2 Branching cones

With notation of Section 2.1.3, we want to describe the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G,{\hat{G}})$ generated by $\operatorname{LR}(G,{\hat{G}})$. We assume that no nonzero ideal of the Lie algebra ${\operatorname{Lie}}(G)$ of $G$ is an ideal of that ${\operatorname{Lie}}({\hat{G}})$ of ${\hat{G}}$: this assumption implies that the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G,{\hat{G}})$ has nonempty interior in $(X(T)\times X(\hat{T}))\otimes{\mathbb{Q}}$.

Consider the natural pairing $\langle\cdot\,,\,\cdot\rangle$ between the one parameter subgroups and the characters of tori $T$ or ${\hat{T}}$. Let $W$ (resp. ${\hat{W}}$) denote the Weyl group of $T$ (resp. ${\hat{T}}$). If $\lambda$ is a one parameter subgroup of $T$ (and thus of ${\hat{T}}$), we denote by $W_{\lambda}$ (resp. ${\hat{W}}_{\lambda}$) the stabilizer of $\lambda$ for the natural action of the Weyl group.

The cohomology group ${\rm H}^{*}(G/P(\lambda),{\mathbb{Z}})$ is freely generated by the Schubert classes $\sigma_{w}$ parameterized by the elements $w\in W/W_{\lambda}$. Assume that $\lambda$ is dominant. Let $w_{0}$ be the longest element of $W$. If $w\in W/W_{\lambda}$, we denote by $w^{\vee}\in W/W_{\lambda}$ the class of $w_{0}w$. By this way $\sigma_{w^{\vee}}$ and $\sigma_{w}$ are Poincaré dual. We consider ${\hat{G}}/{\hat{P}}(\lambda)$, $\sigma_{\hat{w}}$ as above but with ${\hat{G}}$ in place of $G$. Consider also the canonical $G$-equivariant immersion $\iota\,:\,G/P(\lambda)\longrightarrow{\hat{G}}/{\hat{P}}(\lambda)$; and the corresponding morphism $\iota^{*}$ in cohomology.

Recall from [RR11], the definition of Levi-movability for the pair $(\sigma_{w},\sigma_{\hat{w}})$. For the purpose of this paper it is only useful to known that if $(\sigma_{w},\sigma_{\hat{w}})$ is Levi-movable then $\iota^{*}(\sigma_{\hat{w}}).\sigma_{w}$ is a nonzero multiple of the class $[pt]$ of the point. Moreover the converse is true if ${\hat{G}}/{\hat{P}}(\lambda)$ is minuscule.

Consider the set $\operatorname{Wt}_{T}({{\operatorname{Lie}}({\hat{G}})}/{{\operatorname{Lie}}(G)})$ of nontrivial weights of $T$ in ${{\operatorname{Lie}}({\hat{G}})}/{{\operatorname{Lie}}(G)}$ and the set of hyperplanes $H$ of $X(T)\otimes{\mathbb{Q}}$ spanned by some elements of $\operatorname{Wt}_{T}({{\operatorname{Lie}}({\hat{G}})}/{{\operatorname{Lie}}(G)})$. For each such hyperplane $H$ there exist exactly two opposite indivisible one parameter subgroups $\pm\lambda_{H}$ which are orthogonal (for the paring $\langle\cdot,\cdot\rangle$) to $H$. The so obtained one parameter subgroups are called admissible and form a $W$-stable set.

Theorem 6

(see [Res10])

Recall that no nonzero ideal of ${{\operatorname{Lie}}(G)}$ is an ideal of ${{\operatorname{Lie}}({\hat{G}})}$. Then, the cone ${{\mathbb{Q}}_{\geq 0}\operatorname{LR}}(G,{\hat{G}})$ has nonempty interior in $X(T\times{\hat{T}})\otimes{\mathbb{Q}}$.

A dominant weight $(\nu,{\hat{\nu}})$ belongs to ${{\mathbb{Q}}_{\geq 0}\operatorname{LR}}(G,{\hat{G}})$ if and only if

[TABLE]

for any dominant admissible one parameter subgroup $\lambda$ of $T$ and for any pair $(w,{\hat{w}})\in W/W_{\lambda}\times{\hat{W}}/{\hat{W}}_{\lambda}$ such that

1. (i)

$\iota^{*}(\sigma_{\hat{w}})\cdot\sigma_{w^{\vee}}=[pt]\in{\rm H}^{*}(G/P(\lambda),{\mathbb{Z}})$, and 2. (ii)

the pair $(\sigma_{w^{\vee}},\sigma_{\hat{w}})$ is Levi-movable.

Moreover, the inequalities (31) are pairwise distinct and no one can be omitted.

4 Description of $\operatorname{LR}(e,f,e+f)$

4.1 The statement

Theorem 7

Let $\alpha,\,\beta,$ and $\gamma$ be three partitions such that $l(\alpha)\leq e$, $l(\beta)\leq f$ and $l(\gamma)\leq e+f$.

Then $c_{\alpha\,\beta}^{\gamma}\neq 0$ if and only if

[TABLE]

and

[TABLE]

for any $i\in\{1,\dots,e\}$ and $j\in\{1,\dots,f\}$, and

[TABLE]

for any $0<r<e$ and $0<s<f$, for any $I\in{\mathcal{S}}(r,e)$, $J\in{\mathcal{S}}(s,f)$ and $K\in{\mathcal{S}}(r+s,e+f)$ such that

[TABLE]

Moreover, the inequalities (33) or (34) are pairwise distinct and no one can be omitted.

The partitions $\alpha$ and $\beta$ in the statement of Theorem 7 are also partitions of length at most $e+f$. Hence the nonvanishing of $c_{\alpha\beta}^{\gamma}$ is equivalent to $(\alpha,\beta,\gamma)\in{{\mathbb{Q}}_{\geq 0}\operatorname{LR}}(e+f,e+f,e+f)$. But, by the classical Horn conjecture (see e.g. [Ful00]), this cone is characterized by the inequalities

[TABLE]

where $\sharp I^{\prime}=\sharp J^{\prime}=\sharp K^{\prime}$ and

[TABLE]

In some sense, Theorem 7 selects among the inequalities (36) those that remain essential when one imposes $l(\alpha)\leq e$ and $l(\beta)\leq f$.

Each inequality (34) has to be consequence of at least one Horn inequality (36). Indeed, by setting $\tilde{I}=I_{-}\cup\{e+s+1,\dots,e+f\}$ and $\tilde{J}=J_{-}\cup\{f+r+1,\dots,e+f\}$, one can check that, under the assumptions of Theorem 7 and modulo the equality (32), the inequality (34) is equivalent to

[TABLE]

But $\sharp\tilde{I}=\sharp\tilde{J}=\sharp K_{-}=e+f-r-s$. One can check that

[TABLE]

Hence the assumption (35) implies that the condition (38) is an Horn inequality (36) for the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e+f,e+f,e+f)$.

For the proof of Theorem 7, we need to recall some notations and results on Schubert calculus on Grassmannians.

4.2 Schubert Calculus

Let ${\mathbb{G}}(r,n)$ be the Grassmann variety of $r$-dimensional linear subspaces of $V={\mathbb{C}}^{n}$. Let $F_{\bullet}$: $\{0\}=F_{0}\subset F_{1}\subset F_{2}\subset\cdots\subset F_{n}=V$ be a complete flag of $V$. Let $I=\{i_{1}<\cdots<i_{r}\}\in{\mathcal{S}}(r,n)$. The Schubert variety $X_{I}(F_{\bullet})$ in ${\mathbb{G}}(r,n)$ is defined to be

[TABLE]

The Poincaré dual of the homology class of $X_{I}(F_{\bullet})$ is denoted by $\sigma_{I}$. It does not depend on $F_{\bullet}$. The classes $\sigma_{I}$ form a ${\mathbb{Z}}$-basis of the cohomology ring of ${\mathbb{G}}(r,n)$. Recall from the introduction the definition of the partition $\tau^{I}$. Then $\sigma_{I}$ has degree $2|\tau^{I}|$. A first cohomological interpretation of the Littlewood-Richardson coefficients is given by the formula (see e.g. [Man01])

[TABLE]

for any $I,\,J$ in ${\mathcal{S}}(r,n)$.

Let $r$ and $s$ be two integers such that $0<r<e$ and $0<s<f$. Fix an identification ${\mathbb{C}}^{e+f}={\mathbb{C}}^{e}\oplus{\mathbb{C}}^{f}$ and consider the morphism

[TABLE]

The associated comorphism in cohomology is

[TABLE]

By Kuneth’s formula, the family $(\sigma_{I}\otimes\sigma_{J})_{(I,J)\in{\mathcal{S}}(r,e)\times{\mathcal{S}}(s,f)}$ is a basis of $\operatorname{H}^{*}({\mathbb{G}}(r,e)\times{\mathbb{G}}(s,f),{\mathbb{Z}})$. A second cohomological interpretation of the Littlewood-Richardson coefficients is given by the formula

[TABLE]

for any $K\in{\mathcal{S}}(r+s,e+f)$.

4.3 Proof of Theorem 7

By the Knutson-Tao theorem of saturation (see [KT99]), $c_{\alpha\,\beta}^{\gamma}\neq 0$ if and only if $(\alpha,\,\beta,\,\gamma)$ belongs to the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$. It remains to prove that the inequalities (33) and (34) characterize the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$ in a minimal way.

Let us fix bases for the two vector spaces $E$ and $F$ of dimension $e$ and $f$. Consider the group $\hat{G}=\operatorname{SL}(E\oplus F)$, its subgroup $G=S(\operatorname{GL}(E)\times\operatorname{GL}(F))$ and on $E\oplus F$ the basis obtained by concatenating the bases of $E$ and $F$. Let $\hat{T}$ be the maximal torus of $\hat{G}$ consisting in diagonal matrices. It is contained in $G$; set $T={\hat{T}}$. Let $\hat{B}$ be the Borel subgroup of $\hat{G}$ consisting in upper triangular matrices. Set $B=\hat{B}\cap G$. Let $\varepsilon_{i}$ be the character of ${\hat{T}}$ mapping a matrix in ${\hat{T}}$ to its $i^{\rm th}$ diagonal entry. Since $\sum_{i}\varepsilon_{i}=0$, $(\varepsilon_{1},\dots,\varepsilon_{e+f-1})$ is a ${\mathbb{Z}}$-basis of $X({\hat{T}})$.

Let $\alpha,\,\beta$, and $\gamma$ be three partitions of length less or equal to $e$, $f$, and $e+f$. The highest weight of the ${\hat{G}}$-module $S^{\gamma}(E\oplus F)$ is $\tilde{\gamma}=(\gamma_{1}-\gamma_{e+f})\varepsilon_{1}+\cdots+(\gamma_{e+f-1}-\gamma_{e+f})\varepsilon_{e+f-1}$. The highest weight of the $G$-module $S^{\alpha}E\otimes S^{\beta}F$ is $\widetilde{(\alpha,\beta)}=(\alpha_{1}-\beta_{f})\varepsilon_{1}+\cdots+(\alpha_{e}-\beta_{f})\varepsilon_{e}+(\beta_{1}-\beta_{f})\varepsilon_{e+1}+\cdots+(\beta_{f-1}-\beta_{f})\varepsilon_{e+f-1}$. Then, by the formula (27)

[TABLE]

In particular, to determine the inequalities for the cone ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$, it is sufficient to describe ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G,{\hat{G}})$. We do this using Theorem 6. The set of weights of $T$ acting on ${\operatorname{Lie}}({\hat{G}})/{\operatorname{Lie}}(G)$ is the set of weights of $T$ acting on $F^{*}\otimes E$ and their opposite. Explicitly $\operatorname{Wt}_{T}({\operatorname{Lie}}({\hat{G}})/{\operatorname{Lie}}(G))=\pm\{\varepsilon_{i}-\varepsilon_{e+j}\,|\,1\leq i\leq e\quad{\rm and}\quad 1\leq j\leq f\}$. Let $(a_{1},\dots,a_{e},b_{1},\dots,b_{f})\in{\mathbb{Z}}^{e+f}$ be the exponents of a one parameter subgroup $\lambda$ of $T$; they satisfy $\sum_{i}a_{i}+\sum_{j}b_{j}=0$. Then $\langle\lambda,\varepsilon_{i}-\varepsilon_{e+j}\rangle=0$ if and only if $a_{i}=b_{j}$. It follows that if $\lambda$ is admissible then the integers $a_{i}$ and $b_{j}$ take at most two values. If moreover $\lambda$ is dominant then there exist integers $r,s$, and $c>d$ such that $a_{1}=\cdots=a_{r}=b_{1}=\cdots=b_{s}=c$ and $a_{r+1}=\cdots=a_{e}=b_{s+1}=\cdots=b_{f}=d$. If moreover $\lambda$ is indivisible, $c=\frac{e+f-r-s}{(r+s)\wedge(e+f)}$ and $d=\frac{-r-s}{(r+s)\wedge(e+f)}$, where $\wedge$ denotes the gcd. Let $\lambda_{r,s}$ denote the so obtained one-parameter subgroup of $T$. Conversely, one easily checks that $\lambda_{r,s}$ is an admissible dominant one-parameter subgroup of $T$, if $0<r<e$ and $0<s<f$ or if the pair $(r,s)$ is one of the four exceptional ones $\{(1,0),(0,1),(e-1,f),(e,f-1)\}$.

The inclusions $G/P(\lambda_{r,s})\subset{\hat{G}}/{\hat{P}}(\lambda_{r,s})$ associated to the four exceptional cases are ${\mathbb{P}}(E)\subset{\mathbb{P}}(E\oplus F)$, ${\mathbb{P}}(F)\subset{\mathbb{P}}(E\oplus F)$, ${\mathbb{P}}(E^{*})\subset{\mathbb{P}}(E^{*}\oplus F^{*})$ and ${\mathbb{P}}(F^{*})\subset{\mathbb{P}}(E^{*}\oplus F^{*})$. Consider ${\mathbb{P}}(E)\subset{\mathbb{P}}(E\oplus F)$. The restriction of $\sigma_{\{f+i\}}\in\operatorname{H}^{*}({\mathbb{P}}(E\oplus F),{\mathbb{Z}})$ in $\operatorname{H}^{*}({\mathbb{P}}(E),{\mathbb{Z}})$ is $\sigma_{\{i\}}$. Then Theorem 6 implies that

[TABLE]

Modulo the identity (32), this is equivalent to $\gamma_{f+i}\leq\alpha_{i}$. Similarly, we get the three other inequalities (33).

Fix now $0<r<e$ and $0<s<f$. The inclusion $G/P(\lambda_{r,s})\subset{\hat{G}}/{\hat{P}}(\lambda_{r,s})$ is the morphism $\phi_{r,s}$ defined in Section 4.2. Consider $\sigma_{I}\otimes\sigma_{J}\in\operatorname{H}^{*}({\mathbb{G}}(r,e)\times{\mathbb{G}}(s,f),{\mathbb{Z}})$ and $\sigma_{K}\in\operatorname{H}^{*}({\mathbb{G}}(r+s,e+f),{\mathbb{Z}})$ such that $\phi_{r,s}^{*}(\sigma_{K}).(\sigma_{I}\otimes\sigma_{J})^{\vee}=[pt]$. Here the Levi movability is automatic since ${\hat{G}}/{\hat{P}}(\lambda_{r,s})$ is cominuscule. Modulo (32), the inequality (31) of Theorem 6 corresponding to $\sigma_{I}\otimes\sigma_{J}$ and $\sigma_{K}$ is the inequality (34). Then the theorem follows from Theorem 6. $\square$

4.4 Complement on stretched Littlewood-Richardson coefficients

Lemma 1

Let $\alpha,\,\beta,$ and $\gamma$ be three partitions such that $l(\alpha)\leq e$, $l(\beta)\leq f$ and $l(\gamma)\leq e+f$.

Then, the map $n\longmapsto c_{n\alpha\,n\beta}^{n\gamma}$ is polynomial of degree not greater than

[TABLE]

where \left(\begin{array}[]{@{}c@{}}e\\ 2\end{array}\right)=\frac{e(e-1)}{2}.

Proof. Since $c_{\alpha\beta}^{\gamma}=c_{\beta\alpha}^{\gamma}$, we may assume that $e\leq f$. By [DW02], the function ${{\mathbb{Z}}_{\geq 0}}\longrightarrow{{\mathbb{Z}}_{\geq 0}}$, $n\longmapsto c_{n\alpha\,n\beta}^{n\gamma}$ is polynomial.

Recall that $E$ and $F$ are complex vector spaces of dimension $e$ and $f$. Set $G=\operatorname{GL}(E)\times\operatorname{GL}(F)$ and $X={\mathcal{F}l}(E)\times{\mathcal{F}l}(F)\times{\mathcal{F}l}(E\oplus F)$. Consider on $X$ the line bundle ${\mathcal{L}}={\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma}$. Since $c_{n\alpha\,n\beta}^{n\gamma}=\dim(\operatorname{H}^{0}(X,{\mathcal{L}}^{\otimes n})^{G})$, the degree of $c_{n\alpha\,n\beta}^{n\gamma}$ is equal to the dimension of $X^{\rm ss}({\mathcal{L}})//G$.

Consider the map $\pi$ defined in (30). By Chevalley Theorem, since $\pi$ is dominant, for any general $y\in X^{\rm ss}({\mathcal{L}})$, one has

[TABLE]

But, $\pi$ is $G$-invariant and $\pi^{-1}(\pi(y))$ contains $G.y$. Then

[TABLE]

where $G_{y}$ is the stabilizer of $y$ in $G$. But, for any $x\in X$, we have $\dim(G.x)\leq\dim(G.y)$ and

[TABLE]

We now claim that there exists $x$ such that $\dim(G_{x})=1$. Then the lemma follows.

We now prove the claim by constructing explicitly $x$, that is, defining complete flags of $E$, $F$ and $E\oplus F$. Fix bases $(\eta_{1},\dots,\eta_{e})$ and $(\zeta_{1},\dots,\zeta_{f})$ of $E$ and $F$. On $E$ and $F$, we consider the two standard flags $F_{\bullet}^{E}$ and $F_{\bullet}^{F}$ in these bases. Consider on $E\oplus F$, the following base

[TABLE]

and the associated flag $F_{\bullet}^{E\oplus F}$. One easily checks that $x=(F_{\bullet}^{E},F_{\bullet}^{F},F_{\bullet}^{E\oplus F})$ works. $\square$

5 Faces of ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$

5.1 Murnaghan’s face

The cone ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ is contained in the linear subspace of points $(\alpha,\beta,\gamma)\in{\mathbb{Q}}^{e+1}\times{\mathbb{Q}}^{f+1}\times{\mathbb{Q}}^{e+f+1}$ that satisfy $|\alpha|=|\beta|=|\gamma|$. In particular its dimension is at most $2e+2f+1$.

Recall that $\bar{\alpha}=(\alpha_{2}\geq\alpha_{3}\cdots)$, if $\alpha=(\alpha_{1}\geq\alpha_{2}\cdots)$. By Proposition 1, the points $(\alpha,\beta,\gamma)$ in ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ satisfy

[TABLE]

The set of points of ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ such that equality holds in the inequality (41) is a face ${\mathcal{F}}^{M}$ ($M$ stands for Murnaghan) of the cone ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$. Consider the linear map

[TABLE]

Lemma 2

The face ${\mathcal{F}}^{M}$ maps by $\pi$ to ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$. Moreover each fiber of $\pi$ over ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$ contains an unbounded interval.

The cone ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ has dimension $2e+2f+1$ and the face ${\mathcal{F}}^{M}$ has dimension $2e+2f$.

Proof. Assume that equality holds in the formula (41). Assume also that the coordinates of $\alpha$, $\beta$, and $\gamma$ are nonnegative integers. Then

[TABLE]

Thus the face ${\mathcal{F}}^{M}$ maps by $\pi$ on ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$. Conversely let $(\lambda,\mu,\nu)\in\operatorname{LR}(e,f,e+f)$. Let $a$ be an integer and set $b=a+|\lambda|-|\mu|$ and $c=a+|\lambda|-|\nu|$. If $a$ is big enough then $a\geq\lambda_{1}$, $b\geq\mu_{1}$ and $c\geq\nu_{1}$. Therefore $\alpha:=(a,\lambda)$, $\beta=(b,\mu)$ and $\gamma=(c,\nu)$ are three partitions of the same integer such that equality holds in the inequality (41). Thus the equality (42) holds and $(\alpha,\beta,\gamma)$ belongs to ${\mathcal{F}}^{M}$. In particular the fiber $\pi^{-1}(\lambda,\mu,\nu)$ contains an unbounded segment.

Since ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(e,f,e+f)$ has dimension $2e+2f-1$ and the fibers of $\pi$ have dimension at least one, the cone ${\mathcal{F}}^{M}$ has dimension at least $2e+2f$. We had already noticed that ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ has dimension at most $2e+2f+1$. These two inequalities (and the fact that ${\mathcal{F}}^{M}$ is a strict face of the cone ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ ) imply the lemma. $\square$

5.2 Proof of Theorem 2

Let $r$, $s$, $I$, $J$, and $K$ be like in Theorem 2. To such a triple $(I,J,K)$, Theorem 7 associates a codimension one face of $\operatorname{LR}(e,f,e+f)$. Using Lemma 2, this face corresponds to a face ${\mathcal{F}}_{IJK}$ of ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ of codimension two. Explicitly, ${\mathcal{F}}_{IJK}$ is the set of $(\alpha,\beta,\gamma)\in{\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$ such that

[TABLE]

This face ${\mathcal{F}}_{IJK}$ is contained in two codimension one faces, ${\mathcal{F}}^{M}$ and another one ${\mathcal{F}}_{IJK}^{M}$ that we want to determine.

Let $\varphi_{\tau}$ denote the linear form defined by

[TABLE]

where $\tau$ is any rational number. Set also

[TABLE]

By the theory of convex polyhedral cones, there exists $\tau_{0}$ such that for any $\tau>\tau_{0}$, $\varphi_{\tau}$ is nonnegative on the cone and the associated face is ${\mathcal{F}}_{IJK}$, and, $\varphi_{\tau_{0}}$ corresponds to ${\mathcal{F}}^{M}_{IJK}$.

Here, $E$ and $F$ are two linear spaces of dimension $e+1$ and $f+1$ and $G=\operatorname{GL}(E)\times\operatorname{GL}(F)$. Consider the variety

[TABLE]

We identify $\operatorname{Pic}^{G}(X)$ with ${\mathbb{Z}}^{2e+2f+3}$ like in Section 2.2.4.

Geometric description of $\varphi_{\infty}$. The inequality corresponding to ${\mathcal{F}}^{M}$ is $\varphi_{\infty}\geq 0$. By Section 2.2.4, ${\mathcal{F}}^{M}$ generates a face of ${{\mathbb{Q}}_{\geq 0}\operatorname{LR}}(G,X)$. Theorem 5 shows that there exists a well covering pair $(C_{\infty},\lambda_{\infty})$ of $X$ such that $\varphi_{\infty}(\alpha,\beta,\gamma)=-\mu^{{\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma}}(C_{\infty},\lambda_{\infty})$.

To describe such a pair $(C_{\infty},\lambda_{\infty})$, fix decompositions $E=\bar{E}\oplus l$ and $F=\bar{F}\oplus m$, where $\bar{E}$ and $\bar{F}$ are hyperplanes and $l$ and $m$ are lines. Let $\lambda_{\infty}$ be the one-parameter subgroup of $G$ acting with weight $1$ on $\bar{E}$ and $\bar{F}$, and with weight [math] on $l$ and $m$. Let $C_{\infty}$ be the set of points in $X$ such that

• •

the hyperplanes of the complete flags of $E$ and $F$ are respectively $\bar{E}$ and $\bar{F}$,

• •

the line of the partial flag of $E\otimes F$ is $l\otimes m$,

• •

the $(e+f+1)$-dimensional subspace of the partial flag of $E\otimes F$ is $(l\otimes m)\oplus(\bar{E}\otimes m)\oplus(l\otimes\bar{F})$.

One can check that $(C_{\infty},\lambda_{\infty})$ works (see [Res11c] for details).

Geometric description of $\varphi_{\tau}$. Fix decompositions $\bar{E}=E_{+}\oplus E_{-}$ and $\bar{F}=F_{+}\oplus F_{-}$, where $E_{+}$ and $F_{+}$ have dimension $r$ and $s$. Assume that $\tau>1$ and write $\tau=\frac{p}{q}$ with two integers $p$ and $q$ satisfying $p\wedge q=1$ and $q>0$. Let $\lambda_{\tau}$ be the one parameter subgroup of $G$ acting with weight $q+p$ on $E_{+}$ and $F_{+}$, with weight $p$ on $E_{-}$ and $F_{-}$ and with weight [math] on $l$ and $m$. The weight spaces of the action of $\lambda_{\tau}$ on $E\otimes F$ are

[TABLE]

where some “$\otimes m$” and “$l\otimes$” have been forgotten.

To $I^{\vee}=\{e+1-i\,:\,i\in I\}$ is associated an embedding $\iota_{I^{\vee}}$ of ${\mathcal{F}l}(E_{+})\times{\mathcal{F}l}(E_{-})$ in ${\mathcal{F}l}(\bar{E})$. Explicitly

[TABLE]

Similarly we consider $\iota_{J^{\vee}}$ and $\iota_{K}$. Observe that $C_{\infty}$ is canonically isomorphic to ${\mathcal{F}l}(\bar{E})\times{\mathcal{F}l}(\bar{F})\times{\mathcal{F}l}(\bar{E}\oplus\bar{F})$. Consider the embedding $(\iota_{I^{\vee}},\iota_{J^{\vee}},\iota_{K})$ of

[TABLE]

in $C_{\infty}$. Denote by $C_{\tau}$ its image. Using for example [Res11b, Proposition 1 and Theorem 1], one can check that, for $\tau$ big enough, $(C_{\tau},\lambda_{\tau})$ is a well covering pair. Moreover, $\varphi_{\tau}(\alpha,\beta,\gamma)=-q\mu^{{\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma}}(C_{\tau},\lambda_{\tau})$.

For any $\tau>1$, $C_{\tau}$ is an irreducible component of $\lambda_{\tau}$. Moreover, $C^{+}_{\tau}$ and $P(\lambda_{\tau})$ do not depend on $\tau>1$. In Particular, $(C_{\tau},\lambda_{\tau})$ is a well covering pair for any $\tau>1$.

Theorem 5 shows that the face determined by the inequality $\varphi_{\tau}$ only depends on $C_{\tau}$, and so does not depend on $\tau>1$: it is ${\mathcal{F}}_{IJK}$. This implies that $\varphi_{1}\geq 0$ on ${\mathbb{Q}}_{\geq 0}\operatorname{Kron}(e+1,f+1,e+f+1)$. Theorem 2 follows. $\square$

Remark. Let ${\mathcal{F}}_{1}$ denote the face associated to $\varphi_{1}$. Up to now, we have not proved that ${\mathcal{F}}_{1}$ has codimension 1 or equivalently that ${\mathcal{F}}_{1}={\mathcal{F}}_{IJK}^{M}$. This is the aim of Section 7.

6 Proof of Theorem 3

Keeping the notation of Section 5.2, we give a geometric description of $\varphi_{1}$. The weight spaces of the action of $\lambda_{1}$ on $E\otimes F$ are

[TABLE]

The irreducible component $C_{1}$ of $X^{\lambda_{1}}$ containing $C_{\tau}$ (for $\tau>1$) is isomorphic to

[TABLE]

Moreover, $C_{1}^{+}=C_{\tau}^{+}$ and $P(\lambda_{1})=P(\lambda_{\tau})$. In particular the pair $(C_{1},\lambda_{1})$ is well covering.

Let $G^{\lambda_{1}}$ denote the centralizer of $\lambda_{1}$ in $G$. Note that $G^{\lambda_{1}}=\operatorname{GL}(E_{+})\times\operatorname{GL}(E_{-})\times{\mathbb{C}}^{*}\times\operatorname{GL}(F_{+})\times\operatorname{GL}(F_{-})\times{\mathbb{C}}^{*}$. By [Res11c, Theorem 2], $g_{\alpha\beta\gamma}$ is the dimension of

[TABLE]

We have to determine the restriction $({\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma})_{|C_{1}}$ via the identification of $C_{1}$ with a product of flag varieties. Fix a basis of $\bar{E}$ starting with a basis of $E_{+}$ followed by a basis of $E_{-}$. For the group $\operatorname{GL}(\bar{E})$ we consider standard maximal tori and Borel subgroups in this basis. Similarly, we choose subgroups of $\operatorname{GL}(\bar{F})$.

The maximal torus of $\operatorname{GL}(E)$ acts on the fiber in ${\mathcal{L}}^{\alpha}$ over the base point of ${\mathcal{F}l}(E)$ with weight $(\alpha_{e+1},\dots,\alpha_{1})$. The maximal torus $\bar{T}$ of $\operatorname{GL}(\bar{E})$ acts on the fiber in ${\mathcal{L}}^{\alpha}$ over the base point of ${\mathcal{F}l}(\bar{E})$ (embedded in ${\mathcal{F}l}(E)$ like $C_{\infty}$ is embedded in $X$) with weight $(\alpha_{e+1},\dots,\alpha_{2})$. Let $w_{I^{\vee}}$ in the symmetric group $S_{e}$ associated to $I^{\vee}$ ($w_{I^{\vee}}(k)$ is the $k^{\rm th}$ elements of $I^{\vee}$ and $w_{I^{\vee}}(r+k)$ is the $k^{\rm th}$ elements of $I^{\vee}_{-}$). Then $\iota_{I^{\vee}}$ maps the base point of ${\mathcal{F}l}(E_{+})\times{\mathcal{F}l}(E_{-})$ to the image by $w_{I^{\vee}}^{-1}$ of the base point of ${\mathcal{F}l}(\bar{E})$. It follows that $\bar{T}$ acts on the fiber in $\iota_{I^{\vee}}^{*}({\mathcal{L}}^{\alpha}_{|C_{\infty}})$ by the weight $w_{I^{\vee}}^{-1}(\alpha_{e+1},\dots,\alpha_{2})$. After computation this gives

[TABLE]

Similarly

[TABLE]

and

[TABLE]

We deduce that $g_{\alpha\beta\gamma}$ is the multiplicity of the $\operatorname{GL}(E_{+})\times\operatorname{GL}(E_{-})\times\operatorname{GL}(F_{+})\times\operatorname{GL}(F_{-})$-simple module

[TABLE]

in the module

[TABLE]

Now the theorem is obtained by using repeatedly the formulas (2), (22) and (27) to decompose the module (46).

7 Proof of Theorem 4

Recall that the aim is to prove that ${\mathcal{F}}_{1}$ has codimension one. Since ${\mathcal{F}}_{IJK}$ has codimension two and it is contained in ${\mathcal{F}}_{1}$, it remains to prove that ${\mathcal{F}}_{IJK}\neq{\mathcal{F}}_{1}$.

Assume now that $(\alpha,\beta,\gamma)$ belongs ${\mathcal{F}}_{IJK}$. Since ${\mathcal{F}}_{IJK}$ is contained in ${\mathcal{F}}^{M}$, $g_{\alpha\beta\gamma}=c_{\bar{\alpha}\bar{\beta}}^{\bar{\gamma}}$. Then (see eg [DW11, Theorem 7.4]) $g_{\alpha\beta\gamma}=c_{\bar{\alpha}_{I}\bar{\beta}_{J}}^{\bar{\gamma}_{K}}.c_{\bar{\alpha}_{I_{-}}\bar{\beta}_{J_{-}}}^{\bar{\gamma}_{K_{-}}}.$ In particular, Lemma 1 shows that $g_{n\alpha\,n\beta\,n\gamma}$ is a polynomial function of $n$ of degree at most

[TABLE]

where $a=e-r$ and $b=f-s$.

Given an algebraic group $\Gamma$ acting on an irreducible variety $Y$, we denote by $\operatorname{mod}(\Gamma,Y)$ the minimal codimension of the $\Gamma$-orbits. By [Res11a, Lemma 2], for any ${\mathcal{L}}$ in the relative interior of ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G^{\lambda_{1}},C_{1})$, the dimension of $C_{1}^{\rm ss}({\mathcal{L}})//G^{\lambda_{1}}$ is equal to $\operatorname{mod}(G^{\lambda_{1}},C_{1})$.

By [Res10, Theorem 4], there exists a line bundle ${\mathcal{M}}$ on $X$ such that ${\mathcal{M}}_{|C_{1}}$ belongs to the relative interior of ${\mathbb{Q}}_{\geq 0}\operatorname{LR}(G^{\lambda_{1}},C_{1})$. We may assume that ${\mathcal{M}}={\mathcal{L}}^{\alpha}\otimes{\mathcal{L}}^{\beta}\otimes{\mathcal{L}}_{\gamma}$ for three partitions $\alpha$, $\beta$ and $\gamma$. But, by [Res10, Theorem 8], $X^{\rm ss}({\mathcal{M}})//G\simeq C_{1}^{\rm ss}({\mathcal{M}}_{|C_{1}})//G^{\lambda_{1}}$. It follows that ${\mathcal{M}}$ is a point on ${\mathcal{F}}_{1}$ satisfying $\dim(X^{\rm ss}({\mathcal{M}})//G)=\operatorname{mod}(G^{\lambda_{1}},C_{1})$. In particular,

[TABLE]

Regarding the assertions (47) and (48), to prove that ${\mathcal{F}}_{IJK}\neq{\mathcal{F}}_{1}$ it is sufficient to prove the claim: $\operatorname{mod}(G^{\lambda_{1}},C_{1})>d_{max}$. The center of $G^{\lambda_{1}}$ contains a dimension 3 torus acting trivially on $C_{1}$. Hence

[TABLE]

After simplification, we get

[TABLE]

Since $a,b,r,$ and $s$ are positive integers, the claim follows.

8 Proof of Theorem 1

We keep notation of Section 5.2, but now $r=e$ and $I=\{1,\dots,e\}$. In particular $E_{-}$ is trivial and the weight spaces of the action of $\lambda_{\tau}$ of $E\otimes F$ are

[TABLE]

Hence $C_{\tau}\simeq{\mathcal{F}l}(\bar{E})\times{\mathcal{F}l}(F_{+})\times{\mathcal{F}l}(\bar{E}\oplus F_{+})$ for $\tau$ big enough. Then, for any $\tau>0$, $(C_{\tau},\lambda_{\tau})$ is a well covering pair. We conclude like in Section 5.2 that $\varphi_{0}$ is nonnegative on the Kronecker cone. This proves the first assertion of the theorem.

Consider now the limit case $\tau=0$. The weight spaces are

[TABLE]

Hence $C_{0}\simeq{\mathcal{F}l}(\bar{E})\times{\mathcal{F}l}(F_{+})\times{\mathbb{P}}(F_{-}\oplus m)\times{\mathcal{F}l}(1,\dots,e+f-1;\bar{E}\otimes(F_{-}\oplus m)\oplus F_{+})\times{\mathbb{P}}(F_{-}\oplus m)$. Moreover $G^{\lambda_{0}}\simeq\operatorname{GL}(\bar{E})\times{\mathbb{C}}^{*}\times\operatorname{GL}(F_{+})\times\operatorname{GL}(F_{-}\oplus m)$. By [Res11c, Theorem 2]

[TABLE]

The computation of this dimension is made using the formulas (2), (22) and (27) like in Section 6.

9 A final inequality

All but two of the inequalities of Theorem 7 had been extended to the Kronecker coefficients by Theorems 1 and 2. The two exceptions are $\alpha_{i}\leq\gamma_{i}$ and $\beta_{j}\leq\gamma_{j}$. Consider the second one, up to permuting $(\alpha,e)$ and $(\beta,f)$. The extended inequality is

[TABLE]

for any $f+1\geq j\geq 2$.

This inequality is satisfied if $g_{\alpha\beta\gamma}\neq 0$. The proof is obtained by considering $I=\emptyset$ and $J=K=\{j-1\}$ in Section 5.2.