Inequalities for Moment Cones of Finite-Dimensional Representations

Introduction

The study of the convexity properties of the moment map and of its image has a long history in mathematics, starting from Schur and Horn’s observation that the diagonal entries of a $d\times d$ Hermitian matrix are always contained in the convex hull of the spectrum Schur (1923); Horn (1954); cf. Kostant (1973). More generally, Atiyah and Guillemin-Sternberg have shown that, for any torus action on a compact, connected Hamiltonian manifold, the image of the moment map is a convex polytope, called the moment polytope Atiyah (1982); Guillemin and Sternberg (1982a). It can be explicitly computed as the convex hull of the images of torus fixed points. For non-abelian groups, the image of the moment map is no longer convex. Instead, Kirwan’s celebrated convexity theorem asserts that, for the action of an arbitrary compact, connected Lie group on a compact, connected Hamiltonian manifold, the intersection of the image of the moment map with a positive Weyl chamber is a convex polytope Kirwan (1984a). This is the correct generalization of the moment polytope to non-abelian group actions. Mumford has given a different proof in the case of projective subvarieties, which relies on a concrete description in terms of the decomposition of the homogeneous coordinate ring into irreducible representations Ness and Mumford (1984); cf. Guillemin and Sternberg (1982b); Brion (1987). However, no effective general methods are known for the computation of these polytopes (in contrast to the case of torus actions).

Our main contribution is a clean algebraic description of the moment cone in terms of finitely many linear inequalities. To state the result, let $\mathfrak{t}$ denote the Lie algebra of a maximal torus of $T$, $i\mathfrak{t}^{*}_{+}$ a positive Weyl chamber, and $\pi$ the (complex) Lie algebra representation induced by $\Pi$ (see section 2 below for precise definitions). We shall say that $H$ is a Ressayre element if 1) the hyperplane $(-,H)=0$ is spanned by weights of ${M}$ and 2) there exists a vector $\psi\in{M}$ annihilated by $\pi(H)$ such that the “tangent map” at $\psi$,

is an isomorphism; here, ${M}(H<0)$ denotes the direct sum of all negative eigenspaces of $\pi(H)$ and $\mathfrak{n}_{-}(H<0)$ the sum of all root spaces for negative roots $\alpha$ such that $(\alpha,H)<0$ (definition 3.14). Note that there are only finitely many Ressayre elements for any given representation $\pi$. In section 3, we will prove the following result:

The moment cone for the $K$-action on ${M}$ is given by

To prove theorem 1.1, we show that any facet corresponds to a Ressayre element by studying the moment map, which is quadratic, locally up to second order. To show that, conversely, any Ressayre element determines a valid inequality, we use Mumford’s description of the moment cone as in Ressayre (2010a). Indeed, our notion of a Ressayre element is closely related to Ressayre’s notion of a dominant pair. We note that the description in theorem 1.1 will typically contain redundancies. Thus our result differs from Brion (1999), where the non-trivial or “general” faces of the moment polytope are characterized precisely at the cost of requiring a recursive strategy for their computation.

If $H$ is a Ressayre element then the domain and codomain of the tangent map (1.1) necessarily have the same dimension, i.e., $\dim\mathfrak{n}_{-}(H<0)=\dim{M}(H<0)$. We call this the trace condition. Moreover, note that the determinant $\delta_{H}$ of (1.1) (with respect to any fixed pair of bases) is a non-zero polynomial in $\psi\in{M}(H=0)$, the zero eigenspace of $\pi(H)$. In fact, $\delta_{H}$ is a canonical (up to scalar multiplication) lowest weight vector for the action of $K(H=0)$, the centralizer of the torus generated by $H$, on the space of polynomials on ${M}(H=0)$. This implies the following result in section 4, which we call the Horn condition:

is an element of the moment cone $C_{K(H=0)}({M}(H=0))$. In fact, $\delta_{H}$ is a lowest weight vector of weight $-\kappa_{H}$.

Here, $\Omega$ denotes the set of weights of ${M}$ and $R_{G,-}$ denotes the set of negative roots of $G$. By applying theorem 1.1 to the lower-dimensional scenario, the Horn condition can be explicitly stated as a set of linear inequalities that have to be satisfied by $\kappa_{H}$.

Tangent maps and their determinants have been studied in great generality by Ressayre and Belkale from an algebro-geometric point of view Ressayre (2010a, b); Belkale (2010), and our theorems 1.1 and 1.2 can also be deduced from their results. In these works, the non-vanishing of the determinant has been in turn been translated into a cohomological condition. In contrast, we propose that, for the purposes of computing moment cones explicitly, it can be useful to instead test the non-vanishing of the determinant directly—either symbolically, which is easily possible in small dimensions, or numerically by using fast algorithms for polynomial identity testing, as we discuss in section 5 below. The challenge imposed by higher dimensions is rather in finding additional a priori constraints on the facets of the moment cone.

Moment Cones of Finite-Dimensional Representations

For $\operatorname{SU}(d)$, whose complexification is $\operatorname{SL}(d)$, we will always use the maximal torus $T(d)$ that consists of the diagonal unitary matrices of unit determinant. Its Lie algebra will be denoted by $\mathfrak{t}(d)$. We will use as positive roots the $\alpha_{i,j}(H):=H_{ii}-H_{jj}$ with $i<j$, and abbreviate the (positive) roots by $R_{d}$ and $R_{d,+}$, respectively. Finally, we write ${\mathcal{O}}^{d}_{\lambda}$ for the coadjoint $\operatorname{SU}(d)$ orbit through a highest weight $\lambda$.

Now let $\Pi\colon G\rightarrow\operatorname{GL}({M})$ a representation of $G$ on a finite-dimensional Hilbert space ${M}$ that is equipped with a $K$-invariant Hermitian inner product $\braket{-}{-}$, which we take to be antilinear in the first argument. We will oftentimes use Dirac’s notation $\braket{\phi}{A}{\psi}:=\braket{\phi}{A\psi}$ for $\phi,\psi\in{M}$ and $A\in\mathfrak{gl}({M})$. We denote by $\pi\colon\mathfrak{g}\rightarrow\mathfrak{gl}({M})$ the induced Lie algebra representation, by $\Omega\subseteq P_{G}$ the set of weights and write the weight space decomposition as ${M}=\bigoplus_{\omega\in\Omega}{M}_{\omega}$. The $K$-action on ${M}$ admits a canonical (up to conventions) moment map, defined by

for all $\psi\in{M}$ and $X\in i\mathfrak{k}$. Here and in the following, we write $(\varphi,X)=\varphi(X)$ for the duality pairing. The map $\mu_{K}$ is indeed a moment map in the sense of symplectic geometry: it is $K$-invariant and satisfies the basic identity

for all $X\in\mathfrak{k}$, where $\omega_{M}(\phi,\psi)=2\operatorname{Im}\braket{\phi}{\psi}$ denotes the symplectic form that we will use for ${M}$. The moment cone then is defined as intersection of the moment map image with the positive Weyl chamber,

where $R({M})=\operatorname{Sym}({M})^{*}$ denotes the space of polynomials on ${M}$. A representation $V_{G,\lambda}^{*}$ occurs in $R({M})$ if and only if there exists a polynomial $P\in R({M})$ of weight $-\lambda$ that is invariant under the action of the lower unipotent subgroup $N_{-}$. We shall call such a polynomial a lowest weight vector in $R({M})$.

(and $C_{K}$ is a pointed cone with base $\Delta_{K}$). Thus we may equivalently study moment cones of representations or moment polytopes of the corresponding projective spaces.

Throughout this paper, we shall always work with moment cones (but see Walter (2014) for an exposition from the projective point of view). We shall moreover assume that the moment cone is of maximal dimension, i.e., $\dim C_{K}=\dim i\mathfrak{t}^{*}$. This is the case if and only if there exists a vector with finite stabilizer.

Facets of the Moment Cone

Like any polyhedral cone, the moment cone can be described by finitely many linear inequalities $(-,H)\geq 0$. Since we have assumed that $C_{K}$ is of maximal dimension, its facets are of codimension one in $i\mathfrak{t}^{*}$ and their inward-pointing normal vectors may be identified with the defining linear inequalities $(-,H)\geq 0$ of the moment cone. Since the moment cone is obtained by intersecting $\mu_{K}({M})$ with the positive Weyl chamber, which itself is a maximal-dimensional polyhedral cone, some of the facets of $C_{K}$ can be subsets of facets of $i\mathfrak{t}^{*}_{+}$, and we shall call those the trivial facets of the moment cone:

A facet of the moment cone is trivial if it corresponds to an inequality of the form $(-,H_{\alpha})\geq 0$ for some positive root $\alpha\in R_{G,+}$. Otherwise, the facet is called non-trivial.

Non-trivial facets have also been called “general” in the literature Brion (1999). We record the following straightforward observation:

Any non-trivial facet of $C_{K}$ meets the relative interior $i\mathfrak{t}^{*}_{>0}$ of the positive Weyl chamber.

We first consider the moment map $\mu_{T}$, defined as in (2.2) for the action of the maximal torus $T\subseteq K$. Let ${M}=\bigoplus_{\omega\in\Omega}{M}_{\omega}$ be the decomposition of ${M}$ into weight spaces and let $\psi\in{M}$ be a vector decomposed accordingly as $\psi=\sum_{\omega}\psi_{\omega}v_{\omega}$. Then $\mu_{T}$ has the following concrete description:

Observe that $\mu_{T}(\psi)$ is a conic combination of weights. It follows that the “abelian” moment cone $C_{T}$ of ${M}$ is precisely equal to the conical hull of the set of weights; it is maximal-dimensional since it contains $C_{K}$. More generally, if $\Omega^{\prime}\subseteq\Omega$ is a subset of weights and ${M}_{\Omega^{\prime}}:=\bigoplus_{\omega\in\Omega^{\prime}}{M}_{\omega}$ then $C_{T}({M}_{\Omega^{\prime}})=\operatorname{cone}\Omega^{\prime}$. For the next lemma recall that a critical point of a smooth map $f\colon M\rightarrow M^{\prime}$ is a point $m\in M$ where the differential df\big{|}_{m} is not surjective; a critical value is the image $f(m)$ of a critical point. Then the following is well-known (e.g., (Christandl et al., 2014, Remark 4.14)):

The set of critical values of $\mu_{T}$ is equal to the union of the codimension-one conic hulls of subsets of weights.

Let $\psi\in{M}$ with weight decomposition $\psi=\sum_{\omega}\psi_{\omega}v_{\omega}$. By (3.1), $\mu_{T}(\psi)$ is a conic combination of weights in $\Omega_{\psi}:=\{\omega\in\Omega:\psi_{\omega}\neq 0\}$. By the moment map property (2.3) and non-degeneracy of the symplectic form, $\psi$ is a critical point if and only if there exists $0\neq X\in\mathfrak{t}$ such that $\pi(X)\psi=0$ (Guillemin and Sternberg, 1982a, Lemma 2.1), i.e., if and only if $\omega(X)=0$ for all $\omega\in\Omega_{\psi}$. It follows that $\psi$ is a critical point if and only if the conic hull of $\Omega_{\psi}$ is of positive codimension.

In particular, any critical value is contained in a codimension-one conic hull of weights, since we may always add additional weights. Conversely, if $\Omega^{\prime}\subseteq\Omega$ is a subset of weights that spans a conic hull of codimension one then $C_{T}({M}_{\Omega^{\prime}})=\operatorname{cone}\Omega^{\prime}$ consists of critical values. ∎

We now derive a basic necessary condition that cuts down the defining inequalities of the moment cone to a finite set of candidates (cf. (Christandl et al., 2014, Remark 3.6)).

An element $H\in i\mathfrak{t}$ is called admissible if the linear hyperplane $(-,H)=0$ is spanned by a subset of weights in $\Omega$.

The notion of admissibility is invariant under the action of the Weyl group $W_{K}$.

Let $(-,H)\geq 0$ be an inequality corresponding to a non-trivial facet of the moment cone. Then $H$ is admissible.

By lemma 3.2, the intersection of $(-,H)=0$ with the interior of the positive Weyl chamber $i\mathfrak{t}^{*}_{>0}$ is non-empty. Each point in this intersection is a critical value for $(\mu_{K},H)=(\mu_{T},H)$, hence of $\mu_{T}$, and therefore according to lemma 3.3 contained in a linear hyperplane spanned by a subset of weights. Since this is true for all points in the intersection, which contains the relative interior of the facet, it follows that the facet is in fact contained in a single such hyperplane. ∎

2 Description of the Moment Cone by Ressayre Elements

Let us now fix an inequality $(-,H)\geq 0$ corresponding to a non-trivial facet $(-,H)=0$ of the moment cone. Let $\psi\in{M}$ be a preimage of a point $\mu_{K}(\psi)\in i\mathfrak{t}^{*}_{>0}$ on the facet $(-,H)=0$. In the proof of lemma 3.5, we have used that $\psi$ is a critical point of $(\mu_{K},H)$ (equivalently, that $\pi(H)\psi=0$) to gain information on the set of possible facets. To study the function $(\mu_{K},H)$ in the vicinity of such a critical point $\psi$ it is natural to consider the Hessian, which is the quadratic form

For tangent vectors generated by the infinitesimal action of $X,Y\in\mathfrak{k}$, we have the formula

where we have used that $\pi(H)\psi=0$. We now decompose

where ${M}(H<0)=\bigoplus_{\omega:(\omega,H)<0}{M}_{\omega}$ is the sum of the eigenspaces of the Hermitian operator $\pi(H)$ with eigenvalue less than , etc. Then it is plain from (3.2) that the index of the Hessian $Q$, i.e., the dimension of a maximal subspace on which the quadratic form is negative definite, is equal to twice the complex dimension of ${M}(H<0)$.

We will now deduce a second formula for the index by observing that the Hessian is necessarily positive semidefinite on the subspace of those tangent vectors $V$ that are mapped to $i\mathfrak{t}^{*}$ by the differential of the moment map. To see this, consider a curve $\psi_{t}$ with $\psi_{0}=\psi$, $\dot{\psi}_{0}=V$ and $\mu_{K}(\psi_{t})\in i\mathfrak{t}^{*}_{>0}$ for all $t\in(-\varepsilon,\varepsilon)$ (such a curve can always be constructed by using the symplectic cross section (Guillemin and Sternberg, 1984b, Theorem 26.7)). Then, since $(\mu_{K}(\psi),H)=0$ and d(\mu_{K},H)\big{|}_{\psi}\equiv 0,

But $(\mu_{K}(\psi_{t}),H)\geq 0$, since $\mu_{K}(\psi_{t})\in C_{K}$ and $(-,H)\geq 0$ is a valid inequality for the moment cone. Together, this shows that, indeed, $Q(V,V)\geq 0$.

The subspace of all such $V$ can be computed in a different way. For this, let $\mathfrak{r}=\bigoplus_{\alpha\in R_{G,+}}\mathfrak{k}_{\alpha}$ denote the sum of the root spaces of the compact Lie algebra. It follows from the moment map property (2.3) that

The Hessian is positive semidefinite on the symplectic complement

of $M_{0}:=\pi(\mathfrak{r})\psi\subseteq{M}$.

In fact, it is well-known that $M_{0}$ is a symplectic subspace of ${M}$ (see, e.g., (Guillemin and Sternberg, 1982a, Lemma 6.7)). To see this, note that the restriction of the symplectic form to $M_{0}$ is given by

and can therefore be identified with the Kirillov-Kostant-Souriau symplectic form on the coadjoint orbit through $\mu_{K}(\psi)\in i\mathfrak{t}^{*}_{>0}$. As a consequence, we have the decomposition ${M}=M_{0}\oplus M_{0}^{\omega_{M}}$.

The Hessian is block-diagonal with respect to the decomposition ${M}=M_{0}\oplus M_{0}^{\omega_{M}}$.

For all $\pi(R)\psi\in M_{0}$ and $V\in M_{0}^{\omega_{M}}$ we have that

since $[-iH,R]\in\mathfrak{r}$ and therefore $\pi([-iH,R])\psi\in M_{0}$. ∎

The tangent map $\mathfrak{r}\rightarrow{M}$, $R\mapsto\pi(R)\psi$ is injective.

The stabilizer of the coadjoint action of $K$ at any $\lambda\in i\mathfrak{t}^{*}_{>0}$ is $T$, while $\mathfrak{k}=\mathfrak{t}\oplus\mathfrak{r}$. As $\mu_{K}(\psi)\in i\mathfrak{t}^{*}_{>0}$, the claim follows from this and $K$-equivariance of the moment map. ∎

The index of the Hessian is equal to twice the number of positive roots $\alpha\in R_{G,+}$ such that $(\alpha,H)>0$.

Since $Q$ is positive semidefinite on $M_{0}^{\omega_{M}}$ (lemma 3.6) and block-diagonal with respect to the decomposition ${M}=M_{0}\oplus M_{0}^{\omega_{M}}$ (lemma 3.7), it suffices to compute the index of $Q$ on $M_{0}=\{\pi(R)\psi:R\in\mathfrak{r}\}$. For this, recall from (3.3) that $Q(\pi(R)\psi,\pi(S)\psi)=(\mu_{K}(\psi),[[H,R],S])$ for all $R,S\in\mathfrak{r}$. Since the tangent map $R\mapsto\pi(R)\psi$ is injective (lemma 3.8), we may instead consider the form

on $\mathfrak{r}$. Now observe that $\widetilde{Q}$ is block-diagonal with respect to $\mathfrak{r}=\bigoplus_{\alpha\in R_{G,+}}\mathfrak{k}_{\alpha}$, since for all $R\in\mathfrak{k}_{\alpha}$ and $S\in\mathfrak{k}_{\beta}$, $[[H,R],S]\in i\mathfrak{k}_{\alpha\pm\beta}$, while $\mu_{K}(\psi)\in i\mathfrak{t}^{*}$. Therefore, it suffices to compute the index on a single root space $\mathfrak{k}_{\alpha}$. For this, define the “Pauli matrices” $X_{\alpha}:=E_{\alpha}+E_{-\alpha}$ and $Y_{\alpha}:=i(E_{-\alpha}-E_{\alpha})$, which satisfy the commutation relations $[X_{\alpha},Y_{\alpha}]=2iH_{\alpha}$ etc. Then $iX_{\alpha}$ and $iY_{\alpha}$ form a basis of $\mathfrak{k}_{\alpha}$ and

Likewise, $\widetilde{Q}(iY_{\alpha},iY_{\alpha})=-2(\alpha,H)(\mu_{K}(\psi),H_{\alpha})$, while $\widetilde{Q}(iX_{\alpha},iY_{\alpha})=0$. Since $(\mu_{K}(\psi),H_{\alpha})>0$, we conclude that the index of $Q$ is equal to twice the number of positive roots $\alpha$ with $(\alpha,H)>0$. ∎

Since we have already seen above that the index of the Hessian is also equal to twice the complex dimension of ${M}(H<0)$, we obtain the following result, which can also be extracted from (Brion, 1999, Theorem 2):

If $(-,H)\geq 0$ defines a non-trivial facet of the moment cone then

We now study the complexified group action. To this end, we consider the Lie algebra $\mathfrak{n}_{-}=\bigoplus_{\alpha\in R_{G,-}}\mathfrak{g}_{\alpha}$ of the negative unipotent subgroup, which plays a role analogous to $\mathfrak{r}$ for vectors $\psi$ that are mapped into the positive Weyl chamber (compare the following with lemma 3.8).

The tangent map $\mathfrak{n}_{-}\rightarrow{M},X\to\pi(X)\psi$ is injective.

Let $E_{-}:=\sum_{\alpha\in R_{G,+}}z_{\alpha}E_{-\alpha}$ be an arbitrary element in $\mathfrak{n}_{-}$. Since $\pi(E_{\pm\alpha})^{\dagger}=\pi(E_{\mp\alpha})$, we find that $\pi(E_{-})^{\dagger}=\pi(E_{+})$ with $E_{+}:=\sum_{\alpha\in R_{G,+}}\bar{z}_{\alpha}E_{\alpha}$ (the Cartan involution of $E_{-}$). Therefore,

so that by using $\mu_{K}(\psi)\in i\mathfrak{t}^{*}_{>0}$ we find that

In contrast to lemma 3.8, which continues to hold true if $\mu_{K}(\psi)$ is mapped to the relative interior of a different Weyl chamber, it is important in lemma 3.11 to choose the negative unipotent subgroup (relative to the choice of positive Weyl chamber). For example, consider an irreducible $G$-representation ${M}=V_{G,\lambda}$ with highest weight $\lambda\in i\mathfrak{t}^{*}_{>0}$ and highest weight vector $v_{\lambda}$. Then $\mu_{K}(v_{\lambda})=\lambda\in i\mathfrak{t}^{*}_{>0}$ and the “lowering operators” in $\mathfrak{n}_{-}$ indeed act injectively. On the other hand, the “raising operators” in the positive nilpotent Lie algebra $\mathfrak{n}_{+}$ annihilate the highest weight vector (by definition).

We now decompose the Lie algebra $\mathfrak{n}_{-}$ similarly to (3.4),

where $\mathfrak{n}_{-}(H<0)=\bigoplus_{\alpha\in R_{G,-}:(\alpha,H)<0}\mathfrak{g}_{\alpha}$ is the sum of the complex root spaces with negative $H$-weight $(\alpha,H)<0$, etc. We observe that corollary 3.10 can be equivalently stated as

Note that $\pi(\mathfrak{n}_{-}(H<0)){M}(H=0)\subseteq{M}(H<0)$. Thus we obtain the following important result:

Let $\psi\in{M}$ such that $\mu_{K}(\psi)\in i\mathfrak{t}^{*}_{>0}$ is a point on a non-trivial facet of the moment cone corresponding to the inequality $(-,H)\geq 0$. Then $\psi\in{M}(H=0)$ and the tangent map restricts to an isomorphism

The fact that $\psi\in{M}(H=0)$ is just a reformulation of $\pi(H)\psi=0$. By the preceding discussion, the tangent map is well-defined as a map from $\mathfrak{n}_{-}(H<0)$ to ${M}(H<0)$; it is injective by lemma 3.11 and surjective since the dimensions agree according to (3.5). ∎

We now prove a partial converse to proposition 3.12, inspired by the argument of Ressayre Ressayre (2010a).

Suppose there exists $\psi\in{M}(H=0)$ such that the tangent map

is surjective. Then $(-,H)\geq 0$ is a valid inequality for the moment cone.

Its differential at $(1,\psi)$ is the linear map

The assumption implies that this map is surjective. It follows that $\Pi(N_{-}){M}(H\geq 0)\subseteq{M}$ contains a small Euclidean ball around $\psi$. In particular, any $N_{-}$-invariant polynomial that is zero on ${M}(H\geq 0)$ is automatically zero everywhere on ${M}$.

We now prove the inequality. By the description of the moment cone in (2.4), it suffices to show that $(\lambda,H)\geq 0$ for all highest weights $\lambda$ such that $V^{*}_{G,\lambda}\subseteq R({M})$. Recall that the highest weight of $V^{*}_{G,\lambda}$ is $\lambda^{*}=-w_{0}\lambda$, where $w_{0}$ is the longest Weyl group element that flips the positive and negative roots. Consider a lowest weight vector, i.e., a polynomial $P\in R({M})$ that is a weight vector of weight $-\lambda$ and invariant under the action of $N_{-}$. Then $\pi(H)P=-(\lambda,H)P$ and the restriction of $P$ to ${M}(H\geq 0)$ is non-zero by our discussion above. But this restriction is an element of $R({M}(H\geq 0))=\operatorname{Sym}({M}(H\geq 0))^{*}$, the space of polynomials on ${M}(H\geq 0)$. Since all $H$-weights in $R({M}(H\geq 0))$ are non-positive, it follows that $-(\lambda,H)\leq 0$, as we set out to prove. ∎

We remark that proposition 3.13 holds unconditionally without any assumption on the dimension of the moment cone $C_{K}$. We summarize our findings in the following definition and theorem that we had already advertised in the introduction.

An element $H\in i\mathfrak{t}$ is called a Ressayre element if

$H$ is admissible, i.e., the linear hyperplane $(-,H)=0$ is spanned by a subset of weights in $\Omega$, and

there exists $\psi\in{M}(H=0)$ such that the map

This follows directly from lemmas 3.5, 3.12 and 3.13. ∎

Theorem 1.1 gives a complete description of the moment cone of an arbitrary finite-dimensional $K$-representation ${M}$ (under the assumption that $C_{K}$ is of maximal dimension). The set of inequalities thus obtained may still be redundant (i.e., not all inequalities necessarily correspond to facets of the moment cone). In contrast, Ressayre’s well-covering pairs Ressayre (2010a) characterize the facets of the moment polytope precisely. In our language, his condition amounts to requiring that that the generic fiber of the map $N_{-}\times_{N_{-}(H\geq 0)}{M}(H\geq 0)\rightarrow{M}$ is a point (as opposed to only requiring that the map be dominant). Our characterization is also related to (Brion, 1999, Theorem 2), which uses algebraic geometry to characterize non-trivial faces of arbitrary codimension. Unlike proposition 3.13, it relies on an assumption about lower-dimensional moment polytopes, which can in principle be obtained recursively.

Generalized Trace and Horn Condition

We now extract two useful necessary conditions that have to hold for any Ressayre element $H$, and therefore for any non-trivial facet of the moment polytope. We will see in section 7.1 below that they are a generalization of the classical Horn inequalities, which justifies our terminology.

The first condition, which we call the trace condition is the observation that the domain and range of the tangent map (3.6) necessarily have to agree, i.e.,

We note that the right-hand side of (4.1) is invariant under the action of the Weyl group. This suggests that we first compute the dominant admissible $H_{0}$ and then determine those $H\in W_{K}\cdot H_{0}$ which satisfy the trace condition (4.1). For this, we will need the following well-known lemma.

Fix any $\rho_{K}\in i\mathfrak{t}_{+}$ such that $(\alpha,\rho_{K})>0$ for all positive roots $\alpha\in R_{G,+}$, and set $H^{\varepsilon}:=H-\varepsilon\rho_{K}$. Let $w\in W_{K}$ be a Weyl group element such that $H=w\cdot H_{0}$. For all positive roots $\alpha\in R_{G,+}$ and $\varepsilon>0$ small enough,

if and only if $(\alpha,H_{0})=0$ implies that $w\cdot\alpha\in R_{G,-}$. In other words, $H^{\varepsilon}_{0}:=w^{-1}\cdot H^{\varepsilon}\in i\mathfrak{t}_{+}$ if and only if $w_{0}w\cdot\alpha\in R_{G,+}$ for all $\alpha\in R_{G,+}$ with $(\alpha,H_{0})=0$. Since $H^{\varepsilon}$ is regular, this immediately shows that such Weyl group elements $w$ exist and are unique. What is more, regularity also implies that

We obtain the following useful corollary:

The second condition, called the Horn condition, is based on the observation that for any Ressayre element $H$ the determinant polynomial

is non-zero (we take the determinant with respect to any fixed pair of bases). This can be understood to imply a statement about a smaller moment cone. To see this, let $G(H=0)$ denote the identity component of the centralizer of the torus generated by $H$, i.e., the connected subgroup of $G$ with Lie algebra $\mathfrak{g}(H=0)=\mathfrak{h}\oplus\bigoplus_{\alpha:(\alpha,H)=0}\mathfrak{g}_{\alpha}$; denote by $N_{-}(H=0)$ the corresponding negative unipotent subgroup and by $K(H=0)=G(H=0)\cap K$ the maximal compact subgroup. Then $G(H=0)$ acts on ${M}(H=0)$ and thus on the space $R({M}(H=0))=\operatorname{Sym}({M}(H=0))^{*}$ of polynomial functions.

is an element of the moment cone $C_{K(H=0)}({M}(H=0))$. In fact, $\delta_{H}$ is a lowest weight vector of weight $-\kappa_{H}$.

In view of (2.4), it suffices to argue that the determinant polynomial $\delta_{H}$ is a lowest weight vector in $R({M}(H=0))$ of weight $-\kappa_{H}$.

To see this, fix a basis $\psi_{1},\dots,\psi_{k}$ of ${M}(H<0)$ such that each $\psi_{j}$ is a weight vector of weight $\omega_{j}$, and denote by $\alpha_{1},\dots,\alpha_{k}$ the negative roots with $(\alpha,H)<0$, so that $E_{k}:=E_{\alpha_{k}}$ is a basis of $\mathfrak{n}_{-}(H<0)$. Then the determinant polynomial $\delta_{H}$ with respect to this basis can be written as

where we write $\Lambda^{k}A$ for the canonical homomorphism $\Lambda^{k}V\rightarrow\Lambda^{k}W$ induced by a linear map $A\colon V\rightarrow W$. It follows that for any $g\in G$ and $\psi\in{M}(H=0)$,

Computation of Moment Cones

Theorem 1.1 reduces the computation of the moment cone of an arbitrary finite-dimensional representation to an enumeration of all Ressayre elements, which in principle is straighforward: Since there are only finitely many weights, the admissibility condition cuts down the number of possible inequalities down to a finite list of candidates, and for each such candidate $(-,H)\geq 0$, the isomorphism condition can be easily checked. Indeed, we only need to verify the trace condition (4.1) and that the determinant polynomial (4.2) is non-zero. In this way, we obtain a deterministic algorithm to compute the moment cone for an arbitrary representation that can easily be implemented in a computer program onl . We remark that checking whether the determinant polynomial is non-zero can be sped up by using a fast probabilistic algorithm for polynomial identity testing, e.g., based on the Schwartz-Zippel lemma.

In practice, naively enumerating all admissible hyperplanes by considering all $(r_{K}-1)$-element subsets of $\Omega$ quickly becomes infeasible as one considers representations of larger dimensions. In this case, it is useful to first determine the admissible elements up to the Weyl group and to impose further necessary conditions by a more refined analysis of the representation at hand. Then corollary 4.2 can be used to obtain directly only those candidates that satisfy the trace condition. In this way, we may often cut down the number of candidates substantially for which we need to check that $\delta_{H}$ is non-zero. In section 6 below, we illustrate this analysis in the case of the one-body quantum marginal problem.

Quantum Marginal Problem and Kronecker Cone

The state of a quantum system is specified by a unit vector $\psi$ in a Hilbert space ${M}$, which we will always assume to be finite-dimensional. Quantum systems composed of several distinguishable particles are described by the tensor product of the Hilbert spaces of their constituents, ${M}=\bigotimes_{k=1}^{n}{M}_{k}$. The state of the $k$-th subsystem can be described by the reduced density matrix $\rho_{k}$, which is the unique positive semi-definite operator on ${M}_{k}$ such that

for all Hermitian operators $X_{k}$ on ${M}_{k}$ (it is also called the partial trace of $\psi$). The fundamental one-body quantum marginal problem asks which $\rho_{1},\dots,\rho_{n}$ can arise as the reduced density matrices of a quantum state $\psi\in{M}$ Klyachko (2004); Christandl and Mitchison (2006); Daftuar and Hayden (2004); Christandl et al. (2007); Walter (2014). Equivalently, it asks for the compatibility conditions that the $\rho_{k}$ have to satisfy in order for there to exist a global state $\psi$. For two subsystems, it is a straightforward consequence of the singular value decomposition that $\rho_{1}$ and $\rho_{2}$ are compatible if and only if they have the same non-zero eigenvalues (including multiplicities). In general, however, the problem is much more involved; it can be shown that it is a strict generalization of the problem of computing the Horn cone of section 7 Klyachko (2004); Christandl et al. (2012); Walter (2014) and it has been solved in Klyachko (2004); Daftuar and Hayden (2004) by using similar methods Klyachko (1998); Berenstein and Sjamaar (2000). However, a concrete description akin to the Horn inequalities is still elusive.

By Schur-Weyl duality and Mumford’s description (2.4), the moment cone $C(a,b,c)$ can equivalently be defined in terms of the representation theory of the symmetric group Christandl and Mitchison (2006); Klyachko (2004); Christandl et al. (2007, 2014): We have

where $\alpha$, $\beta$, and $\gamma$ vary over the set of Young diagrams with the same number $k$ of boxes and no more than $a$, $b$, and $c$ rows, respectively, and where $g_{\alpha,\beta,\gamma}$ denotes the Kronecker coefficient of the symmetric group, i.e., the multiplicity of the invariant subspace in the corresponding triple tensor product of irreducible $S_{k}-$representations. We will henceforth refer to $C(a,b,c)$ as the Kronecker cone.

It will be convenient to assume without loss of generality that $1<a\leq b\leq c\leq ab$. In this case, $C(a,b,c)$ is maximal-dimensional (see corollary A.2 in the appendix) and our method is directly applicable. In fact, we shall see below that all other cases can be reduced to the case $c=ab$. According to theorem 1.1, the moment cone $C(a,b,c)$ is thus cut out by those $H$ which are Ressayre elements, i.e., which are admissible and satisfy the isomorphism condition. Naively determining the admissible $H$ by enumerating all subsets of $\Omega$ with cardinality $r-1$, determining whether they span a hyperplane and computing the normal vector amounts to considering $\binom{\lvert\Omega\rvert}{r-1}=\binom{abc}{a+b+c-3}$ subsets, which rapidly becomes infeasible (e.g., for $a=b=c=4$, there are over 27 billion such subsets). We therefore need to derive additional constraints to make this approach computationally feasible.

By considering $\operatorname{SU}(a)\times\operatorname{SU}(b)\times\operatorname{SU}(c)\subseteq\operatorname{SU}(ab)\times\operatorname{SU}(c)$, the moment cone for the tripartite problem can now be written in terms of the bipartite cone $C(ab,c)$ and the moment polytopes $\Delta(a,b|\lambda_{AB})$ of the action of $\operatorname{SU}(a)\times\operatorname{SU}(b)$ on the coadjoint $\operatorname{SU}(ab)$-orbits ${\mathcal{O}}^{ab}_{\lambda_{AB}}$:

A first consequence is that the case $c\neq ab$ can always be reduced to $c=ab$. Indeed, (6.2) and (6.3) imply that

if $c<ab$. Thus the moment cone for $c>ab$ is isometric to $C(a,b,ab)$, while for $c<ab$ it is obtained as a projection of the latter. In the case where $c=ab$, it was observed in Manivel (1997); Klyachko (2004) that any normal vector of the moment cone $C(a,b,ab)$ necessarily has a rather special form. We state this result and give a succint alternative proof that does not rely on the results of Klyachko (2004); Berenstein and Sjamaar (2000):

Let $H=(H_{A},H_{B},H_{AB},z)$ be the normal vector of a non-trivial facet of the moment cone $C(a,b,ab)$ such that $(H_{A},H_{B})\neq 0$. Then:

$(H_{A},H_{B})$ is determined by a maximal number of equations of the form $H_{A,i}+H_{B,j}=H_{A,k}+H_{B,l}$,

the components of $H_{AB}$ are precisely all possible partial sums $-H_{A,i}-H_{B,j}$, and

By lemma 3.2, there exists a regular dominant point $\lambda=(\lambda_{A},\lambda_{B},\lambda_{AB})\in i\mathfrak{t}^{*}_{>0}$ in the interior of this facet. In a neighborhood of $\lambda$, the moment cone locally looks like a half-space, so that

is not only a valid inequality that holds for all $(\mu_{A},\mu_{B})\in\Delta(a,b|\lambda_{AB})$, as follows from (6.3), but in fact a facet of the moment polytope $\Delta(a,b|\lambda_{AB})$, since we have assumed that $(H_{A},H_{B})\neq 0$.

While determining the moment polytopes $\Delta(a,b|\lambda_{AB})$ is just as hard as determining the cone $C(a,b,ab)$, this reformulation gives us an additional insight: Recall that the Duistermaat-Heckman measure for the action of the maximal torus $T(a)\times T(b)$ of $\operatorname{SU}(a)\times\operatorname{SU}(b)$ on the coadjoint $\operatorname{SU}(ab)$-orbit ${\mathcal{O}}^{ab}_{\lambda_{AB}}$ is a measure on the abelian moment polytope. Now let $\pi$ denote the projection $i\mathfrak{t}(ab)^{*}\rightarrow i\mathfrak{t}(a)^{*}\oplus i\mathfrak{t}(b)^{*}$. The affine hyperplanes through some $\pi(w\lambda_{AB})$ spanned by subsets of the restricted roots $\pi(\alpha)$, $\alpha\in R_{ab,+}$, partition the abelian moment polytope into a finite number of polyhedral chambers, and it as an immediate consequence of the Heckman formula Harish-Chandra (1957); Heckman (1982) that the measure has a polynomial density function on each chamber (cf. Boysal and Vergne (2009)). The Duistermaat-Heckman measure for the action of $\operatorname{SU}(a)\times\operatorname{SU}(b)$ can be recovered by applying a number of partial derivatives to the measure for $T(a)\times T(b)$ (e.g., Christandl et al. (2014)). It follows that the moment polytope $\Delta(a,b|\lambda_{AB})$, which is the support of the latter measure, is equal to a finite union of chambers; in particular, its non-trivial facets are contained in hyperplanes of the form just described.

Applied to the facet (6.5), it follows that its normal vector $(H_{A},H_{B})$ is defined by a maximal number of equations of the form

for some indices $i,j,k,l$. This shows the first assertion. Moreover, since the facet contains $\pi(w\lambda_{AB})$ for some $w\in S_{ab}$ we obtain that

Following Klyachko, we shall call any $(H_{A},H_{B})$ that satisfies condition 1 of lemma 6.1 an extremal edge if it is in addition dominant and primitive (in the dual of the root lattice). There are only finitely many extremal edges and we shall denote them by $\mathcal{E}_{+}(a,b)$. The extremal edges span the extreme rays of the cubicles, which are the full-dimensional convex cones of elements $(H_{A},H_{B})$ cut out by a maximal set of inequalities of the form $H_{A,i}+H_{B,j}\geq H_{A,k}+H_{B,l}$. In other words, a cubicle is defined as a set of $(H_{A},H_{B})$ with fixed order of the $H_{A,i}+H_{B,j}$ and can therefore be encoded by a standard Young tableaux of rectangular shape $a\times b$ Klyachko (2004). This gives a straightforward way of computationally determining all extremal edges (see table 1).

We remark that standard tableaux that correspond to cubicles have also been called additive in the literature Vallejo (2014); Manivel (2014). Manivel has shown that they can be associated with minimal regular faces of the moment cone for which the corresponding Kronecker coefficients stabilize Manivel (1997, 2014). The corresponding extremal edges determine non-trivial facets of $C(a,b,ab)$ incident to this face Manivel (1997); Klyachko (2004). However, not all non-trivial facets can be obtained in this way.

Let $H=(H_{A},H_{B},H_{C},z)$ be the normal vector of a non-trivial facet of the moment cone $C(a,b,c)$, where $c\leq ab$. Then $(H_{A},H_{B})=0$ or $(H_{A},H_{B})$ is proportional to an element in the $S_{a}\times S_{b}$-orbit of $\mathcal{E}_{+}(a,b)$ (i.e., an extremal edge up to $S_{a}\times S_{b}$ and rescaling).

In view of (6.4), we may obtain finite and complete set of inequalities for $C(a,b,c)$ by taking any facet $H=(H_{A},H_{B},H_{AB},0)$ for $C(a,b,ab)$ and restricting $H_{AB}$ to its first $c$ components (that is, set $H_{C}$ to be the traceless part of $(H_{AB,1},\dots,H_{AB,c})$ and define $z$ accordingly). Such an inequality will not necessarily define a facet of $C(a,b,c)$, but all facets arise in this way. If we started with a non-trivial inequality then the claim follows from lemma 6.1. If we started with a trivial inequality then we either obtain a trivial inequality or $(H_{A},H_{B})=0$. ∎

Lemma 6.1 is efficient in finding candidates for facets of $C(a,b,ab)$, but not necessarily so for $C(a,b,c)$with $c<ab$. Indeed, while we know from the proof of corollary 6.2 that any facet of the latter can be obtained by “restriction” of a facet of the former, for each given facet there are many possible restrictions, as we need to pick a subset of $c$ components out of the $ab$ components of the given $H_{AB}$. If $c\ll ab$ then it can be more efficient to apply corollary 6.2 to all three of $(H_{A},H_{B})$, $(H_{A},H_{C})$ and $(H_{B},H_{C})$ (e.g., in the case of $a=b=c=4$). For this we will use the following lemma:

Let $(H_{A},H_{B})$ be an extremal edge. Then $H_{A}$ and $H_{B}$ are each either zero or primitive in the dual of the root lattice.

Case 2: $\mathfrak{m}=i\mathfrak{t}(b)$. Since the matrix with columns the roots of $A_{b-1}$ is totally unimodular (e.g., (Schrijver, 1986, p. 274, (18))), it follows that $S_{B}$ spans the root lattice. To find a contradiction, suppose that $H_{A}$ is neither zero nor primitive. Then we can write $H_{A}=nH^{\prime}_{A}$ for some $n>1$ and a non-zero element $H^{\prime}_{A}$ in the dual of the root lattice. But then,

for all $(\alpha,\beta)\in S$. Since the $S_{B}$ spans the root lattice, we conclude that also $H_{B}/n$ is an element of the dual of the root lattice. But then $H=(H_{A},H_{B})$ is not primitive, which is the desired contradiction. ∎

Let $H=(H_{A},H_{B},H_{C},z)$ be the normal vector of a non-trivial facet of the moment cone $C(a,b,c)$, where $c\leq ab$. Then $(H_{A},H_{B},H_{C})$ is proportional to an element in the $S_{a}\times S_{b}\times S_{c}$-orbit of

Since the following argument is equivariant under the Weyl group $S_{a}\times S_{b}\times S_{c}$, we may without loss of generality assume that $(H_{A},H_{B},H_{C})$ is dominant

Case 1: Only on component is non-zero, say, $H_{C}\neq 0$ and therefore $(H_{A},H_{B})=0$. We may rescale $H$ such that $H_{C}$ is primitive. Then $(H_{A},H_{C})$ and $(H_{B},H_{C})$ are also primitive and therefore extremal edges by corollary 6.2.

Case 2: At least two components are non-zero, say, $H_{A}$ and $H_{B}$. We may rescale $H$ such that $(H_{A},H_{B})$ is primitive. Since $H_{A}$ and $H_{B}$ are non-zero, lemma 6.3 shows that both $H_{A}$ and $H_{B}$ are individually primitive. It follows that three of $(H_{A},H_{B})$, $(H_{A},H_{C})$ and $(H_{B},H_{C})$ are primitive and therefore extremal edges by corollary 6.2. ∎

By theorem 1.1, any non-trivial facet is necessarily admissible. As admissibility is a Weyl group-invariant property, we immediately obtain the following corollary:

Let $H$ be the normal vector of a non-trivial facet of the moment cone $C(a,b,c)$, where $c\leq ab$. Then $H$ is proportional to an element in the $S_{a}\times S_{b}\times S_{c}$-orbit of

For any given $(H_{A},H_{B},H_{C})\in\mathcal{E}_{+}(a,b,c)$, there are in general several $z$ such that $H=(H_{A},H_{B},H_{C},z)$ is admissible (or no such $z$ at all). The sets $\mathcal{E}_{+}(a,b,c)$ and $\mathcal{E}_{+,\text{adm}}$ consist of dominant, primitive elements, and they can be easily obtained algorithmically from the sets of extremal edges.

Let $H$ be the normal vector of a non-trivial facet of the moment cone $C(a,b,c)$, where $c\leq ab$. Then $H$ is proportional to an element in

where $W(H_{0})$ denotes the set of triples $w=(w_{A},w_{B},w_{C})\in S_{a}\times S_{b}\times S_{c}$ such that $(w_{0}w_{A},w_{0}w_{B},w_{0}w_{C})$ is a triple of shuffles with respect to the components of $H_{0}$ whose lengths sum up to $\dim{M}(H_{0}<0)$.

We remark that it is straightforward to computationally generate all shuffles of a given length by adapting the algorithm of Effler and Ruskey Effler and Ruskey (2003). The upshot of corollary 6.6 then is that the moment cone $C(a,b,c)$ is cut out by those candidates in $\mathcal{E}(a,b,c)$ that are also Ressayre elements (together with the trivial inequalities):

We conclude this section with an illustrative example of the method.

Consider the moment cone $C(d,d,d)$ for any $d>1$. It is not hard to verify that all pairs formed from $(1,\dots,1,1-d)$ and $(d-1,-1,\dots,-1)$ are extremal edges (formally, we should divide by $d$ to work with primitive elements, but we refrain from doing so in the interest of readability). Therefore,

There are two possible values of $z$ which extend the above to an admissible element $H_{0}\in\mathcal{E}_{+,\text{adm}}(d,d,d)$. The first option is $z=-1$. However, the dimension of ${M}(H_{0}<0)$ is $2(d-1)^{2}+d$, and therefore strictly larger than the dimension of $\mathfrak{n}_{-}$. Thus the trace condition (4.1) can never be satisfied!

We now verify that $H$ is a Ressayre element. For this, observe that we have

for any $\psi=\sum_{ijk}\psi_{ijk}e_{ijk}\in{M}(H=0)$. It follows at once from theorem 1.1 that $(H,\lambda)\geq 0$ is a valid inequality for the moment polytope. We have thus obtained the well-known polygonal inequality Higuchi et al. (2003)

in a completely mechanical fashion. By symmetry, the two other inequalities obtained by permuting the subsystems $A$, $B$, and $C$ are also valid. We remark that the moment cone $C_{K(H=0)}({M}(H=0))$ is closely related to the Horn cone for $\operatorname{U}(d-1)$ discussed in section 7 below Christandl et al. (2012); Walter (2014); Littlewood (1958); Murnaghan (1955).

2 Computational Results

To verify that a given $H$ is a Ressayre element, we have implemented a computer program that works for arbitrary representations. To compute the set of candidates $\mathcal{E}(a,b,c)$ in the case of the one-body quantum marginal problem, we have used the strategy explained above. In table 2 we list some results obtained by our program in the symmetric scenario $a=b=c$, which corresponds to three quantum particles with the same number of degrees of freedom. While the moment cones $C(2,2,2)$ and $C(3,3,3)$ had already been computed in Higuchi et al. (2003); Bravyi (2004); Franz (2002); Higuchi (2003) using different methods, the cone $C(4,4,4)$ had been out of reach using current methods. In contrast, our method allows us the computation of $C(4,4,4)$ in a few minutes, since it does not rely on an intermediate computation of the higher-dimensional cone $C(4,4,16)$.

There are further variants of the one-body quantum marginal problem, such as for fermionic systems Coleman (1963); Ruskai (1969), where the facets amount to strengthenings of the classical Pauli exclusion principle. Our theory is also applicable to these scenarios, and it would be interesting to undertake a similar analysis of the facets that would allow the computation of the corresponding moment cones beyond what has been possible in the literature Klyachko and Altunbulak (2008). We refer to (Walter, 2014, §3) for initial investigations, where a family of fermionic inequalities has been proved for all local dimensions by using our method. We remark that we have verified numerically that the pure-state fermionic inequalities list in Klyachko and Altunbulak (2008) are correct (but not their sufficiency).

Horn Cone and Howe-Lee-Tan-Willenbring Invariants

In this section, we consider the representation of $G=\operatorname{GL}(d)\times\operatorname{GL}(d)\times\operatorname{GL}(d)$ on ${M}=\mathfrak{gl}(d)\oplus\mathfrak{gl}(d)$ given by

Any non-negative Hermitian matrix can be written in the form $a^{\dagger}a$; since the spectra of $aa^{\dagger}$ and $a^{\dagger}a$ are equal, the moment cone is equal to

where $\operatorname{spec}X$ denotes the eigenvalues of a Hermitian matrix $X$, ordered non-increasingly. We will call $C(d)$ the Horn cone in $d$ dimensions. As proved in Klyachko (1998); Knutson and Tao (1999), cf. Knutson and Tao (2001); Belkale (2006), $C(d)$ is cut out by the Horn inequalities, which we will recall below.

In the following, we will show that the trace condition and the Horn condition assume their familiar form in the context of Horn’s problem (e.g., Knutson and Tao (2001)), thereby justifying our terminology. For any $I=\{i_{1}<\dots<i_{r}\}\subseteq[d]$ of cardinality $r$, let $w_{I}$ denote the permutation that sends $[r]$ to $I$ and $[r]^{c}$ to the complement $I^{c}=\{i^{c}_{1}<\dots<i^{c}_{d-r}\}$ while preserving the order of each block. Then $E_{I}=w_{I}\cdot E_{r}=w_{I}E_{r}w_{I}^{-1}$ (we identify $w_{I}$ with a permutation matrix). However, $w_{I}$ does not satisfy the conditions of corollary 4.2. Instead, we shall write $E_{I}=\widetilde{w}_{I}\cdot E_{r}$ with $\widetilde{w}_{I}:=w_{I^{c}}w_{0}$, which reverses the order of each block. Then we find that $H_{IJK}$ satisfies the trace condition if and only if

where $H_{r}:=(E_{r},E_{r},E_{r})$. For the left-hand side, we compute

where we have defined $\lambda_{I}:=(d-r-(i_{a}-a))_{a=1}^{r}$, which is a sequence of non-increasing non-negative integers. For the right-hand side, observe that ${M}(H_{r}<0)$ consists of pairs of block matrices of the form $\left(\begin{smallmatrix}0&0\\ *&0\end{smallmatrix}\right)$, hence is isomorphic to $M_{d-r,r}^{2}$, the vector space of pairs of complex $(d-r)\times r$-matrix. We conclude that the trace condition for $H_{IJK}$ amounts to

It is not hard to see that (7.1) is precisely equivalent to Horn’s trace condition.

where we have defined $T_{I}:=\{X\in M_{d-r,r}:\braket{b}{X}{a}=0\text{ if }i_{a}>i^{c}_{b}\}$ etc. A short computation then reveals that the element $\kappa_{H_{IJK}}$ of proposition 1.2 identifies with the highest weight

of $\operatorname{U}(r)^{3}\times\operatorname{U}(d-r)^{3}$, where $\chi_{r}$ denotes the weight of the determinant representation of $\operatorname{U}(r)$. On the other hand, by using the isomorphism

the moment cone $K(H_{IJK}=0)$ on ${M}(H_{IJK}=0)$ gets likewise identified with the direct product of the Horn cones $C(r)\times C(d-r)$. Thus the Horn condition (proposition 1.2) asserts that $(\lambda_{I},\lambda_{J},\lambda_{K}-2(d-r)\chi_{r})\in C(r)$ as well as $(\lambda_{I^{c}},\lambda_{J^{c}},\lambda_{K^{c}}-r\chi_{d-r})\in C(d-r)$. These two conditions are in fact equivalent by a well-known duality of the Littlewood-Richardson coefficients. Therefore, we arrive at a single condition

This condition is not only necessary for $H_{IJK}$ to be a facet, but it is also sufficient for $H_{IJK}$ to be a valid inequality, as is well-known (e.g., Knutson and Tao (2001)). We will show in the next section how in the context of our work this can be deduced from the saturation conjecture and Schubert calculus (in fact, we shall see that (7.5) implies that $H_{IJK}$ is a Ressayre element). In particular, we obtain the familiar recursive definition of Horn’s inequalities: Define $\operatorname{Horn}(d,r)$ to be the set of all triples $I,J,K\subseteq[d]$ of cardinality $r<d$ that satisfy the trace condition (7.1) as well as

for all $s<r$ and all triples $(A,B,C)\in\operatorname{Horn}(r,s)$. Then (7.5) implies that $H_{IJK}$ satisfies the Horn condition if and only if $(I,J,K)\in\operatorname{Horn}(d,r)$. Therefore, our trace and Horn conditions are indeed a generalization of the classical conditions due to Horn.

2 The Determinant Polynomial

In this section, we will show that any $H_{IJK}$ that satisfies the trace and Horn condition is automatically a Ressayre element, i.e., that the determinant polynomial $\delta_{IJK}:=\delta_{H_{IJK}}$ is non-zero. To start, we note that by the saturation property of the Littlewood-Richardson coefficients Knutson and Tao (1999) (cf. Belkale (2006)), the Horn condition (7.5) implies that the following space of $\operatorname{GL}(r)$-invariants is non-zero: