Spectrahedral representation of polar orbitopes

Let $K$ be a compact Lie group and $V$ a finite-dimensional representation of $K$. The orbitope of a vector $x\in V$ is the convex hull $\mathscr O_x$ of the orbit $Kx$ in $V$. We show that if $V$ is polar then $\mathscr O_x$ is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope $\mathscr O_x^o$, which is the convex set polar to $\mathscr O_x$. We prove that $\mathscr O_x^o$ is the convex hull of finitely many $K$-orbits, and we identify the cases in which $\mathscr O_x^o$ is itself an orbitope. In these cases one has $\mathscr O_x^o=c\cdot\mathscr O_x$ with $c>0$. Moreover we show that if $x$ has"rational coefficients"then $\mathscr O_x^o$ is again a spectrahedron. This provides many new families of doubly spectrahedral orbitopes. All polar orbitopes that are derived from classical semisimple Lie can be described in terms of conditions on singular values and Ky Fan matrix norms.


Introduction
Let K be a compact Lie group, and let V be a finite-dimensional real representation of K. The orbitope of a vector x ∈ V , denoted O x , is the convex hull of the orbit Kx in V . Orbitopes are highly symmetric objects that are interesting from many perspectives, like convex geometry, algebraic geometry, Lie theory, symplectic geometry, combinatorial geometry or optimization. We refer to [18] for a broad overview with plenty of explicit examples.
Here our focus will be on properties of orbitopes that are particularly relevant to optimization, and more specifically, to semidefinite programming. We are interested in existence and explicit construction of spectrahedral representations for orbitopes and related convex bodies. For this we consider a particular class of group representations, namely polar representations of connected compact Lie groups. As far as the orbit structure is concerned, all such representations arise from Riemannian symmetric spaces M = G/K as actions of the isotropy group on the tangent space at a point. In other words, each polar representation comes from a Cartan decomposition g = k⊕ p of a real semisimple Lie algebra g, as the adjoint representation of K on p. Dadok [7] showed that these representations have very particular properties. At the same time they comprise all the familiar actions of the classical unitary, orthogonal or symplectic groups on (skew-) hermitian or symmetric matrices.
Our main results are as follows. We prove that every orbitope O x in a polar representation is a spectrahedron, i.e. an affine-linear slice of the psd matrix cone. In fact we produce an explicit linear matrix inequality representation for any such orbitope (Theorem 4.4). So far, this result was known only for a few scattered classes of examples. We also consider the dual convex body O o x , called the coorbitope of x. We prove that O o x always is the convex hull of finitely many K-orbits, and we identify those orbits explicitly (Corollary 6.6). In particular, we isolate the cases Date: October 6, 2020. This work was partially supported by DFG grants SCHE281/10-1 and SCHE281/ 10-2. when O x is a biorbitope (Theorem 6.7), meaning that O o x is an orbitope as well. Remarkably, O x is always self-polar up to positive scaling when it is a biorbitope (Theorem 6.9). Moreover, we show that whenever the orbitope O x can be "defined over the rational numbers Q", the coorbitope O o x is again a spectrahedron, and we find an explicit linear matrix inequality for it (Theorem 7.2). So far, only very few examples of doubly spectrahedral sets were known [20]. Our result provides many new series of sets with this property.
The main and all-important tool for our results is Kostant's convexity theorem [13]. It allows to reduce most questions considered here to a Cartan subspace, and even to a Weyl chamber. In this way the questions become polyhedral in nature.
The paper is organized as follows. Polar representations and orbitopes are recalled in Section 1. The general background on semisimple real Lie algebras and their restricted root systems is summarized in Section 2, as far as it will be needed here. Kostant's theorem is stated in Section 3, together with a few immediate consequences. Spectrahedral representations for polar orbitopes are constructed in Section 4. In Sections 5 and 6 we relate the facial structure of O x to the momentum polytope P x , with an emphasis on maximal faces of O x resp. facets of P x . Since the maximal faces of O x correspond to the extreme points of the polar set O o x , this allows us to identify biorbitopes. Doubly spectrahedral orbitopes are considered in Section 7. Finally, in Section 8 we list all polar orbitopes that are derived from semisimple Lie algebras of classical type. All these orbitopes have descriptions in terms of singular values of matrices over R, C or H. Typically, they consist of intersections of balls of various radii with respect to different Ky Fan matrix norms.
Some of the results presented here are taken from the 2018 doctoral thesis [12] of the first author, written under the guidance of the second.

Polar representations and orbitopes
1.1. We recall the notion of polar representation, following Dadok [7]. Let K be a Lie group with Lie algebra k, and let K → O(V ) be a linear representation of K on a (finite-dimensional) real vector space V , preserving a fixed inner product. For every x ∈ V , the linear subspace a x = (kx) ⊥ of V meets every K-orbit [7, Lemma 1]. A vector x ∈ V is said to be regular if the orbit Kx has maximal dimension. The subspaces a x , for x regular, are called the Cartan subspaces of the representation. The Cartan subspaces are all K-conjugate if, and only if, they are orthogonal to the K-orbits passing through them [7,Prop. 2]. The representation ρ is said to be polar if these equivalent conditions are satisfied.
Every Riemannian symmetric space M gives rise to a polar representation, namely the action of the isotropy group on the tangent space T e (M ) at a point e ∈ M . In other words, let G be a connected real semisimple Lie group with Lie algebra g, let g = k ⊕ p be a Cartan decomposition, and let K ⊆ G be the analytic subgroup corresponding to k. Then K is a maximal compact subgroup of G, and the adjoint action of K on p is an example of a polar representation. This representation is irreducible if and only if g is simple as a Lie algebra. Conversely, as far as the orbit structure is concerned, these are the only examples of polar representations of connected Lie groups: [7,Proposition 6]) Let V be a polar representation of a connected Lie group H. There is a real semisimple Lie algebra g with Cartan decomposition g = k ⊕ p, together with a vector space isomorphism f : V i be the irreducible decomposition of an arbitrary polar representation of K. Then each irreducible summand V i is again a polar representation of K (Dadok [7,Theorem 4] . Indeed, such a direct product decomposition holds for the K-orbit of x by Dadok's theorem, and hence it holds for the convex hulls as well. In other words, every polar orbitope is a cartesian direct product of irreducible polar orbitopes.

1.4.
Recall that a spectrahedron in R n is the solution set of a linear matrix inequality (LMI). So S ⊆ R n is a spectrahedron if there exist complex hermitian matrices A 0 , . . . , A n of some size d × d such that where A 0 means that A is positive semidefinite (all eigenvalues are nonnegative). Note that an LMI with complex hermitian d × d matrices may be converted into an equivalent LMI with real symmetric 2d × 2d matrices, which is why spectrahedra are often defined via real symmetric LMIs.
1.5. Let V be a linear representation (real and finite-dimensional) of a compact Lie group K. Given x ∈ V , the convex hull conv(Kx) of the orbit of x in V is called the (K-) orbitope of x. We usually denote it by O x = conv(Kx), assuming that K and V are understood. The orbitope O x is a compact convex set on which K acts, and whose set of extreme points coincide with the orbit Kx.
We will study orbitopes O x = conv(Kx) in polar representations V of compact connected Lie groups K. Using Proposition 1.2, we can and will always assume that V = p where g = k ⊕ p is a Cartan decomposition of a real semisimple Lie algebra g, and that the action is the adjoint action of the analytic subgroup K of Aut(g).
1.6. If V is a vector space over R, the dual vector space is denoted by V ∨ = Hom(V, R). The convex hull of a set M ⊆ V is written conv(M ). Our notation for matrix groups and matrix Lie algebras tries to follow the conventions in [11]. In particular, SU (n), SO(n), Sp(n) are the classical compact Lie groups, su(n), so(n), sp(n) are their Lie algebras, etc. The diagonal n × n matrix with diagonal entries a 1 , . . . , a n is denoted diag(a 1 , . . . , a n ).

Background on semisimple real Lie algebras
We use standard notation and terminology for semisimple Lie groups and Lie algebras, and we'll recall it briefly here. As a general reference we refer to Knapp's monograph [11], in particular to Chapter 6.
2.1. Let g be a semisimple Lie algebra over R, hence a finite direct sum of simple (nonabelian) Lie algebras over R. Recall that if g is simple then either g has a structure as a (simple) Lie algebra over C, or else g C := g ⊗ C is a simple Lie algebra over C.
Let θ be a Cartan involution on g and g = k ⊕ p the corresponding Cartan decomposition. With respect to the Killing form −, − of g, the decomposition k ⊕ p is orthogonal, and the restriction of −, − to k (resp. p) is negative (resp. positive) definite.

2.2.
Choose a maximal commutative subspace a of p, and let Σ ⊆ a ∨ = Hom(a, R) be the system of restricted roots of (g, a). This is an abstract root system, possibly non-reduced. Fixing an ordering on a ∨ we have the sets Σ + ⊆ Σ of positive restricted roots and Γ = {β 1 , . . . , β n } ⊆ Σ + of simple restricted roots.

2.3.
Let t be a maximal commutative subalgebra of the centralizer Z k (a) of a in k. Then h := it ⊕ a is a commutative subalgebra of g C (the complexification of g) for which h C is a Cartan algebra of g C . The rank of g C will be denoted by l = dim(h), the real rank of g by n = dim(a).
The Killing form of g C restricted to h is a euclidean inner product on h, denoted by −, − . All roots of (g C , h C ) are real-valued on h. We let ∆ ⊆ h ∨ = Hom(h, R) be the root system of (g C , h C ). Extending the ordering on a ∨ suitably to h ∨ (e.g. take the lexicographic order on h ∨ = (a + it) ∨ with a before it [11, p. 377]), we let ∆ + ⊆ ∆ be the positive roots and Π = {α 1 , . . . , α l } ⊆ ∆ + the simple roots.

2.4.
Let G = Aut(g) 0 , the identity component of the automorphism group of g. The Lie algebra of G is naturally identified with g. The analytic subgroup K ⊆ G with Lie algebra k is a maximal compact subgroup of G, and K acts on (k and) p via the adjoint action. We denote this action simply by gx := Ad(g)x, for g ∈ K and x ∈ p. Every element of p is K-conjugate to an element of a. Note that the K-action preserves the quadratic (Killing) form on p. This action is a polar representation of K, and as far as the orbit structure is concerned, every polar representation of a connected Lie group arises in this way (Proposition 1.2).
2.6. From the inner product −, − on h we get linear isomorphisms h ∼ → h ∨ and a ∼ → a ∨ . We use them to transfer the inner product from h to h ∨ and from a to a ∨ . For every restricted root β ∈ Σ let s β : a ∨ → a ∨ denote the root reflection γ → γ − 2 β,γ |β| 2 β. We always write W = s β : β ∈ Σ for the (restricted) Weyl group of (g, a). Via the identification a ∼ → a ∨ we consider W as a reflection group on a as well: It is well known that W is naturally isomorphic to N K (a)/Z K (a) [11, 6.57].
We always denote by the (closed) Weyl chamber for the action of W . So C is a polyhedral convex cone, and every element of a is W -conjugate to a unique element of C.

Kostant's convexity theorem
We assume the setup of Section 2. So g is a semisimple real Lie algebra with Cartan decomposition g = k ⊕ p and maximal abelian subspace a of p. This gives us the system Σ ⊆ a ∨ of reduced roots of (g, a), on which the reduced Weyl group W acts. After fixing an ordering we have the simple positive roots Γ and the Weyl chamber C ⊆ a.
The key technical tool for this paper is Kostant's convexity theorem, together with its consequences. To a large extent, it allows to reduce the study of the polar orbits Kx and their orbitopes O x to a Weyl chamber, whereby the K-action on p gets replaced by the W -action on a. We now recall this theorem.
where µ 1 , . . . , µ n ∈ a ∨ is the dual basis of Γ as in 2.7. For x ∈ a, the convex hull P x := conv(W x) of the (restricted) Weyl group orbit of x will play a central role. In Hamiltonian geometry, P x is called the momentum polytope associated to x [17], a term that we will adopt. According to Kostant, P x is characterized as follows: Recall that the Killing form of g restricts to a euclidean inner product on p. Let π : p → a denote the orthogonal projection from p to a. Kostant's convexity theorem says: We record a few immediate consequences. Recall that O x = conv(Kx) denotes the convex hull of the K-orbit of x in p.
Proof. Theorem 3.3 implies π(O x ) = P x , and hence O x ∩ a ⊆ P x . The reverse inclusion is obvious since W x ⊆ Kx. The second assertion follows from the first, since every K-orbit in p meets a.
Corollary 3.5. If x, y ∈ a are K-conjugate then they are W -conjugate. Every K-orbit in p intersects C in a unique element.
Proof. Both statements are equivalent, so it suffices to prove the second. Let x, y ∈ C be K-conjugate. Then P x = P y by 3.3, so 3.
Example 3.6. Let n ≥ 2 and g = sl(n, R), the real n × n matrices of trace zero. The resulting polar representation is the natural action of the special orthogonal group K = SO(n) on p = sym 0 (n, R), the trace zero symmetric real matrices. The standard choice for a is to take all diagonal matrices in p. The (restricted) Weyl group is W = S n , the symmetric group, acting by permutation of the diagonal elements of x ∈ a. So the momentum polytope P x is the permutahedron of x, namely the convex hull of all permutations of x. Kostant's theorem 3.3 specializes to the (symmetric) Schur-Horn theorem (see [10] 4.3.45 and 4.3.48, [14], [18]). Likewise, the hermitian version of the Schur-Horn theorem arises from 3.3 if we take k = su(n, C) and g = k C , so we get the adjoint action of the special unitary group K = SU (n) on (traceless) hermitian n × n matrices, with the analogous theorem.
In Section 8 we will discuss examples of polar orbitopes in a systematic way.

Polar orbitopes as spectrahedra
4.1. In general, orbitopes under compact connected linear groups K can't be expected to be spectrahedra. Examples are suitable SO(2)-orbitopes like the 4dimensional Barvinok-Novik orbitope [23], or the Grassmann orbitope G 3,6 of dimension 20 [18,Theorem 7.6], where the group K = SO(6) is even semisimple. Using results from [21] it is easy to construct orbitopes under the 2-torus K = SO(2) × SO(2) which are not even linear projections of spectrahedra, for example the convex hull of {(s, In fact it can be shown that in sufficiently high dimension, "most" (in a suitable sense) orbitopes under SO(2) × SO(2) fail to be projected spectrahedra [12].
Given this general situation, we think the following theorem all the more remarkable: Any orbitope in a polar representation V of a connected Lie group K is a spectrahedron.

4.3.
By Proposition 1.2 we may assume that g = k ⊕ p is a Cartan decomposition of a real semisimple Lie algebra, and that K has Lie algebra k and acts on V = p via the adjoint representation. We provide an explicit linear matrix inequality description for any such orbitope.
Fix a, h together with compatible orderings (2.3), and let otherwise notation be as in Section 2. In particular, Π = {α 1 , . . . , α l } is the system of simple roots of (g C , h C ), and Γ = {β 1 , . . . , β n } is the system of simple restricted roots of (g, a). For each j = 1, . . . , n choose an index i = i(j) ∈ {1, . . . , l} with r(α i(j) ) = β j , and let ρ j : g C → End(V j ) be the (complex) irreducible representation of g C with highest weight ω i(j) . So ρ j is the i(j)-th fundamental representation of g C , see 2.7. There exists an hermitian inner product on V j making ρ j (x) self-adjoint for all x ∈ p (Lemma 4.6(a) below). In particular, ρ j (x) has real eigenvalues for every x ∈ p. A more precise version of Theorem 4.2 is: consists of all y ∈ p such that for each j = 1, . . . , n, all eigenvalues of ρ j (y) are less or equal than the largest eigenvalue of ρ j (x). In other words, Upon choosing orthogonal bases of the representation spaces V j , this is an explicit description of O x by linear matrix inequalities, involving hermitian matrices in general. We remark that the fundamental representations ρ j of g C are very well known and understood [24], in particular so for the classical Lie algebras.
Remark 4.5. Theorem 4.4 implies in particular that all faces in a polar orbitope are exposed, since this is true in every spectrahedron. This fact was proved before by Biliotti, Ghigi and Heinzner [1]. Some spectrahedral representations contained in, or closely related to, Theorem 4.4 were constructed by Sanyal, Sottile and Sturmfels [18], namely for symmetric Schur-Horn orbitopes (see 3.6), and also for skew-symmetric Schur-Horn orbitopes (see 8.7) and Fan orbitopes (see 4.8 and 8.5). These latter orbitopes, as considered in [18], do not directly fall under the assumptions of 4.4, since the groups acting there are not connected (full instead of special orthogonal groups). It is not hard, however, to recover the results from [18] in our setup, see Remark 8.5. In Saunderson-Parrilo-Willsky [20], a spectrahedral representation for the convex hull of the special orthogonal group SO(n) (and for its dual convex body) was found, see also Remarks 4.9 and 7.7 below. Otherwise we believe that our result is new.
Before we give the proof of Theorem 4.4, recall the following well-known facts.
(a) There exists an hermitian inner product on V that makes ρ(x) self-adjoint for every x ∈ p.
So let x, y ∈ C. See 2.7 to 2.9 for notation in the following discussion. Since ρ j has highest weight ω i(j) , the largest eigenvalue of ρ j (y) is ω i(j) (y) (Lemma 4.6(b)). Hence y ∈ O(x) if and only if ω i(j) (y) ≤ ω i(j) (x) = c j for j = 1, . . . , n. By Lemma 2.9, the restriction r(ω i(j) ) ∈ a ∨ is a positive scalar multiple of µ j (recall that if and only if µ j (y) ≤ µ j (x) for j = 1, . . . , n. By Proposition 3.2(b) this is equivalent to y ∈ P x . On the other hand, y ∈ P x is equivalent to y ∈ O x by Corollary 3.4.
Example 4.8. We illustrate the statement of Theorem 4.4. Let n ≥ 3, and consider the action of (g, h) ∈ K = SO(n) × SO(n) on x ∈ M n (R) by gxh t . This is a polar representation of K that arises from the split real form of D n , i.e. from the simple Lie algebra Note that g consists of all block matrices with u, v, w ∈ M n (R) and u, v skew-symmetric, and p ⊆ g is the subspace of all symmetric such matrices, i.e. with u = v = 0. As maximal commutative subspace of p we take the space a of all matrices (3) with u = v = 0 and w = diag(x 1 , . . . , x n ) diagonal. Denote such a matrix by x = (x 1 , . . . , x n ). The simple roots By Lemma 3.2, the momentum polytope for x ∈ C is described by P . . , n}. The first fundamental representation ρ 1 of g is the natural representation (2), the higher ones are the exterior powers . Moreover, ρ n−1 and ρ n are the two half-spin representations. So dim(ρ i ) = 2n i for i ≤ n − 2, and dim(ρ n−1 ) = dim(ρ n ) = 2 n−1 . Expressing the ρ i by matrices one arrives at explicit spectrahedral representations of the K-orbitopes O x , for x ∈ M n (R). These representations are closely related to [18,Theorem 4.7], where the group acting is O(n) × O(n) instead of our K. For special x this may be different. We illustrate this remark with just one example, deferring a detailed discussion to a later occasion.
Consider again the action of K = SO(n) × SO(n) on M n (R), as in 4.8, and take x = I n ∈ M n (R), the identity matrix, so x = (1, . . . , 1) in notation of 4.8. The orbitope is O x = conv SO(n), the convex hull of the group SO(n). Due to the special choice of x, the description of the momentum polytope simplifies. For y ∈ C, the condition y 1 ≤ 1 implies k i=1 y i ≤ k for every k = 1, . . . , n. So P x ∩ C is already described by the two inequalities λ 1 (y) ≤ 1 and λ n−1 (y) ≤ n−2 2 . We conclude that O 5. Face correspondence 5.1. As before let g = k ⊕ p be a Cartan decomposition of a semisimple real Lie algebra g. For general setup and notation see Section 2. We continue to denote the orthogonal projection p → a by π. Let x ∈ p, and let P x be the momentum polytope of x (3.1). If Q is any face of P x , then For any w ∈ W there exists g ∈ N K (a) with w = gZ K (a). The projection π : p → a is easily seen to commute with the action of N K (a), and therefore F wQ = gF Q holds. Hence the assignment Q → F Q induces a map from W -orbits of faces of P x to K-orbits of faces of O x .
The following theorem asserts, in particular, that this map is bijective. This fact was originally proved by Biliotti, Ghigi and Heinzner [1, Theorem 1.1]. We give a new proof that we think is considerably easier. Note however that [1] proves a more precise result, implying in particular that the faces F Q of O x are themselves orbitopes under suitable groups.
(a) There exists a face Q of P x and an element g ∈ K such that F = gF Q .
, then there exists w ∈ W such that wQ ⊆ Q ′ . In particular, Q → F Q induces a bijective correspondence between W -orbits of faces of P x and K-orbits of faces of O x , compatible with inclusion of faces.
For the proof observe the following lemma: . Let x ∈ p, let Q be a face of P x , and let y ∈ P x with P y ∩ Q = ∅. Then wy ∈ Q for some w ∈ W .
Proof. We can assume Q = P x , so there is a supporting hyperplane H ⊆ a of P x with Q = H ∩ P x . Since P y ⊆ P x and H ∩ P y is not empty, the hyperplane H is a supporting hyperplane of P y as well. In particular, Q ′ = H ∩ P y is a face of P y , and therefore contains an extreme point y ′ of P y . Thus y ′ ∈ Q ′ ⊆ H ∩ P x = Q, and y ′ = wy for some w ∈ W since P y = conv(W y).

Proof of Theorem 5.2.
(a) Let F be a face of O x . By Theorem 4.2, O x is a spectrahedron, so all faces are exposed. Hence there exist z ∈ p and c ∈ R such that H = {y ∈ p : y, z = c} is a supporting hyperplane of O x with H ∩ O x = F . Upon replacing F with gF for some g ∈ K we can assume z ∈ a, since z is K-conjugate to an element of a. Then H ∩ a is a supporting hyperplane of P x , and so Q := H ∩ P x is a face of P x . Clearly F = F Q .
(b) By (a) it suffices to show: If Q, Q ′ are faces of P x , and if gF Q ⊆ F Q ′ for some g ∈ K, then there exists w ∈ W with wQ ⊆ Q ′ . Let y ∈ relint(Q). Since Q ⊆ F Q we have gy ∈ F Q ′ , and therefore π(gy) ∈ Q ′ . On the other hand, π(gy) ∈ π(Ky) = P y . So Lemma 5.3 applies and shows wy ∈ Q ′ for some w ∈ W . Since y ∈ relint(Q), this implies wQ ⊆ Q ′ .
In particular, if Q, Q ′ are faces of P x for which F Q and F Q ′ are K-conjugate, then Q and Q ′ are W -conjugate.
Recall that a face Q of a polytope P is called a facet if dim(Q) = dim(P ) − 1. If the compact group K acts on V and x ∈ V , the polar set of the orbitope O x = conv(Kx) is called the coorbitope of x [18]. Clearly the group K acts on O o x , but in general O o x won't be a K-orbitope. Below (5.9 and 6.7) we'll identify those cases when this happens.

5.7.
For any irreducible abstract root system (V, Σ) (possibly non-reduced), the Weyl group W acts irreducibly on V . For any 0 = x ∈ V , the polytope P x = conv(W x) therefore contains an open neighborhood of the origin. If (V, Σ) is not necessarily irreducible and x ∈ V , it follows that the polytope P x is full-dimensional if and only if every irreducible component of Σ contains a root α with α(x) = 0. Moreover in this case, 0 is an interior point of P x .
Let K → SO(V ) be a polar representation, and let x ∈ V . The previous discussion implies that when V is irreducible, the origin is an interior point of O x as soon as x = 0. When V is an arbitrary polar representation of K, let V = V i be the decomposition into irreducible K-modules as in 1.3, and let When studying the orbitope O x , we can obviously assume that O x is full-dimensional (or equivalently, 0 is an interior point of O x ), by the previous discussion. : ∀ x ∈ F x, y = 1}. Then F is an exposed face of O o , and F → F restricts to an inclusion-reversing bijection between exposed faces of O and exposed faces of O o , with inverse map G → G.
To prove the proposition we can assume x ∈ C. Let F ac(x) be a set of representatives of the W -orbits of facets of P x . Let z be an exposed extreme point of O o x = {y ∈ p : ∀ g ∈ K gx, y ≤ 1}, and write G z := z = {y ∈ O x : y, z = 1}. By the fact just recalled, G z is a maximal face of O x , and so G z = gF Q for some Q ∈ F ac(x) and some g ∈ K (5.2, 5.5). It is easily checked that G hz = hG z for any h ∈ K. If u ∈ O o x is another exposed extreme point, and if G u = hF Q for some h ∈ K, then G gh −1 u = gh −1 G u = G z , whence gh −1 u = z, so u and z are K-conjugate. This shows that the exposed extreme points of O o x consist of finitely many K-orbits, each of them corresponding to a different W -orbit of facets of P x . Since exposed extreme points are dense within all extreme points (Straszewicz' theorem, e.g. [22] 1.4.7), we conclude that all extreme points of O o x are exposed. For each facet Q of P x we claim conversely that F Q is an (exposed) extreme point of O o x . Indeed, otherwise F Q would be a minimal exposed face of O o x of dimension ≥ 1. But such a face cannot exist, since all extreme points of O o x are exposed. Altogether we have proved the bijection between K-orbits of extreme points of O o x and W -orbits of facets of P x . We will determine these cases explicitly in the next section, after having discussed the facets of P x in more detail.
Note that under the equivalent conditions of 5.9, the coorbitope O o x is a spectrahedron itself, by Theorem 4.2. In Sect. 7 we will uncover many more cases where this holds.

Facets of the momentum polytope
In the previous section, a close relation was established between the faces of the orbitope O x and the faces of the momentum polytope P x . The faces of the latter can be described in terms of root data. We start by recalling this description.
6.1. Let (V, Σ) be an abstract root system (which may be non-reduced), fix an ordering ≤ on V , and let Γ = {β 1 , . . . , β n } be the corresponding system of simple positive roots. Let µ 1 , . . . , µ n be the dual basis of Γ in V , so β i , µ j = δ ij for 1 ≤ i, j ≤ n. Let W be the Weyl group, and let C = {x ∈ V : β i , x ≥ 0, 1 ≤ i ≤ n}, the closed Weyl chamber associated to Γ.
Let x ∈ C be a given point and write P x = conv(W x). A subset I ⊆ Γ is said to be x-connected if every connected component of I contains a root β with β, x = 0. (Of course, connectedness notions refer to the Dynkin graph.) Let W I be the subgroup of W generated by the root reflections s β where β ∈ I. The following result is quoted from Casselman [5, Theorem 3.1], where it is proved in the more general context of arbitrary finite Coxeter groups. As Casselman remarks, the result is already implicit in much older work of Satake [19] and Borel-Tits [3]. A related discussion can also be found in [2, §6] and [1, §4]. Here we are mainly interested in the facets of P x . Assume that P x is fulldimensional in V , or equivalently, that every connected component of Γ contains a root β with β, x = 0 (see 5.7). For facets the theorem gives: Then P x (i) is a facet of P x . Conversely, every facet of P x is W -conjugate to P x (i) for a unique index i ∈ I(x).
Proof. Let i ∈ I(x). Clearly, P x (i) is a face of P x , and is proper since P x is fulldimensional. Let W ′ := W Π βi = s β : β ∈ Π, β = β i . Then conv(W ′ x) ⊆ P x (i) holds since µ i (s βj (y)) = µ i (y) for every j = i. By Theorem 6.2, conv(W ′ x) is a facet of P x , so we have equality. The remaining assertion follows directly from Theorem 6.2 as well.
We will also use the following (well-known) fact: Proof. It is enough to prove this in the case where the root system is simple and 0 = x ∈ C. Since β j (x) ≥ 0 for all j = 1, . . . , n and β j (x) > 0 for at least one j, the lemma follows from the fact that the inverse of the Cartan matrix has strictly positive coefficients [15]. 6.5. Now again consider the adjoint representation of K on p. We apply Corollary 6.3 to the system Σ of restricted roots of (g, a). In this way we are going to identify explicitly the K-orbits of extreme points of the coorbitope O o x (see Proposition 5.8). Let x ∈ C ⊆ a such that P x is full-dimensional, and let I(x) = {i ∈ {1, . . . , n} : Γ {β i } is x-connected} (as in 6.3). For every i ∈ I(x) we have the facet P x (i) = {y ∈ P x : µ i (y) = µ i (x)} of P x . For easier notation, let us write F i instead of F Px(i) = {y ∈ O x : π(y) ∈ P x (i)}. By Theorem 5.2, F i is a maximal proper face of O x , and every maximal proper face is K-conjugate to F i for a unique index i ∈ I(x) (using also 6. x . Using 6.3 we identify these points as follows. Let i ∈ I(x), let h µi ∈ a be the element satisfying h µi , y = µ i (y) for all y ∈ a. Note that C is the cone generated by h µ1 , . . . , h µn . By Lemma 6.4 we have Indeed, the element z := h µi /µ i (x) satisfies y, z = π(y), z = µ i (π(y))/µ i (x) for all y ∈ O x . Since π(y) ∈ P x (3.4), this shows y, z ≤ 1, with equality if and only if π(y) ∈ P x (i). So we have proved: A particularly interesting case arises when |I(x)| = 1, i.e. the polytope P x has only one W -orbit of facets. By 5.9 it is equivalent that the coorbitope O o x is a K-orbitope itself. We will say that O x is a (K-) biorbitope in this case, and we can characterize it as follows: Then O x is a K-biorbitope if, and only if, the Lie algebra g is simple, the restricted root system Γ is not of type D n (n ≥ 4) or E n (n = 6,7,8), and the following holds: There is a simple restricted root β ∈ Γ with γ(x) = 0 for all γ ∈ Γ {β}, and such that Γ {β} is connected.
In other words, the condition is saying that β(x) = 0 for only one simple restricted root β, that β sits at an end of the restricted Dynkin graph Γ, and that Γ has at most one other end. In 8.12 and 8.13 we'll make all biorbitopes explicit for the classical Lie algebras.
Proof. By Corollary 6.6, O x is a biorbitope if and only if P x has only one W -orbit of facets. One sees immediately that this can hold only when the restricted root system Σ is irreducible. Therefore we may assume that the Lie algebra g is simple.
Let Γ 1 = {β ∈ Γ : β(x) = 0}. We say that β ∈ Γ is a boundary root if Γ {β} is connected. If Γ 1 contains a non-boundary root then P x has two non-conjugate facets. Indeed, choose two different boundary roots β i and β j . Then P x (i) and P x (j) are both facets of P x , and are not W -conjugate, according to Corollary 6.3. Exactly the same argument works if Γ 1 contains two different boundary roots β i , β j .
So all facets of P x can only be W -conjugate if Γ 1 consists of just one single boundary root. Conversely, if this is the case then the conjugacy classes of facets of P x correspond precisely to the remaining boundary roots. This proves the equivalence in the theorem, since D n (n ≥ 4) and E n (n = 6,7,8) are precisely the simple root systems with more than two boundary roots. Example 6.8. Let n ≥ 4 and g = so(n, n). For the description of a, C, the α i and λ i see 4.8. The restricted root system is of type D n . If we take x = (1, . . . , 1) as in 4.9, we have α i (x) = 0 for all i = n, so I(x) = {n} is a singleton set. Yet P x has two W -orbits of facets, represented by the facets P x (1) = {y ∈ P x : y 1 = 1} and P x (n − 1) = {y ∈ P x : y 1 + · · · + y n−1 − y n = n − 2}. Hence the orbitope O x = conv SO(n) has two K-orbits of maximal dimensional faces, a fact already proved in [18,Theorem 4.11]. This means that the coorbitope

Doubly spectrahedral orbitopes
Following Saunderson, Parrilo and Willsky [20] we use the term doubly spectrahedral convex sets to refer to convex sets S in R n for which both S and the polar convex set S o are spectrahedra. As remarked in [20], it is a very special phenomenon that the polar set of a spectrahedron is again a spectrahedron. Apart from polyhedra (which have this property for obvious reasons) it seems that only one other distinct family of doubly spectrahedral convex sets is known, namely the homogeneous convex cones (Vinberg [25] and Chua [6], see [20, 6.1]). In addition, the convex hull of the matrix group SO(n) is doubly spectrahedral for every n ≥ 1, by the main theorem of [20]. In fact, explicit spectrahedral representations for both conv SO(n) and SO(n) o were constructed in [20].
Below we show that all polar orbitopes O x with "rational coordinates" are doubly spectrahedral as well. Moreover we'll give explicit linear matrix inequality representations for those orbitopes and their polars. As a particular case, we recover the results from [20], see Remark 7.7 below.
Let g = k ⊕ p be a real semisimple Lie algebra with Cartan decomposition, and consider the adjoint representation of K on p as before. We use notation and conventions from Section 2. In particular, a is a maximal abelian subspace of p, and C ⊆ a is the Weyl chamber with respect to the fixed ordering on a. As before, let Γ = {β 1 , . . . , β n } ⊆ a ∨ be the simple positive restricted roots.
Definition 7.1. Given x ∈ a, we say that the K-orbitope O x = conv(Kx) has rational coordinates if there is b ∈ R such that β j (x) ∈ Qb for j = 1, . . . , n.
Since any two choices of a ⊆ p are conjugate under K [11, 6.51], and since every K-orbit in p intersects a in one full W -orbit (Corollary 3.5), the property of having rational coordinates depends only on the orbit Kx, and neither on the choice of a nor on the particular choice of a representative of Kx in a. Given x ∈ a, let P x = conv(W x) be the momentum polytope of x as before, and let P o x be the polar set of P x in a, i.e. P o x = {y ∈ a : ∀ w ∈ W wx, y ≤ 1}. We have the following lemma: If y ∈ C then wx, y ≤ x, y for every w ∈ W [13, Lemma 3.2]. Therefore, if x, y ≤ 1 then y ∈ P o x . The opposite inclusion is trivial from the definition. Recall that π : p → a denotes orthogonal projection to a. Forming the polar convex body commutes with projection to (or intersection with) a: Proof. For y ∈ a and z ∈ p we have y, π(z) = y, z . From this the lemma follows immediately.

7.5.
Proof of Theorem 7.2. We can assume x ∈ C. Since x has rational coordinates we can assume β j (x) ∈ Q for j = 1, . . . , n, after scaling x with a suitable positive real number. So there exist rational numbers c j ≥ 0 such that x, y = n j=1 c j µ j (y) for all y ∈ a (namely c j = β j (x)). Hence, and by Lemma 2.9, there are an integer k ≥ 1 and an integral dominant weight ω ∈ h ∨ of (g C , h C ) such that k · x, y = ω(y) for all y ∈ a. By the highest weight theorem, there is an irreducible representation ρ of g C with highest weight ω. From Lemmas 7.3 and 7.4 we get Indeed, both sets in (5) are K-invariant, so it suffices to check that their intersections with C coincide. For y ∈ C the largest eigenvalue of ρ(y) is ω(y) = k x, y (Lemma 4.6(b)). So (5) follows indeed from (4), and the theorem is proved.
Remark 7.6. The highest weights of irreducible representations of (g C , h C ) are the nonnegative integral linear combinations of the fundamental weights ω i = 1 2 |α i | 2 λ i (i = 1, . . . , l). The restriction of ω i ∈ h ∨ to a is 0 if r(α i ) = 0, and is 1 2m |α i | 2 µ j if r(α i ) = β j = 0, where m ∈ {1, 2} is the number of simple roots in Π that restrict to β (Lemma 2.9). Since the |α i | 2 are explicit rational numbers, we see how to find, for given x ∈ a with rational coordinates, a real number c > 0 and an integral dominant weight ω of (g C , h C ) such that cx, y = ω(y) for all y ∈ a.
Example 7.7. We illustrate the previous remark with the example already studied in 4.8, so consider the action of K = SO(n) × SO(n) on M n (R) for n ≥ 3. We take the identity matrix x = I n as in 4.9 and are looking for a linear matrix inequality description of the coorbitope O o x = (conv SO(n)) o . The orbitope O x has rational coordinates since α i (x) = 0 for 1 ≤ i < n and α n (x) = 2 (see 4.8). Since x, y = n i=1 y i = 2λ n (y) for y ∈ a, the procedure in 7.6 leads to the spectrahedral representation where ρ n is the n-th fundamental representation. This is in accordance with Saunderson, Parrilo and Willsky [20,Theorem 1.1], c.f. the remark in 4.9.
x is the convex hull of two K-orbits, but not of one (Example 6.8).

Examples
We describe all irreducible polar representations that arise from semisimple Lie algebras of classical type. Roughly, these are the well-known unitary group actions on rectangular matrices, and on (skew-) hermitian resp. (skew-) symmetric square matrices, over K = R, C or H, where H is the skew-field of Hamilton quaternions. (For K = H there is no action on (skew-) symmetric matrices.) In each case we mention a standard choice of a maximal abelian subspace a and of a Weyl chamber C. Using Kostant's results, in particular Proposition 3.2 and Corollary 3.4, this allows us to give explicit descriptions of the respective orbitopes in all cases. Naturally, this uses the description of the (reduced) root systems and of the fundamental weights, for which there are many references (e.g. [11], [24], [16]). We will see that the corresponding orbitopes can be described in terms of Ky Fan norm balls, which in turn are defined using singular values of matrices.

8.1.
First recall the singular value decomposition. Let always K be one of R, C or H, and let U (n, K) denote the unitary group over K, i.e. U (n, K) = {g ∈ M n (K) : gg * = I n } where g * = g t . So U (n, R) = O(n) is the real orthogonal group, U (n, C) = U (n) is the usual (complex) unitary group and U (n, H) = Sp(n) is the symplectic group. Given a (rectangular) matrix x ∈ M m×n (K) where m ≥ n, there exist unitary matrices u ∈ U (m, K) and v ∈ U (n, K) such that uxv * = diag(a 1 , . . . , a n ) with real numbers a 1 ≥ · · · ≥ a n ≥ 0. The a i are uniquely determined by x, they are called the singular values of x and denoted σ i (x) := a i (1 ≤ i ≤ n). For K = R or C this is classical (e.g. [10, 2.6]), here a 2 1 , . . . , a 2 n are the eigenvalues of the psd hermitian matrix x * x. For K = H, essentially the same is true (with eigenvalues replaced by right eigenvalues), but less well-known; see [26, 7.2] and [9, 5.7] for details.

8.2.
Let K = R, C or H, and let V = M m×n (K) with m ≥ n. For k = 1, . . . , n and x ∈ V let x k := σ 1 (x) + · · · + σ k (x), sum of the k largest singular values of x. This defines a matrix norm on the space of matrices, the k-th Ky Fan norm ( [8] and [10], 7.4.8 and 7.4.10). In particular, all balls with respect to any of these norms are convex. Note that the first Ky Fan norm x 1 = σ 1 (x) is the operator norm of x. The last one x n = n i=1 σ i (x) is called the nuclear norm and often denoted x * .
In view of Remark 1.3, we restrict our discussion of classical polar orbitopes to orbitopes that arise from simple real Lie algebras g of classical type.
together with the action of (u, v) ∈ K on x ∈ V = M m×n (K) by uxv * . This is a polar representation, arising from the simple Lie algebra g = so(m, n) resp. g = su(m, n) resp. g = sp(m, n) (assume m + n = 2, 4 if K = R). For a ∼ = R n we can take the space of real matrices that are diagonal in the upper n rows and zero below. If (a 1 , . . . , a n ) is the diagonal part of such a matrix x, let us write x i := a i . The case K = R and g = so(m, n) with m = n is exceptional (see 8.4 below), so let us first discard it. In all other cases the Weyl chamber C consists of all x ∈ a with x 1 ≥ · · · ≥ x n ≥ 0. Moreover, using Lemma 3.2 we see that P x ∩ C = y ∈ C : y 1 + · · · + y k ≤ x 1 + · · · + x k (k = 1, . . . , n) If x ∈ C then clearly x i = σ i (x), the i-th singular value of x. It follows for arbitrary x ∈ V that O x = y ∈ V : y k ≤ x k (k = 1, . . . , n) since both sets are K-invariant and their intersections with C coincide. So O x is an intersection of balls with center 0 with respect to the Ky Fan norms · k (k = 1, . . . , n), the radii of the balls being the norms of x. Note that the K-orbit Kx consists of all matrices with the same singular values as x.
Example 8.4. Now consider the exceptional case K = R and m = n of the previous example, so we have the natural action of K = SO(n) × SO(n) on V = M n (R). Here the Weyl chamber C consists of all x = diag(x 1 , . . . , x n ) ∈ R n with x 1 ≥ · · · ≥ x n−1 ≥ |x n |, and Now the last diagonal entry x n of x ∈ C coincides with the smallest singular value σ n (x) only up to sign. With similar reasoning as in 8.3 we conclude If m > n, it is easy to see that the K ′ -orbit of an arbitrary matrix x ∈ M m×n (R) coincides with the K-orbit of x. For m = n this is true if det(x) = 0, but otherwise K ′ x is the union of two distinct K-orbits, as one sees from the determinant. A spectrahedral representation of the x := conv(K ′ x) was given in [18,Theorem 4.7] in the case m = n (and O ′ x was called a Fan orbitope there). Note that O ′ x = conv O(n) if x = I n . Moreover, a spectrahedral representation of the coorbitope (conv O(n)) o was provided in [18] (Corollary 4.9). Both representations can easily be derived from our discussion of K-(co)orbitopes. Example 8.6. Next consider the actions of the classical compact simple Lie groups on hermitian matrices. Let n ≥ 2, and let K = SO(n) (case K = R) resp. K = SU (n) (case K = C) resp. K = Sp(n) (case K = H). Let V = {x ∈ M n (K) : x = x * }, the space of hermitian n×n-matrices over K, and let V 0 = {x ∈ V : tr(x) = 0}, where for K = H the trace condition has to be replaced by trd(x) = 0 (reduced trace). We let g ∈ K act on x ∈ V by gxg * . Clearly V 0 is K-invariant, and V = V 0 ⊕ R as K-modules. The action of K on V 0 is an irreducible polar representation, resulting from the simple Lie algebra g = sl(n, R) resp. g = su(n, C) resp. g = sl(n, H). We let a ∼ = R n−1 be the space of all real diagonal matrices x = (x 1 , . . . , x n ) with trace zero. The Weyl chamber is C = {x ∈ a : x 1 ≥ · · · ≥ x n }, and for x = (x 1 , . . . , x n ) ∈ a we have The K-orbit Kx consists of all hermitian matrices with the same eigenvalues as x (for K = H one has to speak of right eigenvalues instead of eigenvalues [9]).
In order to describe the x ′ is positive semidefinite (psd ). Of course this doesn't change the orbitope up to an affine-linear isomorphism. So let x ∈ V be a psd hermitian matrix. Then clearly y 0 holds for every y ∈ O x , and the sequence σ(y) = (σ 1 (y), . . . , σ n (y)) of singular values coincides with the sequence of (right) eigenvalues for these y. So we see that where for nonincreasing sequences a, b ∈ R n the majorization relation is defined by for k = 1, . . . , n, with equality for k = n [10, 4.3.41]. In terms of Ky Fan norms this says that O x is the set of all y ∈ V with y 0 and y k ≤ x k for k = 1, . . . , n − 1 and y n = x n (provided that x 0). Example 8.7. Next let the unitary group over K act on skew-hermitian matrices over K by gxg * . For K = C this is essentially the action of SU (n) on hermitian matrices, already considered in 8.6, since a complex matrix x is skew-hermitian if and only if ix is hermitian. For the remaining two cases we have K = R and K = SO(n) (with n ≥ 3), or K = H and K = Sp(n) (with n ≥ 1), and V is the space of skew-hermitian (x + x * = 0) matrices over K of size n. This is an irreducible polar representation of K, namely the adjoint action of K on its Lie algebra V = Lie(K) (so g = Lie(K) C here.) A maximal abelian subspace can be described as follows. If K = R and K = SO(n), put m := ⌊ n 2 ⌋ and let a consist of all real block matrices x = 0x −x 0 wherex = diag(x 1 , . . . , x m ); if n is odd, an extra row (at the bottom) and column (at the right) of zeros has to be added. If K = H and K = Sp(n), let a consist of all diagonal matrices x = (ix 1 , . . . , ix n ) with x 1 , . . . , x n ∈ R.
To describe Weyl chamber and orbitopes, let first K = R and K = SO(n). The Weyl chamber C consists of all x ∈ a with x 1 ≥ · · · ≥ x m ≥ 0 (case n odd), resp. x 1 ≥ · · · ≥ x m−1 ≥ |x m | (case n even). The description of P x ∩ C, for x ∈ C, is analogous to 8.3 resp. 8.4. The singular values of x ∈ C are x 1 , x 1 , . . . , x m , x m , with an extra zero if n is odd. So we get O x = m k=1 y ∈ V : y 2k ≤ x 2k for n = 2m + 1 odd, and {y : y 2k ≤ x 2k } ∩ y : σ n (x) − σ n (y) ≤ x n−2 − y n−2 for n = 2m even. Note that in either case, only the even Ky Fan norms are needed.
If K = H and K = Sp(n), the Weyl chamber C consists of all x ∈ a with x 1 ≥ · · · ≥ x m ≥ 0, and we find again O x = n k=1 y ∈ V : y k ≤ x k for x ∈ V . Example 8.8. There remains the action of the complex unitary group K = U (n) on V = sym(n, C) resp. V = so(n, C) (symmetric resp. skew-symmetric complex matrices) by gxg t (g ∈ K, x ∈ V ). Again this is an irreducible polar representation that arises from the simple Lie algebra g = sp(n, R) (for V = sym(n, C)) resp. g = so * (2n) (for V = so(n, C)).
First let V = sym(n, C). A maximal abelian subspace a consists of all real diagonal matrices x = diag(x 1 , . . . , x n ), and the Weyl chamber is C = {x ∈ a : x 1 ≥ · · · ≥ x n ≥ 0}. For x ∈ C we have P x ∩ C = {y ∈ C : y 1 + · · · + y k ≤ x 1 + · · · + x k (k = 1, . . . , n)}. Since x i = σ i (x) for x ∈ a, we get O x = y ∈ sym(n, C) : y k ≤ x k (k = 1, . . . , n) In the skew-symmetric case V = so(n, C) let m = ⌊ n 2 ⌋. For n even, a maximal abelian subspace a consists of all block matrices x = 0x −x 0 withx = diag(x 1 , . . . , x m ) a real diagonal matrix. For n odd the description is the same, except that one row (at the bottom) and one column (at the right) of zeros has to be added. In either case the Weyl chamber C consists of all x ∈ a with x 1 ≥ · · · ≥ x m ≥ 0. For x ∈ a the singular values of x are x 1 , x 1 , . . . , x m , x m , together with an extra zero if n is odd. Once more we therefore find O x = y ∈ so(n, C) : y k ≤ x k (k = 1, . . . , n) The fact that any (skew-) symmetric complex matrix is unitarily congruent to a real matrix in a as above is known as Youla's theorem (see e.g. [  Proof. Let A ∈ M N (C) be a complex matrix with eigenvalues θ 1 , . . . , θ N . The k-th exterior power Λ k A of A is a square matrix of size N k that depends linearly on A, and whose eigenvalues are the sums θ i1 + · · · + θ i k with 1 ≤ i 1 < · · · < i k ≤ N .
If K = H, we replace quaternions with complex 2 × 2 matrices, to avoid the problem of defining exterior powers of quaternion matrices. Let A ∈ M N (H) be hermitian (A = A * ), with (real) right eigenvalues θ 1 , . . . , θ N . Write A = A 1 + jA 2 with A 1 , A 2 ∈ M N (C), and letÃ ThenÃ is a complex hermitian matrix of size 2N with eigenvalues θ 1 , θ 1 , . . . , θ N , θ N . Let us, for this purpose, define Λ k A to be the complex matrix Λ kÃ (of size 2N k ). Now let K be any of R, C, H, let x ∈ M m×n (K), and let x ∈ M m+n (K) be the hermitian (block) matrix The (right) eigenvalues of x are ±σ i (x), i = 1, . . . , n, together with m−n additional zeros. It follows that x ∈ B k , i.e. x k ≤ 1, if and only if all eigenvalues of Λ k x are ≤ 1. In other words, this shows that B k is described by the linear matrix inequality B k = x ∈ M m×n (K) : Λ k x I .

8.12.
Finally, we record the cases when the orbitope O x is a K-biorbitope. First consider the action 8.6 on hermitian matrices, for K = R, C, H. Up to scaling and translation there is exactly one biorbitope O x of this type, namely for x = diag(1, 0, . . . , 0). The orbit Kx consists of all psd rank one matrices of (reduced) trace 1. Its convex hull has the rank condition removed: O x = {y : y 0, tr(y) = 1}. In case (b) of the action 8.3, the K-orbit Kx is the Stiefel manifold V n (K m ) of orthonormal n-frames in K m , at least for n < m. We therefore call these orbitopes the Stiefel orbitopes. For m = n and K = R, we get tautological orbitopes: O x is the convex hull of SU (n) (case K = C), resp. of Sp(n) (case K = H).