Dimension of the space of unitary equivariant translation invariant tensor valuations

Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.


Introduction
For a real vector space V of finite dimension and an abelian semigroup (A, +), we write K(V) to denote the space of convex bodies in V (i.e., compact convex sets) equipped with the Hausdorff metric. We call an operator Z : K(V) → A a valuation if Z(K ∪ L) + Z(K ∩ L) = Z(K) + Z(L) holds for any K, L ∈ K(V) satisfying that K ∪ L ∈ K(V). Typical choices for the semigroup A are the field of real numbers R, or the vector space V itself, and more generally the space S d (V) of symmetric rank d tensors of V. Also A = K(V) equipped with Minkowski addition leads to many interesting valuations.
One of the principal goals in the theory of valuations is to obtain characterizations of known operators as the only valuations satisfying certain simple geometric and topological properties. The fundamental result in this direction goes back to 1952, when Hadwiger proved that, for V = R n , the linear combinations of intrinsic volumes are the only continuous real-valued valuations being invariant under rigid motions of R n (see [21]).
Hadwiger's result can be generalized in different directions. One of them is to change the group acting on K(V) and classify the continuous real-valued translation invariant valuations that are invariant under the The first two authors were partially supported by National Research, Development and Innovation Office, NKFIH K 119934, K 132002, KH 129630, ANN 121 649. The third author was partially supported by FEDER-MINECO grant PGC2018-095998-B-I00, and the Serra Húnter Programme.
linear action of some group G. The space of such valuations is finitedimensional precisely when G acts transitively on the unit sphere [2]. For the first nontrivial case G = U(m), this was achieved by Alseker [4] and refined by Bernig and Fu [9]. After this breakthough, several other groups have been succesfully studied. We refer the reader to [7,8,10,12,16,34] and references therein for some results in this direction.
Another important generalization of Hadwiger's theorem consists of changing the target space A. The case A = S d (V) of tensor-valued valuations is of particular interest and has been thoroughly studied, specially under equivariance assumptions with respect to orthogonal and special linear groups (see e.g. [1,11,19,20,23]). The space of U(m)-equivariant valuations is much less understood, and is the object of the present paper.
Other current research directions in valuation theory include the following. Real-valued and tensor-valued valuations defined on lattice polytopes have been studied in [14,30]. A very active area is the study of valuations taking values in the space of convex bodies (see e.g., [26,27] and the references in [28]). Also, important results on valuations defined in several function spaces have been recently obtained (cf. e.g., [5,14,15,29]).
In this paper we begin the study of unitary-equivariant valuations on complex vector spaces. To state our results precisely, let us introduce some notation. We denote the space of continuous translation invariant R-valued valuations on K(V) by Val = Val(V). The subspace of khomogeneous valuations (i.e. such that Z(λK) = λ k Z(K) for any convex body K and λ ≥ 0) is denoted Val k . Given a linear action of a closed subgroup G ⊂ GL(V) on a finite dimensional R-vector space W, we say that a valuation Z : K(V) → W is G-equivariant if Z(ϕ(K)) = ϕZ(K) holds for any ϕ ∈ G and K ∈ K(V). The space of W-valued continuous translation invariant G-equivariant valuations is naturally identified with the subspace (Val ⊗ W) G of G-invariants in Val ⊗ W (the symbol ⊗ is used in this paper for tensor products over R).
We will focus on continuous translation invariant and U(m)-equivariant tensor-valued valuations on K(R 2m ) for m ≥ 2. So in our case, V will be C m , viewed as the real vector space R 2m . The group G will be the unitary group U(m) with its defining action on V = C m , and W = S(R 2m ) will be the d th symmetric tensor power over R of V. For the homogenity degree k = 0, . . . , 2m of a valuation, we set ℓ = min{k, 2m − k}. In the scalar valued case, Alesker [4] proved that dim R Val and he provided two different sets of bases for Val In the vector valued case, Wannerer [35] [33,Theorem 6.14]).
Our main result is the determination of the dimension of the space of U(m)-equivariant tensor valued valuations of all ranks: Theorem 1.1. For m ≥ 2, k = 0, . . . , 2m and d ≥ 0, then using the notation f := ⌊ d 2 ⌋ and ℓ := min{k, 2m − k}, the dimension of (Val k ⊗ S d (R 2m )) U(m) is as follows: Once these dimensions are known, a natural goal is to determine a basis for (Val k ⊗S d (R 2m )) U(m) . So far this has been only known for rank d = 0 by Alesker [4]. Here we construct such a basis in the vector valued case (i.e. for d = 1). To this end, we use the area measures introduced by Wannerer [35] (see Section 4 for definitions). We write S(V) to denote the unit sphere in a euclidean vector space V, and Area(V) to denote the space of smooth area measures, which is a certain class of translation invariant valuations taking values in the space of signed measures of S(V). To each Ψ ∈ Area(V), one can assign the smooth vector valued valuation C(Ψ) defined by for any convex body K.
For V = C m , Wannerer [35] gave a complete description of the space Area U(m) of U(m)-equivariant area measures. In particular, he introduced a certain family ∆ k,q ∈ Area U(m) with specially nice properties (see Section 4). Theorem 1.2. For m ≥ 2, a C-vector space basis of the complex vector space (Val k ⊗ R C m ) U(m) is given by the family C(∆ k,q ) where 0 ≤ k < 2m and max(0, k − m) < q ≤ k 2 . One obtains the following immediate corollary:

Branching rules
Let us recall some background material from representation theory. For technical reasons we shall work with complex representations. So given a compact Lie group G, by a G-module we shall mean a finite dimensional complex vector space V endowed with an action of G via linear transformations, such that the corresponding group homomorphism G → GL(V ) is smooth. Write V for the isomorphism class of V . The set of isomorphism classes of G-modules is a commutative monoid with addition given by V + W = V ⊕ W . The Grothendieck group of this monoid consists of formal differences of isomorphism classes of G-modules. It is a free abelian group R G freely generated by τ G , the set of isomorphism classes of irreducible G-modules. In fact R G is a ring, called the representation ring of G, with multiplication given by V · W = V ⊗ W . For ease of language or notation we shall frequently identify the isomorphism class V ∈ R G with a G-module V representing the isomorphism class V . We shall denote by V ↓ G H the restriction of the G-module V to a subgroup H of G.
Let us briefly sketch the strategy in the proof of Theorem 1.1. We will consider the complexifications . Starting from known decompositions into irreducible SO(2m)-modules, and restricting those to U(m), we will determine the decomposition into irreducible summands of the U(m)-modules Val k,C and S d C (R 2m ). Combining (1) and (2), and using Schur's Lemma will then yield the dimension of (Val k ⊗ S d (R 2m )) U(m) .
Next we turn to a parametrization of τ O(2m) based on partitions, where O(2m) is the full orthogonal group. This material can be found for example in [32], [18], [22] (the notation in these sources is different, and they mainly work in the context of complex linear algebraic groups and not with compact Lie groups). Set Denoting by λ ′ 1 the length of the first column of the Young diagram of λ, our [λ] • corresponds to the partition whose Young diagram is obtained by replacing the first column of the Young diagram of λ by a column of length 2m − λ ′ 1 . We mention that the irreducible O(2m)modules are all defined over R, that is, they are complexifications of irreducible real O(2m)-modules.
Next we turn to the unitary group U(m) = O(2m) ∩ GL(C m ), where GL(C m ) is the complex general linear group. We have This is the restriction to the maximal compact subgroup U(m) of the general linear group GL(C m ) of its representation obtained by tensoring by the (−µ 1 ) th power of the determinant representation the Schur module associated with the partition ( . In certain formulae below elements of R U(m) denoted by {µ; λ} where λ ∈ Π + p , µ ∈ Π + q , and p + q > m will also occur. They can be expressed as an integral linear combination of elements in τ U(m) by a repeated application of the following modification rule given by King [24,p. 433], see also [13,Section 3]: In case it is possible to remove a boundary strip of h boxes from the Young diagram of λ, starting at the foot of the first column, and we obtain a Young diagram of a partition, then we denote this partition by λ − h. Otherwise we say that λ − h does where the boundary h-strip removed from the Young diagram of λ ends in the x th column, and the boundary h-strip removed from the Young diagram of µ ends in the y th column. Note that in the special case p + q = m + 1, i.e. when h = 0, the outcome of the above rule is {µ; λ} = −{µ; λ}, which implies {µ; λ} = 0 whenever p + q = m + 1. µ,ν the associated Littlewood-Richardson coefficient (cf. e.g. [17, A.8]). It is determined by the equality For p ≤ m we shall treat Π p as a subset of Π m , by identifying (λ 1 , . . . , λ p ) ∈ Π p with (λ 1 , . . . , λ p , 0, . . . , 0).

Theorem 2.2 (King [24]). For any
where (2β) ′ stands for a partition such that all columns of its Young diagram have even length (i.e. the transpose of a partition with even parts).
The above statement can be found at [24, p. 440 It follows that for each λ, ξ, ν, β ∈ Π m we have So (4) can be rewritten as We note that in the special case when λ, ξ, ν ∈ Π ⌊ m 2 ⌋ , the same formula for the multiplicity of {ξ; ν} as a summand in For the complexification of the symmetric tensor power S d (R 2m ) of the defining O(2m)-module R 2m we have where (g, 2 h ) stands for the partition (g, 2, . . . , 2 h terms ) (here we extended the notation introduced in Section 2 in the obvious way to locally finite dimensional SO(n)-representations in which each irreducible SO(n)module has finite multiplicity).

A basis for vector valued valuations in terms of area measures
In this section we construct a C-vector space basis of (Val⊗ R C m ) U(m) , the complex vector space of U(m)-equivariant translation invariant vector valued valuations on C m for m ≥ 2.
In the scalar valued case, Bernig and Fu [9] constructed a basis of Val U(m) consisting of the so-called hermitian intrinsic volumes µ k,q , defined for 0 ≤ k ≤ 2m and 0, k − m ≤ q ≤ k 2 . These valuations are even and hence characterized by their Klain function [25]. The Klain function of an even valuation ϕ ∈ Val k (V) is a function Kl ϕ on the k-Grassmannian Gr k (V) given by ϕ For k ≤ m, the Klain function of µ k,q is where θ i is the i th elementary symmetric function, and θ 1 , . . . , θ ⌊k/2⌋ are the Kähler angles of the k-dimensional linear subspace E. These angles are characterized as follows. Let ψ E be the endomorphism of E that maps u ∈ E to the orthogonal projection of √ −1u to E. Then ψ E has eigenvalues Given a linear subspace E ⊂ V there exists a restriction map r from Area(V) to Area(E) characterized as follows. Given a Borel set U ⊂ S(V ), let U = (U + E ⊥ ) ∩ S(V). The restriction of Φ ∈ Area(V) is given by (16) r For V = C m , the space Area U(m) k of k-homogeneous U(m)-invariant smooth area measures was described in [34]. We will need the following.
2 such that (i) glob(∆ k,q ) = µ k,q (ii) for every polytope P , and every Borel set U ⊂ S(C m ) where F k is the set of k-dimensional faces, N(P, F ) is the set of outer unit normal vectors to P at points of F , and F is the k-dimensional linear subspace parallel to F .
Given a p-dimensional real subspace E ⊂ C m and the corresponding restriction map r, it follows from (17) that (18) C(r(∆ k,q ))(A) = c m,p,k C(∆ k,q )(A), for c m,p,k = 0 depending only on m, p, k.
It was shown in [34] that the family C(∆ k,q ) with 0, k − m < q ≤ k 2 is R-linearly independent. Since we already know that dim C ((Val k ⊗ R C m ) U(m) ) = ⌊ ℓ(k) 2 ⌋ (see Remark 3.2 (ii)), to prove Theorem 1.2 we only need to show that the above family is in fact C-linearly independent. Proof of Theorem 1.2. Let us first assume k ≥ m. For 1 ≤ r ≤ ⌊ 2m−k 2 ⌋ consider the following element of Area By (14) and (15) we have Kl glob(Ψ k,r ) (F ) = σ r (cos 2 θ 1 , . . . , cos 2 θ ⌊ 2m−k 2 ⌋ ). where the θ i refer to the Kähler angles of F ⊥ .
We show next that the vector valuations C(Ψ k,r ), and hence the C(∆ k,q ) with q > k − m, are linearly independent elements of the complex vector space (Val k ⊗ R C m ) U(m) . Since this space has dimension ⌊ 2m−k 2 ⌋, the statement will follow.