Signature quantization and representations of compact Lie groups

The Kostant Partition Function and Its q Analogues

We start by introducing the Kostant partition function.

Definition 1: The Kostant partition function for a root system Φ, given a choice of positive roots Φ₊, is the function i.e., K(μ) is the number of ways that μ can be written as a sum of positive roots (see ref. 4).

Note that K(μ) can also be computed as the number of integer points inside the polytope We can write down a generating function for the K(μ) that is very similar to Euler's generating function for the number of partitions (see ref. 4, section 25.2): The classical q analogue K̂ _q(μ) of K(μ), according to Lusztig (5), keeps track of how many times the roots appear: corresponding to the generating function The q analogue K_q(μ) that interests us here is the one that counts the integer points of Q _μ according to how many k _α values are nonzero: In terms of generating functions, this translates to

The Representations Ṽ_λ = V_λ–ρ ⊗ V_ρ

We are working in the context of a complex semisimple Lie algebra with root system Φ, choice of positive roots Φ₊, and Weyl group 𝒲; ρ is half the sum of the positive roots (or the sum of the fundamental weights). For a dominant weight λ, we denote by V _λ the irreducible representation of with highest weight λ. We will call a weight λ strictly dominant if λ – ρ is dominant. We will use the notation Λ⁺ for the set of dominant weights, and Λ⁺ _S for the set of strictly dominant weights. For a strictly dominant weight, we define the representation and its character The following theorem of Guillemin et al. (1) provides a formula for the multiplicities of the weights in the weight space decomposition of Ṽ _λ. This formula is very similar to the Kostant multiplicity formula (Eq. 3), but it uses the q = 2 specialization of the q analogue of the Kostant partition function K_q(μ) introduced above, instead of the usual Kostant partition function. The formula for the Ṽ _λ multiplicities further distinguishes itself from the Kostant formula by being free of the ρ factors.

An Analogue of the Kostant Multiplicity Formula for the Ṽ_λ

Theorem 1 [Guillemin–Sternberg–Weitsman ( 1 )]. Let λ be a strictly dominant weight. Then the multiplicity of the weight ν in the tensor product Ṽ _λ = V _λ–ρ ⊗ V _ρ is given by where |ω| is the length of ω in the Weyl group.

Proof: We give a simple proof here using the Weyl character formula. This formula expresses the character χ_λ of V _λ as the quotient where . For ρ, we get the nice expression (ref. 4, lemma 24.3) which means, in particular, that we get Thus, for λ strictly dominant, Extracting the coefficient of e ^ν on both sides gives Eq. 14.

The next step will be to use a formula due to Atiyah and Bott (6, 7) for the characters of the V _λ and Ṽ _λ to break down Ṽ _λ into its irreducible components and find their multiplicities. The Atiyah–Bott formula (6, 7) gives the character of V _μ as

Remark 1: We can deduce this formula from the Weyl character formula (Eq. 15) by first observing that Also, Combining Eq. 21 with Eq. 22 gives and we can translate Weyl's character formula into the Atiyah–Bott formula using this equation.

For any , since characters are invariant under the Weyl group action. Using this and the Atiyah–Bott formula, we can write where, as before, α_I = Σ_{α∈ I} α. This gives Letting λ_I = λ – α_I, we observe that if λ_I is dominant, the Atiyah–Bott formula tells us that is the character χ_{λ I} of the irreducible representation V _{λ I}, so that if all the λ_I are dominant.

Alternatively, we can obtain Eq. 25 from Eq. 18 by observing that for

Finally, since α_I and α_{I ′} can be equal for different subsets I and I′, certain highest weights appear multiple times in the above sums. For the weight μ = λ_I = λ – α_I, we will get V _μ as many times as we can write α_I = λ – μ as a sum of positive roots, where each positive root appears at most once. Hence where the sum is over all μ such that μ = λ_I for some I, and P(ν) is given by

Remark 2: David Vogan pointed out to us that this decomposition is well known and can be deduced from the Steinberg formula. For type A_n, the number of distinct μ's in the above sum is the number of forests of labeled unrooted trees on n + 1 vertices (8, 9).

A Tensor Product Formula for the Ṽ_λ

We will derive here an analogue of the Steinberg formula for the Ṽ _λ. Given two representations Ṽ _λ and Ṽ _μ, the problem is to determine whether their tensor product Ṽ _λ ⊗ Ṽ _μ can be decomposed in terms of Ṽ _ν's. This is readily seen to be the case, as Breaking up V _λ–ρ ⊗ V _ρ ⊗ V _μ–ρ into irreducibles V _γ and tensoring each factor with V _ρ yields factors V _γ ⊗ V _ρ = Ṽ _γ+ρ. Thus, for strictly dominant weights λ and μ, we can write for some nonnegative integers .

Theorem 2. For λ, μ and ν strictly dominant weights, the tensor product multiplicity of Ṽ _ν in Ṽ _λ ⊗ Ṽ _μ is given by

Proof: Starting from the equation , we can use Eq. 18 to write Canceling terms and using Theorem 1 to write down the character yields Substituting γ = ω(λ) + β on the left-hand side and γ = τ(ν) on the right-hand side gives and extracting the coefficient of e ^γ on both sides yields Now, since (γ) vanishes unless τ^–1(γ) is strictly dominant, all the terms in the sum on the right-hand side vanish except for the one where τ is the identity (i.e. the term where γ = ν), and we get the result.

If we denote by N ^ν _λμ the multiplicities of the irreducible representations V _ν in the tensor product V _λ ⊗ V _μ, defined by then we can write down the tensor product multiplicities for the decomposition of Ṽ _λ ⊗ Ṽ _μ into Ṽ _ν's in terms of the as follows: so that for strictly dominant ν,

Remark 3: In type A, there is a combinatorial interpretation for the coefficients N ^ν _λμ in terms of shifted Young tableaux: they are given by a shifted analogue of the Littlewood–Richardson rule (see ref. 10).

Links with Symmetric Functions in Type A

As for the weight multiplicities and Clebsch–Gordan coefficients, a link exists between the character products and symmetric functions in type A, again in terms of Schur functions.

The character of the irreducible polynomial representation V _λ of , where we now think of λ as a partition with k parts (allowing the empty part) is the Schur function s _λ(x ₁,..., x_k). We will call a partition strict if all its parts are distinct (corresponding to a strictly dominant weight). Thus, we have that, for , for any strict partition λ. It is readily checked that the weight ρ corresponds to the partition (k – 1, k – 2,..., 1, 0).

Remark 4: We can also write the characters of Ṽ _λ in terms of Hall–Littlewood polynomials (see ref. 11, [III, 1. and 2.]). The results of the following sections can be deduced from this link with Hall–Littlewood polynomials, but we will rather use the Schur function expression (Eq. 37) for the characters. This makes the proofs a bit more technical but avoids the heavier machinery of Hall–Littlewood polynomials.

A Branching Rule for the Ṽ_λ in Type A

We have seen that the representations Ṽ _λ behave somewhat like irreducible representations, in that tensor products of them can be broken down into direct sums of Ṽ _ν's again and that the multiplicities in those decompositions as well as in the weight space decomposition are given by formulas very similar to those of Kostant and Steinberg in the irreducible case. The Weyl branching rule (see ref. 4, for example) describes how to restrict a representation V _λ from to . This rule can be applied iteratively and provides a way to index one-dimensional subspaces of V _λ by diagrams [Gel'fand–Tsetlin diagrams (12)] that is compatible with the weight space decomposition. It is natural to ask whether the representations Ṽ _λ of are also well behaved under restriction, or, in another words, if there is an analogue of the Weyl branching rule for the Ṽ _λ in type A.

For two partitions μ = (μ₁,..., μ_m) and γ = (γ₁,..., γ_{m –1}), we say that γ interlaces μ, and write γ ◃ μ, if For two such partitions μ and γ such that γ ◃ μ, we define In other words, ▿(μ, γ) is the number of γ_i that are wedged strictly between μ_i and μ_{i +1}.

Theorem 3. The decomposition of the restriction of the representation Ṽ _λ of to into irreducible representations of is given by

Proof: We argue using characters and the fact that those characters can be written in terms of Schur functions. We saw above (Eq. 37) that the character of the representation Ṽ _λ of is the product of Schur functions s _λ–ρ(x ₁,..., x_k) s _ρ(x ₁,..., x_k). We obtain the character of the restriction of Ṽ _λ to by setting the last variable x_k equal to 1. Using well known identities on Schur functions (see ref. 13, section 7.15, for example), we have that and Thus, We recognize the product Π_{1≤ i < j ≤ k –1} (x_i + x_j) as the Schur function s _ρ(x ₁,..., x_{k –1}) (where ρ now corresponds to the partition (k – 2, k – 3, ···, 1, 0) with k – 1 parts) and the product (x_i + 1) as the sum (e ₀ + e ₁ + · · · + e_{k –1}) of elementary symmetric functions in the variables x ₁,..., x_{k –1}. A dual version of the Pieri rule (ref. 13, section 7.15) describes how to break down the product of a Schur function with an elementary symmetric function into Schur functions: where the sum is over all ν obtained from μ by adding a vertical strip of size m, i.e., over the ν such that μ ⊆ ν and the skew-shape ν/μ consists of m boxes, no two of which are in the same row. As we are working in k – 1 variables, the s _ν with more than k – 1 parts vanish, so we can add the further constraint that the vertical strip be confined to the first k – 1 rows (we will say such a vertical strip has height at most k – 1). This gives where the sum is over all the ν that can be obtained from μ by adding a vertical strip of size and height at most k – 1. We can rewrite this as where the sum is over all strict partitions ν such that ν – ρ can be obtained from μ by adding a vertical strip of size and height at most k – 1. Since the s _ν s _ρ are linearly independent, we can lift this to the level of representations to get with the sum over the same set of ν as before.

To compute the multiplicity of a given Ṽ _ν in Ṽ _λ, we define, for strict partitions λ and ν, n(λ, ν) to be the number of ways that ν – ρ can be obtained by adding a vertical strip of size and height at most k – 1 to some partition μ such that μ ◃ λ – ρ, so that Note that δ has two different meanings here: for the group , it corresponds to the partition (k – 1, k – 2,..., 1, 0), while for , it corresponds to the partition (k – 2, k – 3,..., 1, 0). To avoid confusion, we will denote the latter by δ′.

The condition μ ◃ λ – δ means that

Replacing μ_i by μ_i + δ′_i = μ_i + (k – 1 – i) = gives

These equations mean that the ith part of μ′= μ + δ′ is at least as large as the (i + 1)th part of λ and smaller than the ith part of λ. In other words, the skew-shape λ/μ′ is a horizontal strip with at least a box in each row, or equivalently μ′ ◃ λ with the further contraints μ′_i< λ_i for all 1 ≤ i ≤ k – 1. Adding a vertical strip to μ to get v – δ is the same as adding a vertical strip to μ′ to get v, provided that we only allow adding vertical strips to μ′ that result in a strict partition. It is then clear that by adding such a vertical strip to μ′, we get a strict partition v such that λ/v is a horizontal strip. Conversely, it is also clear that for any strict v such that λ/v is a horizontal strip, there is a μ′ such that v can be obtained from μ′ by adding a vertical strip. So the only summands, Ṽ_v for which n(λ, v) ≠ 0 in the decomposition (Eq. 56) are those for which v ◃ λ.

Given such a v, we will compute n(λ, v) by constructing row by row the strict partitions μ′ = μ + δ′ from which we can obtain v. Given v_i, there are three cases to consider for the possible :

1. v_i = λ_i. In this case, since we must have μ′_i < λ_i, it has to be that μ′_i = λ_i – 1 and that we have a box in row i of the vertical strip. So there is only one choice for μ′_i.

2. v_i = λ_{i +1}. Then we must have μ_i = λ_{i +1} ≤ μ′_i ≤ v_i and therefore μ′_i = v_i, so we don't have a box in row i of the vertical strip. Again, there is only one choice for μ′_i in this case.

3. λ_i > v_i > λ_{i +1}. Then we can either have μ′_i = v_i – 1 and have a box from the vertical strip in row i, or have μ′_i = v_i and have no box from the vertical strip in row i. So there are two possibilities for μ′_i in this case.

We have to show that any choice of μ′_i that we make gives rise to a strict partition (by construction, it is clear that μ′ ◃ λ). If for some i we had μ′_i = μ′_{i +1}, then because λ_{i +1} is at least μ′_{i +1} + 1, this would mean that λ_i is at least μ′_i + 2, since λ_i > λ_{i +1}. But then λ/μ′ contains two boxes in the same column: the box after box μ′_iin row i, and the box after box μ′_i= μ′_{i +1} in row i + 1, which contradicts the fact that μ′ ◃ λ (or equivalently, that λ/μ′ is a horizontal strip). Hence we get two choices for each instance of a pattern of the form λ_i > v_i > λ_{i +1}. We called the number of such instances above ▿(λ, v). Since the choices at each row are independent, we have from which the proposed expression for the branching rule follows.

Gel'fand–Tsetlin Theory for the Ṽ_λ

After restricting to , we can further restrict to . From now on, we will assume that all partitions are strict. We can write Denoting by the strict partitions indexing the representations Ṽ of , we can iterate the branching rule until we get to : We will call a sequence of strict partitions of the form λ⁽¹⁾ ◃ ··· ◃ λ^(k) = λ a twisted Gel'fand–Tsetlin diagram for λ, which can be viewed schematically as with and each is a nonnegative integer satisfying and for all 1 ≤ j ≤ i, 1 ≤ i ≤ k – 1. Let be the subspace of Ṽ _λ corresponding to a twisted Gel'fand–Tsetlin diagram 𝒟. This subspace has dimension , where We can also think of as the number of triangles with strict inequalities in the diagram 𝒟.

We show here that lies completely within the same weight space as the weight space decomposition of Ṽ _λ. We will think of the groups as included into one another by identifying with The maximal forces of is its subgroup of invertible matrices T_k, whose Lie algebra will be denoted t_k. We will consider two bases of t_k: let 1 ≤ m ≤ k, and J_m = I_m – I_{m –1} for 1 ≤ m ≤ k.

Consider the element and a representation Ṽ _μ of . We have the representation . For ν ∈ V _μ–ρ and ω ∈ V _ρ, we have since V _μ–ρ has highest weight μ – ρ and V _ρ has highest weight ρ. So gets represented as () I in Ṽ _μ. In general, for we will find that gets represented as () I in Ṽ _λ(m). Therefore, in the basis I ₁,..., I_k, the subspace corresponding to a twisted Gel'fand–Tsetlin diagram 𝒟 has weight or in the usual basis J ₁,..., J _k.

In other words, if or, equivalently, Hence twisted Gel'fand–Tsetlin diagrams for λ correspond to the same weight if all their row sums are the same. So we have proved the following analogue of the Gel'fand–Tsetlin theorem (12).

Theorem 4. Let λ = (λ₁,..., λ_k) be a strictly dominant weight. The dimension of the representation Ṽ _λ of is given by where the sum is over all twisted Gel'fand–Tsetlin diagrams with top row λ. Furthermore, the multiplicity m̃ _λ(β) of the weight β in Ṽ _λ is given by where the sum is over all twisted Gel'fand–Tsetlin diagrams with top row λ and row sums satisfying Eq. 57 (or Eq. 58).

Remark 5: We can also prove that lies completely within a weight space of Ṽ _λ using characters and Schur function identities.