The Heisenberg product seen as a branching problem for connected reductive groups, stability properties

1 Introduction

The Heisenberg product was first introduced by Marcelo Aguiar, Walter Ferrer Santos and Walter Moreira in [1] in order to simplify and unify a diversity of related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto–Reutenauer, convolution, …) defined on various objects: species, representations of symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, … .

In this paper, we are only interested in one of these contexts where they thus defined this Heisenberg product, namely, the one of complex representations (every representation considered throughout the article will be a complex vector space) of symmetric groups. In this particular context, let us fix throughout the whole article two positive integers k and l. Then we denote by 𝔖k (resp. 𝔖l) the symmetric group formed by the permutations of the finite set {1,…,k} (resp. {1,…,l}), and the Heisenberg product can be defined on two complex representations V and W of 𝔖k and 𝔖l, respectively. It is denoted by V⁢♯⁢W and is a direct sum of representations of the groups 𝔖i for i from max⁡(k,l) to k+l. We explain precisely the construction of this product in Section 2.1.

One interesting thing to notice about this product is that, when k=l, V⁢♯⁢W is a direct sum of representations of 𝔖k,𝔖k+1,…,𝔖2⁢k, and the term in this direct sum which is a representation of 𝔖k corresponds simply to the tensor product V⊗W seen as a 𝔖k-module (with 𝔖k acting diagonally). This tensor product of representations of 𝔖k is sometimes referred to as the “Kronecker product” since it gives rise to the Kronecker coefficients when applied to irreducible 𝔖k-modules. As a consequence, the Heisenberg product extends – in a certain way – this so-called Kronecker product.

An important point that we use concerning the representation theory of the symmetric groups is that the irreducible complex representations of the group 𝔖k are known: they are in bijection with the partitions of the integer k, and one can moreover construct them. If λ is a partition of k (denoted λ⊢k), i.e. a finite non-increasing sequence (λ1≥λ2≥…≥λn>0) of positive integers (called parts) whose sum (sometimes called the size of the partition, and denoted by |λ|) is k, there is an explicit construction – several, in fact – giving a representation of 𝔖k over the field ℂ of complex numbers, which happens to be irreducible. We denote this 𝔖k-module by Mλ. We do not detail the construction of Mλ here: two examples of such can for instance be found in [3, Chapter 4].

Considering that every complex representation of a symmetric group decomposes as a direct sum of irreducible ones, it is natural to seek to understand the Heisenberg product of two of the latter. If λ and μ are respectively partitions of k and l, the Heisenberg product Mλ⁢♯⁢Mμ is a direct sum of 𝔖i-modules for i∈{max⁡(k,l),…,k+l}, and then every term in this sum decomposes as a direct sum of irreducible 𝔖i-modules:

The multiplicities aλ,μν in these decompositions are non-negative integers which are called the Aguiar coefficients. They were introduced in [12] by Li Ying, who also proved interesting stability results concerning them. We recall these results in Section 2.2.

The fact is that Li Ying’s stability results look very much like similar results already proven concerning Kronecker coefficients. Let us recall that these particular coefficients are the multiplicities gλ,μ,ν arising in the decomposition

where λ and μ are partitions of k. In [7], we gave some geometric methods allowing to prove stability results for those, as well as for some other similar coefficients. In fact, these techniques can be applied as soon as we are looking at branching coefficients for complex connected reductive groups: if G and G^ are two such groups and if we have a morphism G→G^, the branching problem consists in seeing irreducible complex representations of G^ as G-modules via the previous morphism and in wondering how as such they decompose as a direct sum of irreducible ones. As a consequence, in Section 3.1, we express the Aguiar coefficients as such branching coefficients, obtaining the following result.

As a consequence, we can use in Section 3.2 the same methods as in [7], and obtain some new stability results, generalising in part those of Li Ying. The main one is the following.

In fact, Li Ying obtained the same conclusion as in the previous theorem for the triple (α,β;γ)=((1),(1);(1)). We call the triples satisfying the same property “Aguiar-stable triples”, and we give – also in Section 3.2 – four other explicit examples of such ones.

Finally, in Section 4, we discuss about what we call “stabilisation bounds”: if (α,β;γ) is Aguiar-stable, such a bound is a non-negative integer d0 (depending on partitions λ, μ and ν) such that the sequence (aλ+d⁢α,μ+d⁢βν+d⁢γ)d∈ℤ≥0 is constant for d≥d0. The geometric methods from [7] can here also give means to compute some stabilisation bounds. We detail the computation for the Aguiar-stable triples ((1),(1);(1)), ((2),(1);(2)) and ((2),(1);(3)).

2 Definition and first properties of the Heisenberg product

2.1 Construction

Let us recall that we fixed, already in the introduction, two positive integers k and l and that we consider the two symmetric groups 𝔖k and 𝔖l.

Remark 2.1.

Notice that, for all non-negative integers a and b, 𝔖a×𝔖b can naturally be seen as a subgroup of 𝔖a+b, thanks to the injective group morphism

ιa,b:𝔖a×𝔖b↪𝔖a+b,

(σa,σb)→i↦{σa⁢(i)if⁢i∈{1,…,a},a+σb⁢(i-a)if⁢i∈{a+1,…,a+b}.

Furthermore, for any non-negative integer a, 𝔖a can be considered as a subgroup of 𝔖a×𝔖a through the diagonal embedding Δa:𝔖a↪𝔖a×𝔖a.

Definition 2.2.

Let V and W be two (complex) representations of 𝔖k and 𝔖l, respectively. Let i∈{max⁡(k,l),…,k+l}. One has the following diagram:

We then consider V⊗W, which is a representation of 𝔖k×𝔖l, and its restriction Res𝔖i-l×𝔖k+l-i×𝔖i-k𝔖k×𝔖l⁢(V⊗W) to a representation of 𝔖i-l×𝔖k+l-i×𝔖i-k. Finally, we define (V⁢♯⁢W)i as the representation induced to 𝔖i from

Res𝔖i-l×𝔖k+l-i×𝔖i-k𝔖k×𝔖l⁢(V⊗W).

It is then an 𝔖i-module, and the Heisenberg product of V and W is

V⁢♯⁢W=⊕i=max⁡(k,l)k+l(V⁢♯⁢W)i.

A remarkable thing proven in [1] is that this product is associative.

Definition 2.3.

Let λ⊢k and μ⊢l. The Heisenberg product between the associated irreducible representations of the symmetric group decomposes as

Mλ⁢♯⁢Mμ=⊕i=max⁡(k,l)k+l⊕ν⊢iMν⊕aλ,μν.

The coefficients aλ,μν are called the Aguiar coefficients.

We adopt the convention that, if the weights of the partitions λ, μ and ν are not compatible to define an Aguiar coefficient (|ν|∉{max⁡(|λ|,|μ|),…,|λ|+|μ|}), then aλ,μν=0. Likewise, if V and W are respectively 𝔖k- and 𝔖l-modules and if i∉{max⁡(k,l),…,k+l} is a positive integer, we will say that (V⁢♯⁢W)i is the trivial 𝔖i-module {0}.

Remark 2.4.

As written earlier, the Heisenberg product extends the Kronecker one: when k=l, the lower term (V⁢♯⁢W)k of V⁢♯⁢W is just V⊗W seen as a representation of 𝔖k. As a consequence, when the three partitions λ, μ and ν have the same size, the Aguiar coefficient aλ,μν coincides with the Kronecker coefficient gλ,μ,ν.

3 New stability results by geometric methods

4 Some explicit stabilisation bounds

In this section, we are going to compute stabilisation bounds for three examples of Aguiar-stable triples: ((1),(1);(1)), ((2),(1);(2)), and ((2),(1);(3)). The proof of Theorem 3.6 gives us a sufficient condition to obtain them: let us fix from now on an Aguiar-stable triple (α,β;γ) (we will specialise this triple later) and a triple (λ,μ;ν) of partitions. We also consider vector spaces V1 and V2 as before (of dimension at least 2), and denote V=V1⊕(V1⊗V2)⊕V2 such that

with

We fix finally a basis e¯=(e1,…,en1) of V1 and a basis f¯=(f1,…,fn2) of V2. We now know that every d0∈ℤ≥0 such that, for all d≥d0,

is a stabilisation bound.

One important tool for our computation is a numerical criterion of semi-stability known as the Hilbert–Mumford criterion.

The Hilbert–Mumford criterion can then be stated as follows (see, for example, [8, Lemma 2]).

Following this property, a one-parameter subgroup τ such that μ𝒩⁢(y,τ)>0 will be said to be “destabilising” for y (relative to 𝒩).

We will then begin the computation by considering the projection

We also denote by ℒ¯ the ample line bundle on X¯ whose pull-back by π is ℒ, by e¯*=(e1*,…,en1*) and f¯*=(f1*,…,fn2*) the dual bases of e¯ and f¯, respectively, and set n=min⁡(n1,n2).