A geometric approach to the stabilisation of certain sequences of Kronecker coefficients

Abstract

We give another proof, using tools from Geometric Invariant Theory, of a result due to S. Sam and A. Snowden in 2014, concerning the stability of Kronecker coefficients. This result states that some sequences of Kronecker coefficients eventually stabilise, and our method gives a nice geometric bound from which the stabilisation occurs. We perform the explicit computation of such a bound on two examples, one being the classical case of Murnaghan's stability. Moreover, we see that our techniques apply to other coefficients arising in Representation Theory: namely to some plethysm coefficients and in the case of the tensor product of representations of the hyperoctahedral group. the same integer; if one repetitively increases by 1 the first part of each of these partitions, the corresponding sequence of Kronecker coefficients ends up stabilising. J. Stembridge, in [21], introduced two notions of stability of a triple of partitions in order to generalise this Murnaghan's stability:• stable if g α,β,γ > 0 and, for any triple (λ, µ, ν) of partitions such that |λ| = |µ| = |ν|, the sequence of general term g λ+dα,µ+dβ,ν+dγ is eventually constant.The terminology "weakly stable" is in fact used by L. Manivel in [13]. The notion of a stable triple is made to generalise the Murnaghan's stability: the latter simply means that the triple (1), (1), (1) is stable. By introducing the notion of a weakly stable triple, Stembridge hoped to find a more simple criterion to determine whether a triple is stable. He proved in [21] that a stable triple is weakly stable, and conjectured that the converse is true. S. Sam and A. Snowden proved shortly after, in [19], that it is indeed verified. We also learned during the redaction of this article about a prepublication by P.-E. Paradan [14], who demonstrated this kind of result in a more general context which in particular contains the case of Kronecker coefficients (as well as the plethysm case). In the first part of this article, we give another new proof of this result: Theorem 1.2. If a triple (α, β, γ) of partitions is weakly stable, then it is stable.A question then arises: given a stable triple, can we determine when the associated sequences of Kronecker coefficients do stabilise? There has already been results on this, at least in the case of Murnaghan's stability: for instance, M. Brion -in 1993-and E. Vallejo -in 1999-calculated bounds from which these sequences are necessarily constant. In [1], E. Briand, R. Orellana, and M. Rosas recall the two bounds from Brion and Vallejo, and determine two other ones, still in the case of the stable triple (1), (1), (1) .The interesting aspect of our proof of Theorem 1.2 is that it gives a nice "geometric bound" from which we can be certain that the sequence (g λ+dα,µ+dβ,ν+dγ ) d is constant, if the triple (α, β, γ) is stable. Indeed, the Kronecker coefficients can classically be related to the dimension of spaces of invariant sections from some line bundles: for all triples (α, β, γ) and (λ, µ, ν), there exist a reductive group G acting on a ...

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

“…We give a sketch of the proof. For every details see [Pel18a]. With Corollary 3.5, we can write, with V 1 and V 2 vector spaces of large enough dimension,…”

“…We consider an element Cv 1 , Cv 2 , C(ϕ 1 + ϕ 2 + ϕ) ∈ X. Similarly to the proof of Proposition 3.3 from [Pel18a], we are only interested in the orbits of maximal dimension or of dimension just below that. Then, considering the usual isomorphism (V 1 ⊗ V 2 ) * ≃ Hom(V 1 , V * 2 ), we say that ϕ corresponds to a linear map ϕ ′ : V 1 → V * 2 , on which G acts by conjugation.…”

“…As a consequence we can use in Section 3.2 the same methods as in [Pel18a], and obtain some new stability results, generalising in part those of Li Ying. The main one is: Theorem 1.2.…”

“…, where λ and µ are partitions of k. In [Pel18a] we exposed 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 Ĝ are two such groups and if we have a morphism G → Ĝ, the branching problem consists in seeing irreducible complex representations of Ĝ as G-modules via the previous morphism and in wondering how as such they decompose as a direct sum of irreducible ones.…”

“…Finally, in Section 4 we discuss about what we call "bounds of stabilisation": if (α, β; γ) is Aguiar-stable, such a bound is a non-negative integer d 0 (depending on partitions λ, µ, and ν) such that the sequence (a ν+dγ λ+dα,µ+dβ ) d∈Z ≥0 is constant for d ≥ d 0 . The geometric methods from [Pel18a] can here also give means to compute some bounds of stabilisation. We detail the computation for the Aguiar-stable triples (1), (1); (1) , (2), (1); (2) and…”