Generalized stability of Heisenberg coefficients
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Aequationes Mathematicae
Generalized stability of Heisenberg coefficients Li Ying
Abstract. Stembridge introduced the notion of stability for Kronecker triples, which generalizes Murnaghan’s classical stability result for Kronecker coefficients. Sam and Snowden proved a conjecture of Stembridge concerning stable Kronecker triples, and they also showed an analogous result for Littlewood–Richardson coefficients. Heisenberg coefficients are Schur structure constants of the Heisenberg product which generalize both Littlewood–Richardson coefficients and Kronecker coefficients. We show that any stable triple for Kronecker coefficients or Littlewood–Richardson coefficients also stabilizes Heisenberg coefficients, and we classify the triples stabilizing Heisenberg coefficients. We also follow Vallejo’s idea of using matrix additivity to generate Heisenberg stable triples. Mathematics Subject Classification. 05E05, 05E10, 20C30. Keywords. Representation theory, Kronecker coefficient, Representation stability.
1. Introduction We assume familiarity with the basic results in the (complex) representation theory of symmetric groups and symmetric functions (see [6,15]). There are two famous structure constants, Kronecker coefficients and Littlewood–Richardson coefficients, which can be defined in terms of representations of symmetric groups. Given a partition λ of n (written as λ n, or λ has size n), let Vλ be the associated irreducible representation of the symmetric group Sn . The λ is the multiplicity of Vλ in the irreducible decomKronecker coefficient gμ,ν Sn ×Sn (Vμ ⊗ Vν ) (viewing Sn as a subgroup of Sn × Sn via the position of ResSn canonical diagonal embedding), the Kronecker product of Vμ and Vν . That is, λ = ResSSnn ×Sn (Vμ ⊗ Vν ) , Vλ , gμ,ν
where λ, μ, and ν are partitions of n, and , denotes the scalar product that makes the irreducible representations orthonormal. The Littlewood–Richardson
L. Ying
AEM
coefficient cλμ,ν is the multiplicity of Vλ in the irreducible decomposition of S (Vμ ⊗ Vν ), the induction product of Vμ and Vν . That is, IndSn+m n ×Sm S
cλμ,ν = IndSn+m (Vμ ⊗ Vν ) , Vλ , n ×Sm where λ n + m, μ n, and ν m for some positive integers n and m. We view partitions as vectors so we define addition, subtraction, and scalar multiplication on them. While the Littlewood–Richardson coefficient is wellstudied and has several beautiful combinatorial interpretations (see [4,6,12]), no explicit combinatorial description for the Kronecker coefficient is known. In 1938 Murnaghan [10] discovered a remarkable stability property for Kronecker coefficients. He stated without proof that for any partitions λ, μ, and ν of the λ+(n) same size, the sequence gμ+(n),ν+(n) is eventually constant. There are many proofs with different flavours for this fact, see [2,5,19]. Stembridge [18] vastly generalized this result by introducing the concept of a stable triple. α Definition 1.1. A triple (α, β, γ) of partitions of the same size with gβ,γ >0 is a K-triple. It is K-stable if, for any other triple of partitions (λ, μ, ν) with λ+nα |λ|
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