The stability of the Kronecker product of Schur functions

Abstract

In the late 1930s Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.

On the complexity of computing Kronecker coefficients

“…This result is also comparable to the result of [BOR2,Theorem 1.4], where, for example, in the case λ = µ = ν they coincide, but in general is also slightly weaker.…”

“…The reduced Kronecker coefficients, see e.g. [BDO,BOR2], are defined as (11)ḡ(λ, µ, ν) = g (n − |λ|, λ), (n − |µ|, µ), (n − |ν|, ν) for n large enough.…”

Deligne categories and reduced Kronecker coefficients

“…A special case of the second result when k = l = 0 recovers a formula due to Briand, Orellana and Rosas (see [3]) which describes how to recover the (standard) Kronecker coefficients from the reduced Kronecker coefficients.…”

“…The reduced Kronecker coefficients (sometimes also called "stable Kronecker coefficients") were originally defined as the stabilizing values of such sequences (see [3], for example). From now on we will use the more common term "reduced Kronecker coefficient" instead of "Deligne-Kronecker coefficient".…”

“…Also, given a Young diagram λ, denote byλ the Young diagram obtained from λ by removing the top row (thus given λ n, S n (Xλ) = λ). With these definitions, the second statement of Proposition 5.2 can also be reformulated as follows (in this form it appears in [3]): Proposition 5.3 Let n ∈ Z + , and let λ, μ, τ be three partitions of n. Then…”