A diagrammatic approach to Kronecker squares

In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition ν¯ of d there is a polynomial kν¯ with rational coefficients in variables xC, where C runs over the set of isomorphism classes of connected skew diagrams of size at most d, such that for all partitions λ of n, the Kronecker coefficient g(λ,λ,(n−d,ν¯)) is obtained from kν¯(xC) substituting each xC by the number of partitions α contained in λ such that λ/α is in the class C. Some results of our method extend to arbitrary Kronecker coefficients. We present two applications. The first is a contribution to the Saxl conjecture, which asserts that if ρk=(k,k−1,…,2,1) is the staircase partition, then the Kronecker square χρ⊗χρ contains every irreducible character of the symmetric group as a component. Here we prove that for any partition ν¯ of size d there is a piecewise polynomial function sν¯ in one real variable such that for all k, one has g(ρk,ρk,(|ρk|−d,ν¯))=sν¯(k). The second application is a proof of a new stability property for arbitrary Kronecker coefficients.