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 k ν ¯ with rational coefficients in variables x C , 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 k ν ¯ ( x C ) substituting each x C 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 ρ k = ( k , k − 1 , … , 2 , 1 ) is the staircase partition, then the Kronecker square χ ρ ⊗ χ ρ 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 s ν ¯ in one real variable such that for all k, one has g ( ρ k , ρ k , ( | ρ k | − d , ν ¯ ) ) = s ν ¯ ( k ) . The second application is a proof of a new stability property for arbitrary Kronecker coefficients.