An elementary approach to the quasipolynomiality of the Kronecker coefficients

Marni Mishna, Mercedes Rosas, Sheila Sundaram

Published 2018-11-25Version 1

The Kronecker coefficients are the structure constants for the decomposition into irreducibles of the tensor product of representations of the symmetric group. In this work we study the piecewise quasipolynomial nature of the Kronecker function using tools from polyhedral geometry. By bounding the lengths of the partitions, we can write the Kronecker function in terms of coefficients of vector partition functions. We give exact formulas in the small cases, and prove that it is always possible. An additional advantage of this approach is that asymptotic estimates for dilations are computable using techniques of analytic combinatorics in several variables.