Vector Partition Functions And Kronecker Coefficients

The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group GL(nm) into irreducibles for the subgroup GL(n) x GL(m). In this work we study the quasipolynomial nature of the Kronecker function using elementary tools from polyhedral geometry. We write the Kronecker function in terms of coefficients of a vector partition function. This allows us to define a new family of coefficients, the atomic Kronecker coefficients. Our derivation is explicit and self-contained, and gives a new exact formula and an upper bound for the Kronecker coefficients in the first nontrivial case.