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 $GL(n)\times GL(m)$. 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.