The complexity of computing Kronecker coefficients

Abstract

Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group S n. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem KRONCOEFF of computing Kronecker coefficients is very difficult. More specifically, we prove that KRONCOEFF is #P-hard and contained in the complexity class GapP. Formally, this means that the existence of a polynomial time algorithm for KRONCOEFF is equivalent to the existence of a polynomial time algorithm for evaluating permanents. © 2008 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.