On the computation of Kronecker coefficients


A Kronecker coefficient $g(\lambda, \mu, \nu)$ is a non-negative integer that depends on three partitions $\lambda$, $\mu$, $\nu$ of a natural number $n$. It is the multiplicity of an irreducible representation $V^\nu$ of the symmetric group of degree n in the tensor (or Kronecker) product $V^\lambda \otimes V^\mu$ of two other irreducible representations of the same group.

The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients but are still very far to be fully grasped.

It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes. In this talk we present a new family of polytopes that permit efficient computations on Kronecker coefficients associated to partitions with few parts and provides insight in the behavior of Murnaghan stability.