Symmetric Groups

Let (μ, ν, λ) be a triple of partitions. The Kronecker coefficient is the tensor product multiplicity for the irreducible representations of the symmetric group:

We knew from [1] that it can be computed by counting lattice points in at most m! fibre (rational) polytopes inside the polyhedral cone . More precisely, let μ, ν (resp. λ) be partitions of length ≤l (resp. ≤m ). Then

where λ^ω is the vector in Z^m such that λ^ω(i)= λ(i)-i+ω(i), S_m(λ) = {ω∈S_m | λ^ω is non-negative}, and | - | is the lattice-point counting. The polytope can be explicitly presented as

{g | Hg ≥ 0, g^Tσ = (μ,ν,λ)}

Here we provide the matrix H and σ for l=m=3 and l=m=4. You can plug the H and σ matrices into your favorite lattice point counting programs (eg., Latte, Polymake, and Normaliz). I also wrote a Sage code based on this. This is by far the fastest algorithm for computing Kronecker coefficients.

1. J. Fei, Cluster Algebras, Invariant Theory, and Kronecker Coefficients II, arXiv.