On the complexity of computing Kronecker coefficients

TL;DR

This paper investigates the computational complexity of Kronecker coefficients, providing bounds based on partition parameters and showing efficient positivity decision algorithms under certain conditions.

Contribution

It establishes explicit bounds on the complexity of computing Kronecker coefficients and introduces efficient algorithms for positivity testing when partitions have a bounded number of parts.

Findings

Bounds are linear in log N when M=O(1)

Positivity can be decided in O(log N) time for bounded parts

Similar bounds hold when M=e^{O(ell)}

Abstract

We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction on $M$. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of $S_n$ are also considered.