Computing Multiplicities of Lie Group Representations

TL;DR

This paper presents a polynomial time algorithm for computing multiplicities of Lie group representations, with implications for geometric complexity theory and the P vs. NP problem, especially for cases with bounded Young diagram rows.

Contribution

It introduces a finite difference formula-based algorithm that computes representation multiplicities efficiently, extending to Kronecker coefficients with bounded rows, and analyzes their complexity implications.

Findings

Algorithm computes multiplicities in polynomial time for bounded row cases.

Kronecker coefficients are computed efficiently when the number of rows is limited.

Asymptotic growth rates do not provide new complexity-theoretic obstructions.

Abstract

For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.