Certifying Numerical Decompositions of Compact Group Representations

TL;DR

This paper introduces a rigorous, efficient algorithm for certifying that matrices approximate projections onto irreducible subspaces in group representations, improving reliability and performance over existing heuristics.

Contribution

The authors develop a new certification algorithm with proven guarantees for compact groups, reducing computational complexity and interfacing with existing software like RepLAB.

Findings

The algorithm provides rigorous certification guarantees for group representation decompositions.

Complexity is reduced to O(max{n^3 log n, D d^2 log d}) compared to previous methods.

Implementation demonstrates practical efficiency and integration with existing tools.

Abstract

We present a performant and rigorous algorithm for certifying that a matrix is close to being a projection onto an irreducible subspace of a given group representation. This addresses a problem arising when one seeks solutions to semi-definite programs (SDPs) with a group symmetry. Indeed, in this context, the dimension of the SDP can be significantly reduced if the irreducible representations of the group action are explicitly known. Rigorous numerical algorithms for decomposing a given group representation into irreps are known, but fairly expensive. To avoid this performance problem, existing software packages -- e.g. RepLAB, which motivated the present work -- use randomized heuristics. While these seem to work well in practice, the problem of to which extent the results can be trusted arises. Here, we provide rigorous guarantees applicable to finite and compact groups, as well as a software implementation that can interface with RepLAB. Under natural assumptions, a commonly used previous method due to Babai and Friedl runs in time O(n^5) for n-dimensional representations. In our approach, the complexity of running both the heuristic decomposition and the certification step is O(max{n^3 log n, D d^2 log d}), where d is the maximum dimension of an irreducible subrepresentation, and D is the time required to multiply elements of the group. A reference implementation interfacing with RepLAB is provided.