Generalised Burnside and Dixon algorithms for irreducible projective representations

TL;DR

This paper extends classical algorithms to compute irreducible projective representations of finite groups, enabling efficient calculations without constructing the representation group, especially useful for floating-point computations.

Contribution

It introduces generalized Burnside and Dixon algorithms for projective representations, bypassing the need for representation group construction and suitable for floating-point arithmetic.

Findings

Algorithms successfully compute characters of all irreducible projective representations.

Methods enable splitting projective representations into irreducible components.

Approach is effective even when exact Schur multiplier values are unavailable.

Abstract

Based on the recently proposed character theory of projective representations of finite groups proposed, we generalise several algorithms for computing character tables and matrices of irreducible linear representations to projective representations. In particular, we present an algorithm based on that of Burnside to compute the characters of all irreducible projective representations of a finite group with a given Schur multiplier, and transpose it to exact integer arithmetic following Dixon's character table algorithm. We also describe an algorithm based on that of Dixon to split a projective representation into irreducible subspaces in floating-point arithmetic, and discuss how it can be used to compute matrices for all projective irreps with a given multiplier. Our algorithms bypass the construction of the representation group of the Schur multiplier, which makes them especially attractive for floating-point computations, where exact values of the multiplier are not necessarily available.