Generalised Burnside and Dixon algorithms for irreducible projective representations

2025-05-20

AI Summary

This paper addresses the computational problem of irreducible projective representations of finite groups. Methodologically, it introduces a universal algorithm that avoids constructing the Schur covering group. Specifically, it systematically generalizes two classical algorithms to the projective setting: (i) the Burnside character algorithm—extended using projective character theory and exact integer arithmetic—to compute all irreducible projective characters and their representing matrices for a given Schur multiplier; and (ii) the Dixon numerical decomposition algorithm—adapted to floating-point arithmetic—to decompose projective representations into irreducible subspaces. The key contribution is circumventing Schur covering group construction, thereby overcoming the traditional reliance on exact knowledge of the Schur multiplier in floating-point environments. Experimental results demonstrate that this dual-path framework significantly enhances the stability, efficiency, and applicability of projective representation computation.

Technology Category

Knowledge Representation and Reasoning: ApplicationsReasoning under Uncertainty: Uncertainty RepresentationsMachine Learning: Representation Learning

Application Category

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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.

Problem

• Generalize algorithms for irreducible projective representations
• Compute character tables without Schur multiplier construction
• Split projective representations into irreducible subspaces efficiently

Innovation

• Generalize Burnside algorithm for projective representations
• Transpose Dixon's algorithm to exact integer arithmetic
• Bypass representation group for floating-point computations