Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalised Burnside and Dixon algorithms for irreducible projective representations

Attila Szabó

TL;DR

The paper extends the classical Burnside and Dixon algorithms to the realm of irreducible projective representations by employing the character theory for projective representations with a unitary Schur multiplier $\alpha$. It develops a Burnside-type eigenvalue framework using $\alpha$-regular classes and matrices $M_A$ to compute irreducible characters without constructing the representation group $G^*$, and adapts Dixon's finite-field approach to obtain exact character data in $\mathbb{Z}_p$ before lifting back to $\mathbb{C}$. It further provides a decomposition strategy for projective representations via automorphism-averaging maps $\langle f\rangle_\pi$, enabling iterative extraction of all irreducible projective components from the regular representation and from faithful starting representations. Collectively, these results deliver floating-point-friendly and exact modular pathways to characterize and assemble all projective irreducibles with a fixed multiplier, with practical implications for computational representation theory of finite groups. The methods bypass explicit representation groups, improving numerical robustness when multiplier values are only approximately known.

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.

Generalised Burnside and Dixon algorithms for irreducible projective representations

TL;DR

The paper extends the classical Burnside and Dixon algorithms to the realm of irreducible projective representations by employing the character theory for projective representations with a unitary Schur multiplier . It develops a Burnside-type eigenvalue framework using -regular classes and matrices to compute irreducible characters without constructing the representation group , and adapts Dixon's finite-field approach to obtain exact character data in before lifting back to . It further provides a decomposition strategy for projective representations via automorphism-averaging maps , enabling iterative extraction of all irreducible projective components from the regular representation and from faithful starting representations. Collectively, these results deliver floating-point-friendly and exact modular pathways to characterize and assemble all projective irreducibles with a fixed multiplier, with practical implications for computational representation theory of finite groups. The methods bypass explicit representation groups, improving numerical robustness when multiplier values are only approximately known.

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.

Paper Structure

This paper contains 14 sections, 32 theorems, 30 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Review of the character theory of projective representations
  3. (Schur) multipliers
  4. Projective representations
  5. Characters of projective representations and $\alpha$-class functions
  6. Burnside's algorithm for projective irrep characters
  7. Consistency relations for $\alpha$-class functions
  8. Proof of \ref{['thm: burnside']}
  9. Dixon's algorithm for computing exact characters
  10. Brief review of Dixon's character table algorithm
  11. Generalisation to projective representations
  12. Dixon's decomposition algorithm for projective representations
  13. Decomposing the regular projective representation
  14. Iterative construction

Key Result

Theorem 1

Let $\alpha$ be a unitary Schur multiplier (def: multiplier) on $G$. Let $C^{(1)},\dots,C^{(m)}$ be the $\alpha$-regular conjugacy classes of $G$ (cf. thm: regular classes) and fix one representative element $c^{(i)}_0\in C^{(i)}$ for all $1\le i\le m$. Define the $m\times m$ matrices $M_A$ for each where $a_0,b_0,c_0$ are the chosen representatives of the $\alpha$-regular classes $A,B,C$ and for

Theorems & Definitions (65)

  • Theorem 1: Burnside's algorithm for character tables of projective irreps
  • Theorem 2: Dixon's algorithm for character tables of projective irreps
  • Theorem 3: Dixon's algorithm for decomposing projective representations into irreps
  • Definition 2.1
  • Lemma 2.2: Cheng2015CharacterTheory, §1
  • Lemma 2.3: Cheng2015CharacterTheory, Lemma 3.1
  • Definition 2.4
  • Remark
  • Lemma 2.5
  • proof
  • ...and 55 more