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Unisingular representations of rank 1 finite simple groups of Lie type

Marco Antonio Pellegrini, Lorenzo Schena

TL;DR

This work resolves the complex unisingular representations for all rank-1 finite simple groups of Lie type by explicitly classifying when an irreducible character $oldsymbol{ ho}$ fails to be unisingular, using the eigenvalue-multiplicity formula $M_{oldsymbol{ ho}}(g)= rac{1}{|g|} ext{ }igl(oldsymbol{ ho}(I)+oldsymbol{ ho}(g)+oldsymbol{ ho}(g^{2})+igr)$ and Clifford theory to reduce to centerless quotients. For $ ext{PSL}_2(q)$ and $ ext{PSU}_3(q)$ (including their projective covers $ ext{PGL}_2(q)$ and $ ext{PGU}_3(q)$), the paper isolates precise degree-exception cases (e.g., $ oldsymbol{ ho}(1) eq q$, $(q-1)/2$ when $q ot o 4$, etc.) that yield non-unisingular characters, while all other irreducibles are unisingular. It then shows that every nonlinear irreducible character of the Suzuki groups $ ext{Sz}(q^2)$ with $q^2=2^{2m+1}$ and of the small Ree groups ${}^2 ext{G}_2(3^{2m+1})$ is unisingular (with a few low-rank exceptions for the smallest groups). Finally, using GAP4, it fully classifies almost simple sporadic groups with non-unisingular representations, providing a comprehensive catalog of the exceptional nonlinear cases in the sporadic realm.

Abstract

A representation $Φ: G \to \mathrm{GL}_n(\mathbb{F})$ of a finite group $G$ is called unisingular if the matrix $Φ(g)$ admits $1$ as an eigenvalue for any $g\in G$. In this paper, we determine all the complex irreducible unisingular representations of the finite simple groups of Lie type of rank $1$ and of the almost simple sporadic groups.

Unisingular representations of rank 1 finite simple groups of Lie type

TL;DR

This work resolves the complex unisingular representations for all rank-1 finite simple groups of Lie type by explicitly classifying when an irreducible character fails to be unisingular, using the eigenvalue-multiplicity formula and Clifford theory to reduce to centerless quotients. For and (including their projective covers and ), the paper isolates precise degree-exception cases (e.g., , when , etc.) that yield non-unisingular characters, while all other irreducibles are unisingular. It then shows that every nonlinear irreducible character of the Suzuki groups with and of the small Ree groups is unisingular (with a few low-rank exceptions for the smallest groups). Finally, using GAP4, it fully classifies almost simple sporadic groups with non-unisingular representations, providing a comprehensive catalog of the exceptional nonlinear cases in the sporadic realm.

Abstract

A representation of a finite group is called unisingular if the matrix admits as an eigenvalue for any . In this paper, we determine all the complex irreducible unisingular representations of the finite simple groups of Lie type of rank and of the almost simple sporadic groups.

Paper Structure

This paper contains 12 sections, 12 theorems, 114 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The linear groups $\mathrm{PSL}_2(q)$
  4. The group $\mathrm{PGL}_2(q)$ with $q$ even
  5. The group $\mathrm{PGL}_2(q)$ with $q$ odd
  6. The group $\mathrm{PSL}_2(q)$
  7. The unitary groups $\mathrm{PSU}_3(q)$
  8. The group $\mathrm{PGU}_3(q)$ with $\gcd(q+1,3)=1$
  9. The group $\mathrm{PGU}_3(q)$ with $\gcd(q+1,3)=3$
  10. The group $\mathrm{PSU}_3(q)$
  11. Suzuki groups and Small Ree groups
  12. Sporadic groups

Key Result

Theorem 1.1

Let $G$ be a rank $1$ finite simple group of Lie type. An irreducible character $\chi$ of $G$ is unisingular if and only if none of the following cases occurs:

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • ...and 5 more