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The eigenvalue one property of finite groups, II

Gerhard Hiss, Rafał Lutowski

TL;DR

The paper resolves the Dekimpe–De Rock–Penninckx conjecture (DDRP) by proving that finite groups of Lie type in even characteristic are not minimal HL1-counterexamples, thereby establishing an eigenvalue-one property for irreducible real representations of odd dimension. The authors develop a comprehensive framework combining large-degree bounds, restriction methods, and Deligne–Lusztig theory (including the generalized Jordan decomposition and semisimple characters), together with detailed automorphism and centralizer analyses, to treat groups of type PSL, PSU, $E_6$, and $P ext{O}_8^+$, among others. They provide explicit bounds and a case-splitting strategy that covers linear, unitary, orthogonal, and exceptional groups (notably $E_6$ and its twisted form), supplemented by computational tools (GAP/Chevie) for intractable subcases. The results yield a new sufficient condition for closed flat manifolds to be $R_ ext{inf}$-manifolds and underscore the broader relevance of the eigenvalue-one phenomenon in the representation theory of finite groups of Lie type.

Abstract

We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an $R_{\infty}$-manifold.

The eigenvalue one property of finite groups, II

TL;DR

The paper resolves the Dekimpe–De Rock–Penninckx conjecture (DDRP) by proving that finite groups of Lie type in even characteristic are not minimal HL1-counterexamples, thereby establishing an eigenvalue-one property for irreducible real representations of odd dimension. The authors develop a comprehensive framework combining large-degree bounds, restriction methods, and Deligne–Lusztig theory (including the generalized Jordan decomposition and semisimple characters), together with detailed automorphism and centralizer analyses, to treat groups of type PSL, PSU, , and , among others. They provide explicit bounds and a case-splitting strategy that covers linear, unitary, orthogonal, and exceptional groups (notably and its twisted form), supplemented by computational tools (GAP/Chevie) for intractable subcases. The results yield a new sufficient condition for closed flat manifolds to be -manifolds and underscore the broader relevance of the eigenvalue-one phenomenon in the representation theory of finite groups of Lie type.

Abstract

We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an -manifold.
Paper Structure (28 sections, 36 theorems, 85 equations, 2 tables)

This paper contains 28 sections, 36 theorems, 85 equations, 2 tables.

Table of Contents

  1. Introduction
  2. General preliminaries
  3. The elementary cases
  4. The groups
  5. Restricting to a maximal unipotent subgroup
  6. Preliminaries for the remaining groups
  7. The $\sigma$-setup for the remaining groups
  8. The dual groups
  9. Automorphisms, I
  10. Semisimple characters
  11. Bounds on orders of centralizers and elements
  12. Bounds on orders of centralizers and elements
  13. The proof for the eight-dimensional orthogonal group
  14. The linear and unitary groups
  15. On the natural representation of the linear and unitary groups
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $a \in \mathbb{R}$ and $m \in \mathbb{Z}$ with $a, m \geq 2$. Then the following inequalities hold. (a) We have (b) We have

Theorems & Definitions (76)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Lemma 4.4
  • proof
  • ...and 66 more