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Permutation groups and symmetric Hecke algebras

Jiawei He, Xiaogang Li

TL;DR

The paper investigates when endomorphism algebras $End_{KG}(K\Omega)$ of permutation modules are symmetric, framing this as symmetry of abstract Hecke algebras. It defines $p$-$S$-permutation and $S$-permutation groups and establishes broad, concrete sufficient criteria—covering subdegree coprimality, regular abelian $p'$-subgroups, rank-$3$ configurations, dihedral structures, and BN-pair quotients—that guarantee symmetry across many natural classes of permutation groups. By connecting End$_{KG}(K\Omega)$ to coherent algebras and Schur rings, the authors provide both general theory and explicit counterexamples (e.g., rank-$3$ dihedral subgroups and certain $GL(3,q)$-BN-pair cases) that delineate the boundaries of these symmetry properties. The work extends prior results, offers a unified framework, and poses several open questions on minimality, primitivity, and when all Schur rings over a group are symmetric, with potential implications for representation theory and combinatorial algebraic structures.

Abstract

The endomorphism algebras of the permutation modules for transitive permutation groups, known as Hecke algebras, are fundamental objects in representation theory. While group algebras are known to be symmetric over any field, it is natural to ask whether this property extends to Hecke algebras. To study this, we introduce the new concepts of $p$-$S$-permutation groups (for a prime $p$) and $S$-permutation groups. A \emph{ $p$-$S$-permutation group} is a transitive permutation group whose associated Hecke algebra is symmetric over every field of characteristic $p$. An \emph{ $S$-permutation group} is a transitive permutation group that is a $p$-$S$-permutation group for all primes $p$. In this paper, we study Hecke algebras from a group-theoretical perspective and we show that several classes of permutation groups are $p$-$S$-permutation groups and $S$-permutation groups in our sense. This result represents a substantial extension of earlier work by Li and He. (Transform Groups, 30(4), 2025), and reframes the question of determining when the algebra \(\End_{KG}(KΩ)\) is symmetric within a more general theoretical framework.

Permutation groups and symmetric Hecke algebras

TL;DR

The paper investigates when endomorphism algebras of permutation modules are symmetric, framing this as symmetry of abstract Hecke algebras. It defines --permutation and -permutation groups and establishes broad, concrete sufficient criteria—covering subdegree coprimality, regular abelian -subgroups, rank- configurations, dihedral structures, and BN-pair quotients—that guarantee symmetry across many natural classes of permutation groups. By connecting End to coherent algebras and Schur rings, the authors provide both general theory and explicit counterexamples (e.g., rank- dihedral subgroups and certain -BN-pair cases) that delineate the boundaries of these symmetry properties. The work extends prior results, offers a unified framework, and poses several open questions on minimality, primitivity, and when all Schur rings over a group are symmetric, with potential implications for representation theory and combinatorial algebraic structures.

Abstract

The endomorphism algebras of the permutation modules for transitive permutation groups, known as Hecke algebras, are fundamental objects in representation theory. While group algebras are known to be symmetric over any field, it is natural to ask whether this property extends to Hecke algebras. To study this, we introduce the new concepts of --permutation groups (for a prime ) and -permutation groups. A \emph{ --permutation group} is a transitive permutation group whose associated Hecke algebra is symmetric over every field of characteristic . An \emph{ -permutation group} is a transitive permutation group that is a --permutation group for all primes . In this paper, we study Hecke algebras from a group-theoretical perspective and we show that several classes of permutation groups are --permutation groups and -permutation groups in our sense. This result represents a substantial extension of earlier work by Li and He. (Transform Groups, 30(4), 2025), and reframes the question of determining when the algebra \(\End_{KG}(KΩ)\) is symmetric within a more general theoretical framework.
Paper Structure (15 sections, 23 theorems, 33 equations, 1 table)

This paper contains 15 sections, 23 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminology
  4. Chevalley groups and groups with a split $BN$-pair
  5. Hecke algebras
  6. Coherent algebras and Schur rings
  7. Coherent algebras
  8. Schur rings
  9. Symmetric algebra, Frobenius algebras and self-injective algebras
  10. Preliminary results
  11. Proof of Theorem \ref{['main']}
  12. Criterion for $p$-$S$-permutation groups
  13. Criterion for $S$-permutation groups
  14. Examples
  15. Further questions

Key Result

Theorem 1.2

Let $G$ be a transitive permutation group on a set $\Omega$ of cardinality $n$. Then the following statements hold.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 20 more