Permutation groups and symmetric Hecke algebras

TL;DR

This paper introduces new classes of permutation groups called $p$-$S$-permutation and $S$-permutation groups, studying when their associated Hecke algebras are symmetric over various fields, extending prior work in the field.

Contribution

The paper defines $p$-$S$-permutation and $S$-permutation groups and shows several classes of groups belong to these categories, broadening understanding of symmetry properties of Hecke algebras.

Findings

Several classes of permutation groups are $p$-$S$-permutation groups.

Several classes of permutation groups are $S$-permutation groups.

Extension of earlier results by Li and He.

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 \(\End_{KG}(K\Omega)\) is symmetric within a more general theoretical framework.