Permutation groups and symmetric Hecke algebras

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.

💡 Research Summary

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The paper investigates when the endomorphism algebra of a permutation module, (\End_{KG}(K\Omega)), is a symmetric algebra. Since group algebras are always symmetric over any field, the authors ask whether the same holds for Hecke algebras associated with transitive permutation groups. To formalise this question they introduce two new notions. A transitive permutation group (G) on a set (\Omega) is called a (p)-S‑permutation group (for a prime (p)) if (\End_{KG}(K\Omega)) is symmetric for every field (K) of characteristic (p). If the condition holds for all primes, (G) is called an S‑permutation group.

The main result, Theorem 1.2, provides a list of sufficient conditions for a group to be a (p)-S‑permutation group (six conditions) and for a group to be an S‑permutation group (six conditions). The (p)-S‑conditions include: (i) all subdegrees of (G) are coprime to (p); (ii) (p) does not divide any non‑trivial subdegree; (iii) (G) possesses an abelian regular (p’)-subgroup; (iv) the degree (n=|\Omega|) satisfies (n<2p); (v) (p) does not divide (n) and (G) has rank 3; (vi) (\Omega) is a coset space (G(q)/P) where (G(q)) is a finite group with a split BN‑pair at characteristic (q\neq p) and (P) is a parabolic subgroup whose order is coprime to (p).

The S‑conditions are more restrictive and guarantee symmetry for all characteristics. They state that a group is an S‑permutation group if it satisfies any of the following: (1) it is 3/2‑transitive (the point stabiliser acts half‑transitively on the remaining points); (2) it has a cyclic regular subgroup whose order is the product of two distinct primes; (3) it has an abelian regular subgroup whose order is coprime to every subdegree; (4) it is a rank 3 group satisfying a specific number‑theoretic condition involving its subdegrees and a parameter (\lambda(G)); (5) it is a dihedral group whose order is not divisible by 8; (6) (\Omega) is a coset space (G(q)/B) where (G(q)) has a split BN‑pair at characteristic (q\neq p) and the Weyl group is not isomorphic to (S_{3}).

The proofs combine modular representation theory, Green correspondence, and structural group theory. The authors use the identification (\End_{KG}(K\Omega)\cong H_{K}(G,H)) (the Hecke algebra associated with the pair ((G,H)) where (H) is a point stabiliser) and analyse when this algebra is Frobenius or, more strongly, symmetric. Lemma 2.2 shows that for groups with a split BN‑pair in characteristic (p), the corresponding Hecke algebra is a Frobenius algebra; since Frobenius algebras are symmetric under the given hypotheses, this yields condition (vi). Other conditions are handled by examining the decomposition of the permutation module, the existence of regular subgroups, and rank considerations. For instance, when the degree satisfies (n<2p), the permutation module splits into a direct sum of trivial and projective modules, forcing the endomorphism algebra to be symmetric.

Two illustrative examples are provided. First, regular permutation groups are shown to be S‑permutation groups because the point stabiliser is trivial, satisfying condition (iii) and (2). Second, all 3/2‑transitive groups (including the classical groups (\operatorname{PSL}(2,q)) and (\operatorname{P\Gamma L}(2,q)) with (q=2p)) satisfy condition (1) and thus are S‑permutation groups.

The paper concludes with several open problems, notably the question of whether every Schur ring over a given group is a symmetric algebra, and the extension of the classification to groups lacking a split BN‑pair.

Overall, the work provides a systematic, group‑theoretic framework for understanding when Hecke algebras of permutation groups are symmetric, unifying and extending earlier scattered results and opening new avenues for research in modular representation theory and algebraic combinatorics.