A remark on the permutation representations afforded by the embeddings of $O_{2m}^\pm(2^f)$ in $Sp_{2m}(2^f)$

TL;DR

This paper explores the permutation representations of symplectic groups over finite fields, revealing an isomorphism between modules induced by their natural action and coset actions of orthogonal subgroups.

Contribution

It establishes a new isomorphism between permutation modules of symplectic groups and those induced by orthogonal subgroup cosets over finite fields.

Findings

Permutation module over complex numbers for symplectic group action

Isomorphism with module from orthogonal subgroup cosets

Insight into permutation representations of finite classical groups

Abstract

We show that the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on its natural module is isomorphic to the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on the union of the right cosets of $O_{2m}^+(2^f)$ and $O_{2m}^-(2^f)$.