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

Abstract

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