On a Bruhat decomposition related to the Shalika subgroup of $GL(2n)$

Let $F$ be a non-archimedean local field or a finite field. In this article, we obtain an explicit and complete set of double coset representatives for $S\mathrm{\backslash }G{L}_{2n}(F)\mathrm{/}Q$ where $S$ is the Shalika subgroup and $Q$ a maximal parabolic subgroup of the group $G{L}_{2n}(F)$ of invertible $2n\times 2n$ matrices. We compute the cardinality of $S\mathrm{\backslash }G{L}_{2n}(F)\mathrm{/}Q$ and also give an alternate perspective on the double cosets arising intrinsically from certain subgroups which are relevant for applications in representation theory. Finally, if $Q$ is a maximal parabolic subgroup of the type $(r,2n-r),$ we prove that $S\mathrm{\backslash }G{L}_{2n}(F)\mathrm{/}Q$ is in one to one correspondence with $\mathrm{\Delta }{S}_n\mathrm{\backslash }{S}_{2n}\mathrm{/}{S}_r\times S_{2n-r}$ leading to a Bruhat decomposition. The results and proofs discussed in this article are valid over any arbitrary field $F$ even though our motivation is from representation theory of $p$-adic and finite linear groups.