We parametrize the space of double cosets group GL(n,k) with respect to two subgroups T−, T+ block strictly triangular matrices. In appendix, we consider quasi-regular representation GL(n,C) in L2 on T−∖GL(n,C), observe that it admits an additional symmetries, find joint spectrum, and is multiplicity free.

In this paper, we prove that a Cayley digraph Γ = Cay(G, S) is a nontrivial lexicographical product if and only if there is a nontrivial subgroup H of G such that S \ H is a union of some double cosets of H in G.

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig’s G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [GKP].

In this project, we are going to study some results concerning the character rings and Heche Algebras in Sk and GL(k, F ). In next section, we will introduce some notation system in Sk and GL(k, F ) first, and with that, we will deduce some interesting and important results in this field, such as the results about the length function on W and the double B − B cosets, where W is the Weyl group o...

There are several results concerning double cosets of Young subgroups of symmetric groups which are often proved as corollaries of more general results about Coxeter groups. In this note we give a selection of these results and give them the short combinatorial proofs they deserve. Our replacement for the machinery of Coxeter groups consists of some simple combinatorial arguments involving tabl...

There are several results concerning double cosets of Young subgroups of symmetric groups which are often proved as corollaries of more general results about Coxeter groups. In this note we give a selection of these results and give them short combinatorial proofs. The replacement for the machinery of Coxeter groups consists of some simple combinatorial arguments involving tableaux. Inevitably ...