Quasi-Parabolic Subgroups of G(m, 1, r)

We study the properties of cosets and double cosets of quasi-parabolic subgroups via tableaux. We characterize these double cosets by n × n arrays of non-negative integer sequences. For each double coset, we construct and count all shortest and longest representatives and find their lengths. We also find the number of double cosets in G(m, 1, r) of each pair of quasi-parabolic subgroups.