Abstract

We consider quasi-parabolic subgroups of the Weyl group W(D_n) of type D_n, which are intersections of W(D_n) with quasi-parabolic subgroups of the Weyl group W(B_n) of type B_n (see [J. Du, L. Scott, The q-Schur² algebra, Trans. Amer. Math. Soc. 352 (2000) 4325–4353] and [C.K. Mak, Quasi-parabolic subgroups of G(m,1,r), J. Algebra 246 (2001) 471–490]). We study the properties of cosets of these subgroups in W(D_n). A length function formula of type D_n is derived. A complete set of right coset representatives of these subgroups is constructed. We show that each of these representatives is of minimum length (with respect to both type B_n and type D_n length functions) in the coset it belongs to. Characterizations of these representatives via certain tableaux are given. Finally, a complete set of double coset representatives of quasi-parabolic subgroups in W(D_n) is also obtained, and we show that each of these representatives is of minimum length with respect to type B_n length functions in the double coset it belongs to.

Article Outline

1. Introduction

2. The Weyl groups W(B_n) and W(D_n)

3. A length function formula

4. Quasi-parabolic subgroups and their cosets

References

Research supported by the URF of Victoria University of Wellington and the National Natural Science Foundation of China (Project 10401005) and the Program NCET. The author wishes to thank the School of Mathematics, Statistics and Computer Science at VUW for their hospitality during his visit in 2005. He also greatly appreciates the referee’s careful reading, corrections and many valuable suggestions.