tailieunhanh - Báo cáo toán học: "The Garnir relations for Weyl groups of type Cn"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Garnir relations for Weyl groups of type Cn. | The Garnir relations for Weyl groups of type Cn Himmet Can Department of Mathematics Faculty of Arts Sciences Erciyes University 38039 Kayseri Turkey can@ Submitted Nov 10 2007 Accepted May 16 2008 Published May 26 2008 Mathematics Subject Classifications 20F55 20C30 Abstract The Garnir relations play a very important role in giving combinatorial constructions of representations of the symmetric groups. For the Weyl groups of type Cn having obtained the alternacy relation we give an explicit combinatorial description of the Garnir relation associated with a A-tableau in terms of root systems. We then use these relations to find a K-basis for the Specht modules of the Weyl groups of type Cn. Introduction Although a great deal of progress has been made in generalizing the representation theory of symmetric groups to Weyl groups very little has been done using the combinatorial approach. The first attempt at providing such a generalization has been given by Morris 14 where the basic combinatorial concepts such as tableau tabloid etc. which were successful for symmetric groups as exemplified in the work of James 13 were interpreted in the context of root systems of Weyl groups. In recent years a further development of these ideas has appeared in Halicioglu and Morris 10 and Halicioglu 8 . In this alternative approach the Weyl groups of type An and Cn are used to motivate a possible generalization to Weyl groups in general. For the construction of a basis for the Specht modules of Weyl groups Halicioglu 8 has considered the root systems of simply laced type only . An Dn E6 E7 E8 and their parabolic subsystems. Later the present author 4 extended these ideas to deal with the root systems of type Cn. Having obtained the perfect systems Halicioglu 8 and the present author 4 conclude that the set of standard A- polytabloids is a basis. But they do not prove that standard A- polytabloids span the Specht module SA A . Inspired by the work of Peel 15 .