On Involutions In Weyl Groups

The thesis focuses on the study of involutions in Weyl groups.The study of(twist-ed)involutions has both combinatorial and geometric background.It originates from the study of the Bruhat order on symmetric varieties and is closely related to the geom-etry of symmetric orbit closure,(e.g.,Schubert calculus).There have been a number of important works on algebraic and combinatorial properties of involutions in Coxeter groups.Many of them have arisen independently in different geometric and combina-torial contexts.A well-known classical fact of Matsumoto says that any two reduced expressions for an element in a Weyl group W can be transformed into each other through a series of basic braid transformations.On the other hand,every involution in Weyl group can be expressed as a twisted product(which is similar but different from the usual product)of some simple reflections,and reduced I*-expression and twisted length can be defined in a similar way.It is natural to expect an analogue of Matsumoto's theory for involutions and their reduced I*-expressions.We consider the involutions in Weyl groups and study the braid I*-transformations between the reduced I*-expressions of involutions.If W is the Weyl group of type An,Bn,Dn and F4,we explicitly describe a finite set of basic braid I*-transformations for all n simultaneously,and show that any two reduced I*-expressions for a given involution can be transformed into each other through a series of basic braid I*-transformations.As an application,we prove a conjecture of Lusztig for the involutions in the Weyl groups of type A.The thesis is organized as follows:In Chapter 1,we introduce the background of twisted involutions and a conjecture of Lusztig,recall some preliminary and known results on reduced I*-expressions for twisted involutions and outline the main results and the structure of the thesis.In Chapter 2,we begin with introducing a new list of braid I*-transformations for the Weyl group of type A which consist of the usual basic braid transformations plus some natural "right end transformations".Then we prove in Theorem 2.1.1 that any two reduced I*-expressions for an involution in Cn can be transformed into each other through a series of braid I*-transformations.This key result will play a central role in the proof of Theorem 2.3.1.In Section 2,we use the Young seminormal bases theory for the semisimple Iwahori-Hecke algebra of type An-1 to show that the dimension of HQ(u)X0 is bigger or equal than the number of involutions in W.In Section 3,as an application,we prove Lusztig,s Conjecture 1.2.1 in the case when*= idw and W is symmetric group Cn for any n ? N.In Chapter 3,we consider the type D case by a similar method as used in the type A case.In this case,the basic braid I*-transformations consist of the usual basic braid transformations plus some natural "right end transformations" and exactly one extra transformation,which is a new phenomenon which does not happen in type A.As an application,we verify partially Lusztig's Conjecture 1.2.1 for the involutions in Weyl groups of type D.In Chapter 4,we deal with the type B case by a similar method as used in the type D case.In this case,the basic braid I*-transformations consist of the usual basic braid transformations plus some natural "right end transformations" and exactly one extra transformation.As an application,we verify partially Lusztig's Conjecture 1.2.1 for the involutions in Weyl groups of type B.In Chapter 5,we are concerned with exceptional Weyl groups by a different method.In type F4 case,the basic braid I*-transformations consist of the usual basic braid transformations plus some natural "right end transformations" and two extra transfor-mations.In type G2 case,the basic braid I*-transformations only consist of two "right end transformations",which is trivial.For other types,we haven't found an effective way other than the method we used in type F4 case.A summary of this thesis and some unsolved problems are presented at the end.