Some Studies On The Representations Of Finite Coxeter Groups And Hecke Algebras With Unequal Parameters

This thesis is concerned with some important knowledge of the repre-sentations of Coxeter groups and some relative properties of r-polynomials in Hecke algebra(we use H-algebra for short) with unequal parameters. This paper is composed of four parts.In chapter one, we briefly introduce some definitions, notations in Cox-eter groups and H-algebra with unequal parameters in this thesis.In chapter two, we first give the properties of character tables of finite groups, then we get the elements and conjugacy classes of B3 and some parabolic subgroups of B3, Next, we get the character tables of B3 and these parabolic subgroups of B3 Further more, we have the corresponding induction-restriction table.In chapter three, we introduce the special double cosets of reflection groups and give the general method of these double cosets, then we solve out some special double cosets of b3, B4 and D4.In chapter four, We first introduce some special properties of r-polynomials in H-algebra with unequal parameters, then we have the necessary and suf-ficient condition of one of these properties of r-polynomials.The main results in this thesis are the deep into study under some known results, we improved and generalized some former researches. Though the work of this paper, we have further knowledge for some properties of Coxeter groups and H-algebra with unequal parameters.