Last update: 8 June 2014
This is a typed copy of Peter Norbert Hoefsmit's thesis Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type published in August, 1974.
The results in this thesis are concerned with the irreducible complex representations of a finite group with a and Weyl group of classical type, which appear in the induced permutation representation from a Borel subgroup of These representations were constructed by Steinberg in [Ste1951] for parametrized by partitions of and he showed an elegant formula to hold for their degrees in terms of the hook lengths of a Young diagram. Some special representations of the generic ring of a Coxeter system and the degrees of the corresponding irreducible constituents of were also obtained by Kilmoyer (cf. [CIK1971]).
In this thesis we construct explicitly all the irreducible representations of the generic ring corresponding to a Coxeter system of classical type, i.e. of type and which specialize to irreducible representations of the Hecke algebra affording the induced representation . The method employed involves a generalization of Young's construction of the semi-normal matrix representations of the symmetric group. This construction also enables us to compute the degrees of the irreducible constituents of In the case that the Weyl group of is of type we obtain a formula for the degrees determined by the hook lengths of pairs of Young diagrams comparable to Steinberg's formula for Indeed, Steinberg's formula is recovered as a special case.
Here is a survey of the contents of this thesis. Chapter 1 contains the necessary preliminaries about the representation theory of the symmetric group. In Chapter 2 the generic ring corresponding to a Coxeter system is introduced. The irreducible representations of the generic ring if a Coxeter system of type defined over the polynomial ring are constructed in Section 2.2. They are parametrized by pairs of partitions with and are rational representations, i.e. they are define over The representations of the generic rings and defined over corresponding to Coxeter systems of type and are obtained in section 2.3 as corollaries by considering appropriate specialized algebras of For the specialization the representation of obtained specialize to give the semi-normal matrix representations of the symmetric group.
Chapter 3 is concerned with the degrees of the irreducible constituents of It is first shown in Section 3.1 that the representations obtained are of parabolic type, i.e. they appear with multiplicity one in some permutation representation where is a parabolic subgroup of The generic degree of an irreducible character of the generic algebra is introduced, such that the degree of the corresponding irreducible character of is obtained by specializing In Section 3.2 an interesting induction formula is derived for where is an irreducible character of This formula, and lengthy computations, enables us to prove in Section 3.4 an explicit formula for as a rational function in the indeterminates and in terms of the hook lengths of pairs of Young diagrams. The generic degrees of all the irreducible characters of and almost all the irreducible characters of are obtained as a corollary.