## Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type

Last update: 8 June 2014

## Notes and References

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.

## Introduction

The results in this thesis are concerned with the irreducible complex representations of a finite group $G$ with a $BN\text{-pair}$ and Weyl group $W$ of classical type, which appear in the induced permutation representation ${1}_{B}^{G}$ from a Borel subgroup $B$ of $G\text{.}$ These representations were constructed by Steinberg in [Ste1951] for $GL\left(n,q\right)\text{,}$ parametrized by partitions of $n$ 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 ${1}_{B}^{G}$ 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 ${A}_{n},$ ${B}_{n}$ $\text{(}n\ge 2\text{),}$ and ${D}_{n}$ $\text{(}n\ge 4\text{),}$ which specialize to irreducible representations of the Hecke algebra ${H}_{C}\left(G,B\right)$ affording the induced representation ${1}_{B}^{G}$. 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 ${1}_{B}^{G}\text{.}$ In the case that the Weyl group of $G$ is of type ${B}_{n},$ we obtain a formula for the degrees determined by the hook lengths of pairs of Young diagrams comparable to Steinberg's formula for $GL\left(n,q\right)\text{.}$ 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 $𝒜\left({B}_{n}\right)$ if a Coxeter system of type ${B}_{n},$ defined over the polynomial ring $ℚ\left[x,y\right]$ are constructed in Section 2.2. They are parametrized by pairs of partitions $\left(\alpha \right),$ $\left(\beta \right)$ with $|\alpha |+|\beta |=n$ and are rational representations, i.e. they are define over $K=ℚ\left(x,y\right)\text{.}$ The representations of the generic rings $𝒜\left({A}_{n}\right)$ and $𝒜\left({D}_{n}\right)\text{,}$ defined over $ℚ\left[x\right],$ corresponding to Coxeter systems of type ${A}_{n}$ and ${D}_{n}$ are obtained in section 2.3 as corollaries by considering appropriate specialized algebras of $𝒜\left({B}_{n}\right)\text{.}$ For the specialization $x\to 1,$ the representation of $𝒜\left({A}_{n}\right)$ 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 ${1}_{B}^{G}\text{.}$ 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 ${1}_{P}^{G}\text{,}$ where $P$ is a parabolic subgroup of $G\text{.}$ The generic degree ${d}_{\chi }$ of an irreducible character $\chi$ of the generic algebra is introduced, such that the degree of the corresponding irreducible character of $G$ is obtained by specializing ${d}_{\chi }\text{.}$ In Section 3.2 an interesting induction formula is derived for ${d}_{\chi }\text{,}$ where $\chi$ is an irreducible character of $𝒜\left({B}_{n}\right)\text{.}$ This formula, and lengthy computations, enables us to prove in Section 3.4 an explicit formula for ${d}_{\chi }$ as a rational function in the indeterminates $x$ and $y\text{,}$ in terms of the hook lengths of pairs of Young diagrams. The generic degrees of all the irreducible characters of $𝒜\left({A}_{n}\right)$ and almost all the irreducible characters of $𝒜\left({D}_{n}\right)$ are obtained as a corollary.