UBC Theses and Dissertations
Representations of Hecke Algebras of finite groups with BN-Pairs of classical type Hoefsmit, Peter Norbert
Let G be a finite group with BN-pair and Coxeter system (W, R). Let A be the generic ring corresponding to (W, R) in the sense of Tits, defined over the polynomial ring D = Q[u[sub r], r ε R]. Let k be any field of characteristic zero. For the homomorphism φ : D —> k defined by φ(u[sub r]) = q[sub r], q[sub r] the index parameters of G, the specialzed algebra Aφ,[sub k] is isomorphic to the Hecke algebra H[sub k](G, B) of G with respect to a Borel subgroup B of G, while for the specialization defined by φ(u[sub r]) = 1, r ε R, Aφ,[sub k] is isomorphic to the group algebra kW. As G the Hecke algebra H[sub k](G, B) affords the induced representation 1[sup G sub B], the G irreducible representations of G appearing in 1[sup G sub B] can be obtained from the representations of H[sub k](G, B). In this thesis, we obtain all the irreducible representations, defined over the quotient field of D, of the generic ring corresponding to a Coxeter system of classical type. The method employed involves a generalization of Young's construction of the semi-normal matrix representations of the symmetric group. We also obtain an explicit formula for the generic degree of these representations in terms of the hook lengths of Young diagrams. Thus the degrees of all the irreducible constituents of 1[sup G sub B] are obtained for the families of Chevalley groups A[sub l](q), B[sub l](q), A[sup 1 sub 2l](q²), A[sup 1 sub 2l-1](q²), D[sup 1 sub l](q²) and for D[sub l](q), l odd. Also, most of the degrees are obtained for D[sub l](q), l even.
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