Young characters on Coxeter basis elements of Iwahori-Hecke algebras and a Murnaghan-Nakayama formula.

*(English)*Zbl 0834.20013M. Geck and the author [Adv. Math. 102, No. 1, 79-94 (1993; Zbl 0816.20034)], defined the character table of a generic Iwahori-Hecke algebra associated to a finite Weyl group. This motivates the study of the character tables of all Iwahori-Hecke algebras associated to irreducible Weyl groups. The character tables of the Iwahori-Hecke algebras of exceptional types have been determined by M. Geck [Habilitationsschrift RWTH Aachen (1993)] with the exception of type \(E_8\). In [Invent. Math. 106, 461-488 (1991; Zbl 0758.05099)] A. Ram has proved an explicit formula for the character values of Iwahori- Hecke algebras of type \(A_n\) by rewriting solutions of the Quantum Yang-Baxter equation and by using Schur polynomials.

In this paper the author investigates representations induced from subalgebras corresponding to parabolic subgroups of a finite Weyl group \(W\). Let \(S' \subseteq S\). Then the \(T_s\), \(s\in S'\), generate a subalgebra \(H'\) of \(H\). If \(V\) is the module arising from the index representation of \(H'\) defined by \(T_s \mapsto q_s\), \(s \in S'\), then \(V \otimes_{H'} H\) is called a Young module. The corresponding character is called a Young character. The author proves a formula for the Young character values on Coxeter basis elements of the associated Iwahori-Hecke algebra \(H\). In the second section the author derives from these the values of the irreducible characters on Coxeter basis elements of type \(A_n\). This enables the author to give an elementary proof of a Murnaghan-Nakayama formula for the character table of Iwahori-Hecke algebras of type \(A\) similar to that by A. Ram [loc. cit.], by means of the Littlewood-Richardson rule. It is hoped that in a subsequent paper a corresponding formula for the Iwahori-Hecke algebras of type \(B_n\) and \(D_n\), will be provided.

In this paper the author investigates representations induced from subalgebras corresponding to parabolic subgroups of a finite Weyl group \(W\). Let \(S' \subseteq S\). Then the \(T_s\), \(s\in S'\), generate a subalgebra \(H'\) of \(H\). If \(V\) is the module arising from the index representation of \(H'\) defined by \(T_s \mapsto q_s\), \(s \in S'\), then \(V \otimes_{H'} H\) is called a Young module. The corresponding character is called a Young character. The author proves a formula for the Young character values on Coxeter basis elements of the associated Iwahori-Hecke algebra \(H\). In the second section the author derives from these the values of the irreducible characters on Coxeter basis elements of type \(A_n\). This enables the author to give an elementary proof of a Murnaghan-Nakayama formula for the character table of Iwahori-Hecke algebras of type \(A\) similar to that by A. Ram [loc. cit.], by means of the Littlewood-Richardson rule. It is hoped that in a subsequent paper a corresponding formula for the Iwahori-Hecke algebras of type \(B_n\) and \(D_n\), will be provided.

Reviewer: Chen Chengdong (Shanghai)

##### MSC:

20C30 | Representations of finite symmetric groups |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

20G05 | Representation theory for linear algebraic groups |