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Young characters on Coxeter basis elements of Iwahori-Hecke algebras and a Murnaghan-Nakayama formula. (English) Zbl 0834.20013
M. 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.

##### MSC:
 20C30 Representations of finite symmetric groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20G05 Representation theory for linear algebraic groups
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