The Hecke--Coxeter complex and the Euler characteristic of a Hecke algebra

Details The Hecke--Coxeter complex and the Euler characteristic of a Hecke algebra The Hecke--Coxeter complex and the Euler characteristic of a Hecke algebra

2011-10-22

arxiv.org/abs/1110.4981v2

Mathematics

Representation Theory

17 pages

Scientific paper

For any Hecke algebra $H=H_q(W,S)$ associated to a Coxeter group $(W,S)$ and a distinguished element $q\in R$ of a commutative ring with unit $R$ we introduce a finite chain complex of left $H$-modules $(C,\partial)$ which reflects many properties of the Coxeter complex of $(W,S)$, i.e., it is acyclic if $(W,S)$ is non-spherical (cf. Thm. A), and $H$ is of type FP under suitable conditions on the distinguished element $q\in R$ (cf. Prop. B). There exists a canonical trace function $\mu:H\to R$ (cf. Prop. 5.1). This trace function $\mu$ evaluated on the Hattori-Stallings rank of $(C,\partial)$ can be considered as the Euler characteristic $\chi$ of $H$. It will be shown that for generic values of $q$ the Euler characteristic coincides with the reciprocal of the Poincar\'e series of $(W, S)$ evaluated in $q$ (cf. Thm. C).

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