Modular representation theory over a ring of higher dimension with applications to Hecke algebras. (English) Zbl 0833.20054
Let \((W,S)\) be a Coxeter system, \(r = (r_s)_{s\in S}\) a system of indeterminates such that \(r_s = r_{s'}\) if and only if \(s\) and \(s'\) are \(W\)-conjugate and different \(r_s\)’s are algebraically independent, \(R = \mathbb{Z}[r]\), and \(H(R)\) a free \(R\)-module with a basis \(\{T_W\}_{w \in W}\). Let \(F\) be a field, \(O : R \to F\) a ring homomorphism, and assume \(W\) finite. The author shows that \(H(R) \otimes F\) is semisimple if and only if the specialization by \(O\) of every generic degree divided by the Poincaré polynomial is well-defined.
Reviewer: Chiu Sen (Shanghai)
MSC:
20G05 | Representation theory for linear algebraic groups |
20C20 | Modular representations and characters |
20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
20C30 | Representations of finite symmetric groups |