Dimensions of the projective indecomposable modules over classical 0- Hecke algebras (Q1200685)

Let \(W\) be a classical Weyl group and \(\Pi\) be the corresponding system of simple roots. For \(w\in W\), let \(R(w)=\{\alpha\in\Pi\mid \ell(ws_ \alpha)<\ell(w)\}\). To any Weyl group one can associate a 0-Hecke algebra \(H(0)\). Norton pointed out that the projective indecomposable modules \(P(J)\) over the 0-Hecke algebra are parametrized by a subset \(J\) of \(\Pi\) and \(\dim P(J)=|\{w\in W; R(w)=J\}|\) [\textit{P. N. Norton}, J. Aust. Math. Soc., Ser. A 27, 337-357 (1978; Zbl 0407.16019)]. In this paper we compute the number of elements in the set \(\{w\in W;\;R(w)=J\}\) applying the canonical expression of an element in the classical Weyl group given by the author [J. Tongji Univ. 15, No. 3, 381-390 (1987; Zbl 0669.20038)]. And we obtain the dimension formulas of the modules \(P(J)\).