The composition factors of the principal indecomposable modules over the 0-Hecke algebra of type \(E_6\) (Q5958881)

Let \((W,S)\) be the finite Weyl group with \(S\) as its Coxeter generating set. For \(w\in W\), let \(R(w)=\{s\in S\mid l(ws)<l(w)\}\) and \(L(w)=\{s\in S\mid l(sw)<l(w)\}\), where one denotes by \(l(w)\) the minimal length of an expression of \(w\) as a product of simple reflections. To any Weyl group, one can associate a corresponding finite dimensional algebra called 0-Hecke algebra \(H\). \textit{P. N. Norton} pointed out [in J. Aust. Math. Soc., Ser. A 27, 337-357 (1978; Zbl 0407.16019)] that the principal indecomposable modules and the irreducible modules over the 0-Hecke algebra \(H\) are parametrized by a subset \(J\) of \(S\). One denotes by \(U(\widehat J)\) and \(M(\widehat J)\) the principle indecomposable module and the irreducible module parametrized by \(J\) respectively, where \(\widehat J=S-J\). For two subset \(J,L\) of \(S\), let \(C_{JL}=\) the number of times \(M(\widehat L)\) is a composition factor of \(U(\widehat J)\). In [loc. cit.] \textit{P. N. Norton} showed that \(C_{JL}=|Y_L\cap(Y_J)^{-1}|\) where \(Y_L=\{w\in W\mid R(w)=\widehat L\}\) and \((Y_J)^{-1}=\{w\in W\mid L(w)=\widehat J\}\). In this paper, the authors describe explicitly \(C_{JL}\) for the 0-Hecke algebra of type \(E_6\) by the canonical expression of every element in the Weyl group of type \(E_6\).