The composition factors of the principal indecomposable modules over the 0-Hecke algebra of type E-6

Let (W S) be the finite Weyl group with S as its Coxeter generating set. Fo r w is an element of W, let R(w) = {s(i) is an element of S \l(ws(i)) < l(w )} and L(w) = {s(i) is an element of S \l(s; w) < l(w)}, where we denote by l(w) the minimal length of an expression of tv as a product of simple refl ections. To any Weyl group, one can associate a corresponding finite dimens ional algebra called 0-Hecke algebra H, where K is any field. [Norton J Aus tral. Math. Soc. 27A (1979), 337-357] pointed out that the principal indeco mposable modules and the irreducible modules over the 0-Hecke algebra H are parametrized by a subset J of S. We denote by U((J) over cap) and M((J) ov er cap) the principal indecomposable module and the irreducible module para metrized by J, respectively, where (J) over cap = S - J. For two subset, J, L of S, let C-JL denote the number of times M((L) over cap) is a compositi on factor of U((J) over cap). Norton showed that C-JL = \Y-L boolean AND (Y -J)(-1)\, where Y-L = {w is an element of W \R(w) = (L) over cap} and (Y-J) (-1) = {w is an element of W \L(w) = (J) over cap. In this paper, we descri be explicitly C-JL for the 0-Hecke algebra of type E-6 by applying the cano nical expression of every element in the Weyl group of type E-6. (C) 2001 A cademic Press.