Cartan matrix over the 0-Hecke algebra of type F4

Abstract

Let (W,S) be the finite Weyl group with S as its Coxeter generating set. For w∈W, let R(w)={si∈S∣l(wsi)<l(w)} and L(w)={si∈S∣l(siw)<l(w)}, where we denote 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 where K is any field. Norton [J. Austral. Math. Soc. Ser. A 27 (1979) 337–357] pointed out that the principal indecomposable modules and the irreducible modules over the 0-Hecke algebra H parametrized by a subset J of S. We denote by U(Ĵ) and M(Ĵ) respectively the principal indecomposable module and the irreducible module parametrized by J. For two subset J, L of S, let CJL= the number of times M(L) is a composition factor of U(Ĵ). Norton [J. Austral. Math. Soc. Ser. A 27 (1979) 337–357] shown that CJL=|YL∩(YJ)−1| where YL={w∈W∣R(w)=L} and (YJ)−1={w∈W∣L(w)=J}. In this article, we describe explicitly CJL for the 0-Hecke algebra of type F4 by applying the canonical expression of every element in the Weyl group of type F4. Thus we determine the Cartan matrix over the 0-Hecke algebra of type F4.