The Loewy Length of the Descent Algebra of Type <Emphasis Type="Italic">D</Emphasis>

In this article the Loewy length of the descent algebra of D _2m + 1 is shown to be m + 2, for m ≥ 2, by providing an upper bound that agrees with the lower bound in Bonnafé and Pfeiffer (2006). The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D _2m + 1 is at most m + 1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group W is identified with an algebra associated to the reflection arrangement of W.