Strong minuscule elements in the finite Weyl groups

Abstract

We introduce the notion of a strong minuscule element, which is a dominant minuscule element $w$ in the Weyl group for which there exists a unique (dominant) integral weight $\Lambda$ such that $w$ is $\Lambda$-minuscule. We prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root (in this paper, all simple roots in the simply-laced cases are treated as short roots). In addition, we enumerate the strong minuscule elements explicitly. As an application of this enumeration, we determine the dimension of certain Demazure module in the finite-dimensional irreducible module whose highest weight is a minuscule weight.