# Strong minuscule elements in the finite Weyl groups

## TL;DR

This paper introduces the concept of strong minuscule elements in finite Weyl groups, characterizes their associated weights, enumerates them explicitly, and applies this to determine dimensions of specific Demazure modules.

## Contribution

It defines strong minuscule elements, links them to fundamental weights, provides explicit enumeration, and applies findings to Demazure module dimension calculations.

## Key findings

- Strong minuscule elements are associated with fundamental weights of short simple roots.
- Explicit enumeration of all strong minuscule elements in finite Weyl groups.
- Determination of Demazure module dimensions for certain minuscule weights.

## Abstract

We introduce the notion of a strong minuscule element, and prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root. In addition, we enumerate the strong minuscule elements explicitly, and then as an application of this enumeration, determine the dimension of certain Demazure modules in the finite-dimensional irreducible modules whose highest weights are minuscule weights.