On the minuscule representation of type $B_n$

Abstract

We study the action of the Weyl group of type $B_n$ acting as permutations on the set of weights of the minuscule representation of type $B_n$ (also known as the spin representation). Motivated by a previous work, we seek to determine when cycle structures alone reveal the irreducibility of these minuscule representations. After deriving formulas for the simple reflections viewed as permutations, we perform a series of computer-aided calculations in GAP. We are then able to establish that, for certain ranks, the irreducibility of the minuscule representation cannot be detected by cycle structures alone.