We study the action of the Weyl group of type acting as permutations on the set of weights of the minuscule representation of type (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.
"On the minuscule representation of type $B_n$." Involve 11 (5) 721 - 733, 2018. https://doi.org/10.2140/involve.2018.11.721