#### Vol. 11, No. 5, 2018

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On the minuscule representation of type $B_n$

### William J. Cook and Noah A. Hughes

Vol. 11 (2018), No. 5, 721–733
##### 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.

##### Keywords
Lie algebra, minuscule representation, Weyl group
Primary: 17B10
Secondary: 20F55

GAP code

##### Milestones
Received: 23 April 2014
Revised: 6 November 2017
Accepted: 20 November 2017
Published: 2 April 2018

Communicated by Ravi Vakil
##### Authors
 William J. Cook Department of Mathematical Sciences Appalachian State University Boone, NC United States Noah A. Hughes Department of Mathematics University of Connecticut Storrs, CT United States