On the minuscule representation of type Bn

Cook, William J.; Hughes, Noah A.

doi:10.2140/involve.2018.11.721

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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

DOI: 10.2140/involve.2018.11.721

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

Mathematical Subject Classification 2010

Primary: 17B10

Secondary: 20F55

Supplementary material

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

Vol. 11, No. 5, 2018