Vol. 11, No. 5, 2018

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ISSN: 1944-4184 (e-only)
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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 Bn acting as permutations on the set of weights of the minuscule representation of type Bn (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