## Involve: A Journal of Mathematics

- Involve
- Volume 11, Number 5 (2018), 721-733.

### On the minuscule representation of type $B_n$

William J. Cook and Noah A. Hughes

#### 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.

#### Article information

**Source**

Involve, Volume 11, Number 5 (2018), 721-733.

**Dates**

Received: 23 April 2014

Revised: 6 November 2017

Accepted: 20 November 2017

First available in Project Euclid: 12 April 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1523498539

**Digital Object Identifier**

doi:10.2140/involve.2018.11.721

**Mathematical Reviews number (MathSciNet)**

MR3784022

**Zentralblatt MATH identifier**

06866579

**Subjects**

Primary: 17B10: Representations, algebraic theory (weights)

Secondary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]

**Keywords**

Lie algebra minuscule representation Weyl group

#### Citation

Cook, William J.; Hughes, Noah A. On the minuscule representation of type $B_n$. Involve 11 (2018), no. 5, 721--733. doi:10.2140/involve.2018.11.721. https://projecteuclid.org/euclid.involve/1523498539

#### Supplemental materials

- GAP code. Supplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.