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

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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.

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


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References

  • N. Bourbaki, Lie groups and Lie algebras, Chapters 7–9, Springer, 2005.
  • R. W. Carter, Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics 96, Cambridge University Press, 2005.
  • W. J. Cook, C. Mitschi, and M. F. Singer, “On the constructive inverse problem in differential Galois theory”, Comm. Algebra 33:10 (2005), 3639–3665.
  • K. Erdmann and M. J. Wildon, Introduction to Lie algebras, Springer, 2006.
  • “GAP – Groups, Algorithms, and Programming”, version 4.8.7, 2017, http://www.gap-system.org.
  • R. M. Green, “Representations of Lie algebras arising from polytopes”, Int. Electron. J. Algebra 4 (2008), 27–52.
  • R. M. Green, Combinatorics of minuscule representations, Cambridge Tracts in Mathematics 199, Cambridge University Press, 2013.
  • J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9, Springer, 1972.

Supplemental materials

  • GAP code.