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Decomposition of exterior and symmetric squares in characteristic two
- Source :
- Linear Algebra Appl. 624 (2021), 349-363
- Publication Year :
- 2021
-
Abstract
- Let $V$ be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element $e \in \mathfrak{sl}(V)$, we describe the Jordan normal form of $e$ on the $\mathfrak{sl}(V)$-modules $\wedge^2(V)$ and $S^2(V)$. In the case where $e$ is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element $u \in \operatorname{SL}(V)$, the Jordan normal form of $u$ on $\wedge^2(V)$ and $S^2(V)$. A recursive formula for the Jordan block sizes of $u$ on $\wedge^2(V)$ was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672). We show that their proof can be adapted to give a similar formula for the Jordan block sizes of $u$ on $S^2(V)$.<br />Comment: to appear in Linear Algebra and its Applications
- Subjects :
- Mathematics - Representation Theory
Mathematics - Group Theory
Subjects
Details
- Database :
- arXiv
- Journal :
- Linear Algebra Appl. 624 (2021), 349-363
- Publication Type :
- Report
- Accession number :
- edsarx.2101.01365
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.laa.2021.04.018