Lowest $\mathfrak {sl}(2)$-types in $\mathfrak {sl}(n)$-representations
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- by Hassan Lhou and Jeb F. Willenbring
- Represent. Theory 21 (2017), 20-34
- DOI: https://doi.org/10.1090/ert/492
- Published electronically: March 13, 2017
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Abstract:
Fix $n \geq 3$. Let $\mathfrak {s}$ be a principally embedded $\mathfrak {sl}_2$-subalgebra in $\mathfrak {sl}_n$. A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer $b(n)$ such that for any finite dimensional irreducible $\mathfrak {sl}_n$-representation, $V$, there exists an irreducible $\mathfrak {s}$-representation embedding in $V$ with dimension at most $b(n)$. We prove that $b(n)=n$ is the sharpest possible bound. We also address embeddings other than the principal one.
The exposition involves an application of the CartanāHelgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the ābranching algebraā introduced by Roger Howe, Eng-Chye Tan, and the second author.
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Bibliographic Information
- Hassan Lhou
- Affiliation: Department of Mathematical Sciences, University of Wisconsin - Milwaukee, 3200 North Cramer Street, Milwaukee, Wisconsin 53211
- Email: hlhou@uwm.edu
- Jeb F. Willenbring
- Affiliation: Department of Mathematical Sciences, University of Wisconsin - Milwaukee, 3200 North Cramer Street, Milwaukee, Wisconsin 53211
- MR Author ID: 662347
- Email: jw@uwm.edu
- Received by editor(s): September 12, 2016
- Received by editor(s) in revised form: October 23, 2016
- Published electronically: March 13, 2017
- Additional Notes: The second author was supported by the National Security Agency grant # H98230-09-0054.
- © Copyright 2017 American Mathematical Society
- Journal: Represent. Theory 21 (2017), 20-34
- MSC (2010): Primary 17B10; Secondary 05E10, 22E46
- DOI: https://doi.org/10.1090/ert/492
- MathSciNet review: 3622114