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Lowest $ \mathfrak{sl}(2)$-types in $ \mathfrak{sl}(n)$-representations


Authors: Hassan Lhou and Jeb F. Willenbring
Journal: Represent. Theory 21 (2017), 20-34
MSC (2010): Primary 17B10; Secondary 05E10, 22E46
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 Cartan-Helgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the ``branching algebra'' introduced by Roger Howe, Eng-Chye Tan, and the second author.


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Additional 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
Email: jw@uwm.edu

DOI: https://doi.org/10.1090/ert/492
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.
Article copyright: © Copyright 2017 American Mathematical Society