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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Lowest $\mathfrak {sl}(2)$-types in $\mathfrak {sl}(n)$-representations
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by Hassan Lhou and Jeb F. Willenbring PDF
Represent. Theory 21 (2017), 20-34 Request permission

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.

References
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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
  • 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