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A lower bound for the dimension of a highest weight module

Authors: Daniel Goldstein, Robert Guralnick and Richard Stong
Journal: Represent. Theory 21 (2017), 611-625
MSC (2010): Primary 17B10, 22E46
Published electronically: December 19, 2017
MathSciNet review: 3738091
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Abstract: For each integer $t>0$ and each simple Lie algebra $\mathfrak {g}$, we determine the least dimension of an irreducible highest weight representation of $\mathfrak {g}$ whose highest weight has width $t$. As a consequence, we classify all irreducible modules whose dimension equals a product of two primes. This consequence, which was in fact the driving force behind our paper, answers a question of N. Katz.

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Additional Information

Daniel Goldstein
Affiliation: Center for Communications Research, San Diego, California 92121
MR Author ID: 709300

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
MR Author ID: 78455

Richard Stong
Affiliation: Center for Communications Research, San Diego, California 92121
MR Author ID: 167705

Keywords: Complex Lie algebras, representation, highest weight
Received by editor(s): April 1, 2016
Received by editor(s) in revised form: August 11, 2017
Published electronically: December 19, 2017
Additional Notes: The second author was partially supported by NSF grants DMS-1302886 and DMS-1600056.
Article copyright: © Copyright 2017 Institute for Defense Analyses