Branching from the general linear group to the symmetric group and the principal embedding
Journal article
Authors/Editors
Strategic Research Themes
Publication Details
Author list: Heaton A., Sriwongsa S., Willenbring J.F.
Publication year: 2021
Volume number: 4
Issue number: 2
Start page: 189
End page: 200
Number of pages: 12
eISSN: 2589-5486
Languages: English-Great Britain (EN-GB)
Abstract
Let S be a principally embedded sl2-subalgebra in sln for n > 3. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sln-representation, V , there exists an irreducible S-representation embedding in V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou), they prove that b(n) = n is the sharpest possible bound, and also address embeddings other than the principal one. These results concerning embeddings may be interpreted as statements about plethysm. Then, in turn, a well known result about these plethysms can be interpreted as a “branching rule”. Specifically, a finite dimensional irreducible representation of GL(n, C) will decompose into irreducible representations of the symmetric group when it is restricted to the subgroup consisting of permutation matrices. The question of which irreducible representations of the symmetric group occur with positive multiplicity is the topic of this paper, applying the previous work of Lhou, Zuckerman, and the third author. © The journal and the authors, 2021.
Keywords
Branching, Howe duality, Plethysm, Principal embedding
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