Geodesic
Branching from the General Linear Group to the Symmetric Group and the Principal Embedding
Heaton, Alexander1 ; Sriwongsa, Songpon2 ; Willenbring, Jeb F.3
1 Max Planck Institute for Mathematics in the Sciences Leipzig and Technische Universität Berlin, Germany
2 Department of Mathematics, Faculty of Science King Mongkut’s University of Technology Thonburi (KMUTT) Bangkok 10140, Thailand
3 Department of Mathematical Sciences University of Wisconsin-Milwaukee United States
Algebraic Combinatorics, Tome 4 (2021) no. 2, pp. 189-200 Cet article a éte moissonné depuis la source Numdam

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Let S be a principally embedded 𝔰𝔩 2 -subalgebra in 𝔰𝔩 n 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 𝔰𝔩 n -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.

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DOI : 10.5802/alco.138
Classification : 05E10, 22E46
Keywords: Branching, Howe duality, plethysm, principal embedding.

Heaton, Alexander 1 ; Sriwongsa, Songpon 2 ; Willenbring, Jeb F. 3

1 Max Planck Institute for Mathematics in the Sciences Leipzig and Technische Universität Berlin, Germany
2 Department of Mathematics, Faculty of Science King Mongkut’s University of Technology Thonburi (KMUTT) Bangkok 10140, Thailand
3 Department of Mathematical Sciences University of Wisconsin-Milwaukee United States
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Heaton, Alexander; Sriwongsa, Songpon; Willenbring, Jeb F. Branching from the General Linear Group to the Symmetric Group and the Principal Embedding. Algebraic Combinatorics, Tome 4 (2021) no. 2, pp. 189-200. doi: 10.5802/alco.138

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