# ALGEBRAIC COMBINATORICS

Branching from the General Linear Group to the Symmetric Group and the Principal Embedding
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 189-200.

Let $S$ be a principally embedded ${\mathrm{𝔰𝔩}}_{2}$-subalgebra in ${\mathrm{𝔰𝔩}}_{n}$ for $n\ge 3$. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer $b\left(n\right)$ such that for any finite-dimensional irreducible ${\mathrm{𝔰𝔩}}_{n}$-representation, $V$, there exists an irreducible $S$-representation embedding in $V$ with dimension at most $b\left(n\right)$. In a 2017 paper (joint with Hassan Lhou), they prove that $b\left(n\right)=n$ 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 $\mathrm{GL}\left(n,ℂ\right)$ 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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.138
Classification: 05E10,  22E46
Keywords: Branching, Howe duality, plethysm, principal embedding.
@article{ALCO_2021__4_2_189_0,
author = {Heaton, Alexander and Sriwongsa, Songpon and Willenbring, Jeb F.},
title = {Branching from the General Linear Group to the Symmetric Group and the Principal Embedding},
journal = {Algebraic Combinatorics},
pages = {189--200},
publisher = {MathOA foundation},
volume = {4},
number = {2},
year = {2021},
doi = {10.5802/alco.138},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.138/}
}
Heaton, Alexander; Sriwongsa, Songpon; Willenbring, Jeb F. Branching from the General Linear Group to the Symmetric Group and the Principal Embedding. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 189-200. doi : 10.5802/alco.138. https://alco.centre-mersenne.org/articles/10.5802/alco.138/

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