Branching from the General Linear Group to the Symmetric Group and the Principal Embedding

Heaton, Alexander; Sriwongsa, Songpon; Willenbring, Jeb F.

doi:10.5802/alco.138

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

Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 189-200.

Abstract

Let $S$ be a principally embedded ${𝔰𝔩}_{2}$ -subalgebra in ${𝔰𝔩}_{n}$ for $n \geq 3$ . 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.

Received: 2019-02-06
Revised: 2020-06-16
Accepted: 2020-06-21
Published online: 2021-04-12

DOI: 10.5802/alco.138

Classification: 05E10, 22E46
Keywords: Branching, Howe duality, plethysm, principal embedding.

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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, Volume 4 (2021) no. 2, pp. 189-200. doi : 10.5802/alco.138. https://alco.centre-mersenne.org/articles/10.5802/alco.138/

References
Cited by

[1] Etingof, Pavel; Golberg, Oleg; Hensel, Sebastian; Liu, Tiankai; Schwendner, Alex; Vaintrob, Dmitry; Yudovina, Elena Introduction to representation theory, Student Mathematical Library, 59, American Mathematical Society, Providence, RI, 2011, viii+228 pages (With historical interludes by Slava Gerovitch) | DOI | MR | Zbl

[2] King, R. C. Branching rules for $G L (N) \supset 𝔖_{m}$ and the evaluation of inner plethysms, J. Mathematical Phys., Volume 15 (1974), pp. 258-267 | DOI | MR | Zbl

[3] Kostant, Bertram The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., Volume 81 (1959), pp. 973-1032 | DOI | MR | Zbl

[4] Lhou, Hassan; Willenbring, Jeb F. Lowest $𝔰𝔩 (2)$ -types in $𝔰𝔩 (n)$ -representations, Represent. Theory, Volume 21 (2017), pp. 20-34 | DOI | MR | Zbl

[5] Loehr, Nicholas A.; Remmel, Jeffrey B. A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., Volume 33 (2011) no. 2, pp. 163-198 | DOI | MR | Zbl

[6] Nishiyama, Kyo Restriction of the irreducible representations of ${GL}_{n}$ to the symmetric group $𝔖_{n}$ (http://rtweb.math.kyoto-u.ac.jp/home_kyo/preprint/glntosn.pdf) | Zbl

[7] Orellana, Rosa; Zabrocki, Mike Products of symmetric group characters, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 299-324 | DOI | MR | Zbl

[8] Scharf, Thomas; Thibon, Jean-Yves A Hopf-algebra approach to inner plethysm, Adv. Math., Volume 104 (1994) no. 1, pp. 30-58 | DOI | MR | Zbl

[9] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | DOI | MR | Zbl

[10] Weyl, Hermann The classical groups, Their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, xiv+320 pages (Fifteenth printing, Princeton Paperbacks) | MR | Zbl

[11] Willenbring, Jeb F.; Zuckerman, Gregg J. Small semisimple subalgebras of semisimple Lie algebras, Harmonic analysis, group representations, automorphic forms and invariant theory (Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap.), Volume 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 403-429 | DOI | MR | Zbl

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