Let $S$ be a principally embedded ${\mathrm{\U0001d530\U0001d529}}_{2}$-subalgebra in ${\mathrm{\U0001d530\U0001d529}}_{n}$ 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 ${\mathrm{\U0001d530\U0001d529}}_{n}$-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 $\mathrm{GL}(n,\u2102)$ 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:

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/

[1] Introduction to representation theory, Student Mathematical Library, 59, American Mathematical Society, Providence, RI, 2011, viii+228 pages (With historical interludes by Slava Gerovitch) | Article | MR 2808160 | Zbl 1242.20001

[2] Branching rules for $GL\left(N\right)\supset {\U0001d516}_{m}$ and the evaluation of inner plethysms, J. Mathematical Phys., Volume 15 (1974), pp. 258-267 | Article | MR 331999 | Zbl 0276.20008

[3] The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math., Volume 81 (1959), pp. 973-1032 | Article | MR 0114875 | Zbl 0099.25603

[4] Lowest $\mathrm{\U0001d530\U0001d529}\left(2\right)$-types in $\mathrm{\U0001d530\U0001d529}\left(n\right)$-representations, Represent. Theory, Volume 21 (2017), pp. 20-34 | Article | MR 3622114 | Zbl 1419.17014

[5] A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., Volume 33 (2011) no. 2, pp. 163-198 | Article | MR 2765321 | Zbl 1229.05275

[6] Restriction of the irreducible representations of ${\mathrm{GL}}_{n}$ to the symmetric group ${\U0001d516}_{n}$ (http://rtweb.math.kyoto-u.ac.jp/home_kyo/preprint/glntosn.pdf)

[7] Products of symmetric group characters, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 299-324 | Article | MR 3944530 | Zbl 1414.05298

[8] A Hopf-algebra approach to inner plethysm, Adv. Math., Volume 104 (1994) no. 1, pp. 30-58 | Article | MR 1272068 | Zbl 0830.20027

[9] 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) | Article | MR 1676282 | Zbl 0928.05001

[10] 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 1488158 | Zbl 1024.20501

[11] 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 | Article | MR 2401818 | Zbl 1390.17017