# ALGEBRAIC COMBINATORICS

Plethysms of symmetric functions and representations of ${\mathrm{SL}}_{2}\left(\mathbf{C}\right)$
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 27-68.

Let ${\nabla }^{\lambda }$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of ${\mathrm{SL}}_{2}\left(\mathbf{C}\right)$. We make a systematic study of when there is an isomorphism ${\nabla }^{\lambda }{\mathrm{Sym}}^{\ell }\phantom{\rule{-0.166667em}{0ex}}E\cong {\nabla }^{\mu }{\mathrm{Sym}}^{m}\phantom{\rule{-0.166667em}{0ex}}E$ of representations of ${\mathrm{SL}}_{2}\left(\mathbf{C}\right)$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\lambda$ and $\mu$ are conjugate partitions and when one of $\lambda$ or $\mu$ is a rectangle. We give a complete classification when $\lambda$ and $\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell =m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ${\nabla }^{\lambda }{\mathrm{Sym}}^{\ell }\phantom{\rule{-0.166667em}{0ex}}E$ is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new $q$-binomial identity in this setting.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.150
Classification: 05E05,  05E10,  20C30,  22E46,  22E47
Keywords: Plethysm, Hermite Reciprocity, Hook Content Formula
@article{ALCO_2021__4_1_27_0,
author = {Paget, Rowena and Wildon, Mark},
title = {Plethysms of symmetric functions and representations of $\protect \mathrm{SL}\_2({\protect \bf C})$},
journal = {Algebraic Combinatorics},
pages = {27--68},
publisher = {MathOA foundation},
volume = {4},
number = {1},
year = {2021},
doi = {10.5802/alco.150},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.150/}
}
Paget, Rowena; Wildon, Mark. Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 27-68. doi : 10.5802/alco.150. https://alco.centre-mersenne.org/articles/10.5802/alco.150/

[1] Bowman, Chris; Paget, Rowena The uniqueness of plethystic factorisation, Trans. Amer. Math. Soc., Volume 373 (2020) no. 3, pp. 1653-1666 | Article | MR 4068277 | Zbl 07174533

[2] Brion, Michel Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Math., Volume 80 (1993) no. 4, pp. 347-371 | Article | MR 1243152 | Zbl 0823.20039

[3] Cagliero, Leandro; Penazzi, Daniel A new generalization of Hermite’s reciprocity law, J. Algebraic Combin., Volume 43 (2016) no. 2, pp. 399-416 | Article | MR 3456495 | Zbl 1331.05222

[4] Cayley, Arthur An Introductory Memoir upon Quantics, Philos. Trans. Roy. Soc. Lond., Volume 144 (1854), pp. 245-258

[5] Cheung, Man-Wai; Ikenmeyer, Christian; Mkrtchyan, Sevak Symmetrizing tableaux and the 5th case of the Foulkes conjecture, J. Symbolic Comput., Volume 80 part 3 (2017), pp. 833-843 | Article | MR 3574536 | Zbl 07003729

[6] Craven, David A. Symmetric group character degrees and hook numbers, Proc. Lond. Math. Soc. (3), Volume 96 (2008) no. 1, pp. 26-50 | Article | MR 2392314 | Zbl 1165.20008

[7] de Boeck, Melanie; Paget, Rowena; Wildon, Mark Plethysms of symmetric functions and highest weight representations (2018) (https://arxiv.org/abs/1810.03448)

[8] Foulkes, Herbert O. Plethysm of $S$-functions, Philos. Trans. Roy. Soc. London. Ser. A., Volume 246 (1954), pp. 555-591 | MR 0062098 (15,926f) | Zbl 0055.24802

[9] Fulmek, Markus; Krattenthaler, Christian The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis. I, Ann. Comb., Volume 2 (1998) no. 1, pp. 19-41 | Article | MR 1682918 | Zbl 0917.05004

[10] Fulton, William Young tableaux, London Mathematical Society student texts, 35, CUP, 1997 | MR 1464693 | Zbl 0878.14034

[11] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991, xvi+551 pages | Article | MR 1153249 | Zbl 0744.22001

[12] Giannelli, Eugenio On the decomposition of the Foulkes module, Arch. Math. (Basel), Volume 100 (2013) no. 3, pp. 201-214 | Article | MR 3032652 | Zbl 1271.20007

[13] Hermite, Charles Sur la théorie des fonctions homogènes à deux indéterminées, Cambridge and Dublin Math. J., Volume 9 (1854), pp. 172-217

[14] Ikeda, Takeshi; Naruse, Hiroshi Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., Volume 361 (2009) no. 10, pp. 5193-5221 | Article | MR 2515809 | Zbl 1229.05287

[15] Kac, Mark Can one hear the shape of a drum?, Amer. Math. Monthly, Volume 73 (1966) no. 4, part II, pp. 1-23 | Article | MR 201237 | Zbl 0139.05603

[16] King, Ronald C. Young tableaux, Schur functions and $\mathrm{SU}\left(2\right)$ plethysms, J. Phys. A, Volume 18 (1985) no. 13, pp. 2429-2440 | Article | MR 805476 | Zbl 0575.22017

[17] 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 | Article | MR 2765321 | Zbl 1229.05275

[18] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | MR MR1354144 (96h:05207) | Zbl 0824.05059

[19] MacMahon, Percy A. Memoir on the Theory of the Partition of Numbers. — Part I, Phil. Trans. Roy. Soc. Lond. A: Maths. Phys. Eng. Sci., Volume 187 (1896), pp. 619-673

[20] Manivel, Laurent An extension of the Cayley–Sylvester formula, European J. Combin., Volume 28 (2007) no. 6, pp. 1839-1842 | Article | MR 2340387 | Zbl 1122.05095

[21] McDowell, Eoghan Tensor products of modular representations of ${\mathrm{SL}}_{2}\left({𝔽}_{p}\right)$ and a random walk on their indecomposable summands (2019) (https://arxiv.org/abs/1904.13263)

[22] Morales, Alejandro H.; Pak, Igor; Panova, Greta Hook formulas for skew shapes I. $q$-analogues and bijections, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 350-405 | Article | MR 3718070 | Zbl 1373.05026

[23] Naruse, Hiroshi Schubert calculus and hook formula, 2014 (http://www.emis.de/journals/SLC/wpapers/s73vortrag/naruse.pdf)

[24] Paget, Rowena; Wildon, Mark Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions, Proc. Lond. Math. Soc. (3), Volume 118 (2019) no. 5, pp. 1153-1187 | Article | MR 3946719 | Zbl 07068310

[25] Peyton Jones, Simon The Haskell 98 Language and Libraries: The Revised Report, Journal of Functional Programming, Volume 13 (2003) no. 1, pp. 0-255 (http://www.haskell.org/definition/) | MR 1989220 | Zbl 1067.68041

[26] Springer, Tonny A. Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, Berlin-New York, 1977, iv+112 pages | MR 0447428

[27] Stanley, Richard P. Enumerative combinatorics. Volume 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

[28] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR 2868112 | Zbl 1247.05003

[29] Sylvester, James J. On the calculus of forms, otherwise the theory of invariants, Cambridge and Dublin Maths. J., Volume 9 (1854), pp. 85-103

[30] Wolfram Research, Inc. Mathematica, Version 12.0 (Champaign, IL, 2019)

[31] Wybourne, Brian G. Hermite’s reciprocity law and the angular-momentum states of equivalent particle configurations, J. Math. Phys., Volume 10 (1969), pp. 467-471 | Article