Let denote the Schur functor labelled by the partition and let be the natural representation of . We make a systematic study of when there is an isomorphism of representations of . Generalizing earlier results of King and Manivel, we classify all such isomorphisms when and are conjugate partitions and when one of or is a rectangle. We give a complete classification when and each have at most two rows or columns or is a hook partition and a partial classification when . As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when 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 -binomial identity in this setting.
Revised: 2020-03-12
Accepted: 2020-09-10
Published online: 2021-02-16
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/item/ALCO_2021__4_1_27_0/} }
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/item/ALCO_2021__4_1_27_0/
[1] The uniqueness of plethystic factorisation, Trans. Amer. Math. Soc., Volume 373 (2020) no. 3, pp. 1653-1666 | Article | MR 4068277 | Zbl 07174533
[2] 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] 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] An Introductory Memoir upon Quantics, Philos. Trans. Roy. Soc. Lond., Volume 144 (1854), pp. 245-258
[5] 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] 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] Plethysms of symmetric functions and highest weight representations (2018) (https://arxiv.org/abs/1810.03448)
[8] Plethysm of -functions, Philos. Trans. Roy. Soc. London. Ser. A., Volume 246 (1954), pp. 555-591 | MR 0062098 (15,926f) | Zbl 0055.24802
[9] 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] Young tableaux, London Mathematical Society student texts, 35, CUP, 1997 | MR 1464693 | Zbl 0878.14034
[11] 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] 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] Sur la théorie des fonctions homogènes à deux indéterminées, Cambridge and Dublin Math. J., Volume 9 (1854), pp. 172-217
[14] 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] 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] Young tableaux, Schur functions and plethysms, J. Phys. A, Volume 18 (1985) no. 13, pp. 2429-2440 | Article | MR 805476 | Zbl 0575.22017
[17] 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] 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] 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] 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] Tensor products of modular representations of and a random walk on their indecomposable summands (2019) (https://arxiv.org/abs/1904.13263)
[22] Hook formulas for skew shapes I. -analogues and bijections, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 350-405 | Article | MR 3718070 | Zbl 1373.05026
[23] Schubert calculus and hook formula, 2014 (http://www.emis.de/journals/SLC/wpapers/s73vortrag/naruse.pdf)
[24] 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] 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] Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, Berlin-New York, 1977, iv+112 pages | MR 0447428
[27] 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] Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR 2868112 | Zbl 1247.05003
[29] On the calculus of forms, otherwise the theory of invariants, Cambridge and Dublin Maths. J., Volume 9 (1854), pp. 85-103
[30] Mathematica, Version 12.0 (Champaign, IL, 2019)
[31] Hermite’s reciprocity law and the angular-momentum states of equivalent particle configurations, J. Math. Phys., Volume 10 (1969), pp. 467-471 | Article