We prove a recursive formula for plethysm coefficients of the form , encompassing those which arise in a long-standing conjecture of Foulkes. This also generalises results on plethysms due to Bruns–Conca–Varbaro and de Boeck–Paget–Wildon. From this we deduce a stability result and resolve two conjectures of de Boeck concerning plethysms, as well as obtain new results on Sylow branching coefficients for symmetric groups for the prime 2. Further, letting denote a Sylow 2-subgroup of , we show that almost all Sylow branching coefficients of corresponding to the trivial character of are positive.
Revised:
Accepted:
Published online:
Keywords: character deflation, plethysm, Sylow branching coefficients
Law, Stacey 1; Okitani, Yuji 2
@article{ALCO_2023__6_2_321_0, author = {Law, Stacey and Okitani, Yuji}, title = {On plethysms and {Sylow} branching coefficients}, journal = {Algebraic Combinatorics}, pages = {321--357}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {2}, year = {2023}, doi = {10.5802/alco.262}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.262/} }
TY - JOUR AU - Law, Stacey AU - Okitani, Yuji TI - On plethysms and Sylow branching coefficients JO - Algebraic Combinatorics PY - 2023 SP - 321 EP - 357 VL - 6 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.262/ DO - 10.5802/alco.262 LA - en ID - ALCO_2023__6_2_321_0 ER -
%0 Journal Article %A Law, Stacey %A Okitani, Yuji %T On plethysms and Sylow branching coefficients %J Algebraic Combinatorics %D 2023 %P 321-357 %V 6 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.262/ %R 10.5802/alco.262 %G en %F ALCO_2023__6_2_321_0
Law, Stacey; Okitani, Yuji. On plethysms and Sylow branching coefficients. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 321-357. doi : 10.5802/alco.262. https://alco.centre-mersenne.org/articles/10.5802/alco.262/
[1] The classification of multiplicity-free plethysms of Schur functions, Trans. Amer. Math. Soc., Volume 375 (2022) no. 7, pp. 5151-5194 | DOI | MR | Zbl
[2] Rectangular symmetries for coefficients of symmetric functions, Electron. J. Combin., Volume 22 (2015) no. 3, p. Paper 3.15, 18 pages | DOI | MR | Zbl
[3] Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Math., Volume 80 (1993) no. 4, pp. 347-371 | DOI | MR | Zbl
[4] Relations between the minors of a generic matrix, Adv. Math., Volume 244 (2013), pp. 171-206 | DOI | MR | Zbl
[5] Even partitions in plethysms, J. Algebra, Volume 328 (2011), pp. 322-329 | DOI | MR | Zbl
[6] On the structure of Foulkes modules for the symmetric group, Ph. D. Thesis, University of Kent (2015)
[7] Plethysms of symmetric functions and highest weight representations, Trans. Amer. Math. Soc., Volume 374 (2021) no. 11, pp. 8013-8043 | DOI | MR | Zbl
[8] The distribution of the number of summands in the partitions of a positive integer, Duke Math. J., Volume 8 (1941), pp. 335-345 | DOI | MR | Zbl
[9] Character deflations and a generalization of the Murnaghan-Nakayama rule, J. Group Theory, Volume 17 (2014) no. 6, pp. 1035-1070 | DOI | MR | Zbl
[10] Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form, J. London Math. Soc., Volume 25 (1950), pp. 205-209 | DOI | MR | Zbl
[11] Characters of odd degree of symmetric groups, J. Lond. Math. Soc. (2), Volume 96 (2017) no. 1, pp. 1-14 | DOI | MR | Zbl
[12] Restriction of odd degree characters and natural correspondences, Int. Math. Res. Not. IMRN (2017) no. 20, pp. 6089-6118 | MR | Zbl
[13] On permutation characters and Sylow -subgroups of , J. Algebra, Volume 506 (2018), pp. 409-428 | DOI | MR | Zbl
[14] Sylow branching coefficients for symmetric groups, J. Lond. Math. Soc. (2), Volume 103 (2021) no. 2, pp. 697-728 | DOI | MR | Zbl
[15] Sylow branching coefficients and a conjecture of Malle and Navarro, Bull. Lond. Math. Soc., Volume 54 (2022) no. 2, pp. 552-567 | DOI | MR
[16] Restricting irreducible characters to Sylow -subgroups, Proc. Amer. Math. Soc., Volume 146 (2018) no. 5, pp. 1963-1976 | DOI | MR | Zbl
[17] Character theory of finite groups, Dover Publications, Inc., New York, 1994, xii+303 pages | MR
[18] Character restrictions and multiplicities in symmetric groups, J. Algebra, Volume 478 (2017), pp. 271-282 | DOI | MR | Zbl
[19] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, xxviii+510 pages | MR
[20] The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978, v+156 pages | DOI
[21] The plethysm at hook and near-hook shapes, Electron. J. Combin., Volume 11 (2004) no. 1, p. Research Paper 11, 26 pages | DOI | MR | Zbl
[22] On Problems in the Representation Theory of Symmetric Groups, Ph. D. Thesis, University of Cambridge (2019)
[23] 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
[24] Gaussian maps and plethysm, Algebraic geometry (Catania, 1993/Barcelona, 1994) (Lecture Notes in Pure and Appl. Math.), Volume 200, Dekker, New York, 1998, pp. 91-117 | MR | Zbl
[25] Effective constructions in plethysms and Weintraub’s conjecture, Algebr. Represent. Theory, Volume 17 (2014) no. 2, pp. 433-443 | DOI | MR | Zbl
[26] Character tables and Sylow subgroups revisited, Group theory and computation (Indian Stat. Inst. Ser.), Springer, Singapore, 2018, pp. 197-206 | DOI
[27] Combinatorics and representations of finite groups, Vorlesungen aus dem Fachbereich Mathematik der Universität GH Essen [Lecture Notes in Mathematics at the University of Essen], 20, Universität Essen, Fachbereich Mathematik, Essen, 1993, ii+94 pages | MR
[28] 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 | DOI | MR | Zbl
[29] Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR
[30] Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295-319 | Zbl
[31] On symmetrized Kronecker powers and the structure of the free Lie ring, Amer. J. Math., Volume 64 (1942), pp. 371-388 | DOI | MR | Zbl
[32] Some observations on plethysms, J. Algebra, Volume 129 (1990) no. 1, pp. 103-114 | DOI | MR | Zbl
Cited by Sources: