ALGEBRAIC COMBINATORICS

Specht module branching rules for wreath products of symmetric groups
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 609-628.

We review a class of modules for the wreath product ${S}_{m}\wr {S}_{n}$ of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a pair of branching rules for this family of modules. These branching rules describe the behaviour of these wreath product Specht modules under restriction to the wreath products ${S}_{m-1}\wr {S}_{n}$ and ${S}_{m}\wr {S}_{n-1}$. In particular, we see that these restrictions of wreath product Specht modules have Specht module filtrations, and we obtain combinatorial interpretations of the multiplicities in these filtrations.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.223
Classification: 05E10
Keywords: branching rules, Specht modules, symmetric groups
Green, Reuben 1

1 Pembroke College St Aldate’s Oxford OX1 1DW
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Green, Reuben. Specht module branching rules for wreath products of symmetric groups. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 609-628. doi : 10.5802/alco.223. https://alco.centre-mersenne.org/articles/10.5802/alco.223/

[1] Benson, D. J. Representations and Cohomology, Cambridge Studies in Advanced Mathematics, 1, Cambridge University Press, 1991

[2] Chuang, J.; Tan, K. M. Representations of wreath products of algebras, Math. Proc. Camb. Philos. Soc., Volume 135 (2003) no. 3, pp. 395 - 411 | DOI | MR | Zbl

[3] Geetha, T.; Goodman, F. M. Cellularity of wreath product algebras and A-Brauer algebras, J. Algebra, Volume 389 (2013), pp. 151 -190 | DOI | MR | Zbl

[4] Graham, J. J.; Lehrer, G. I. Cellular algebras., Invent. Math., Volume 123 (1996) no. 1, pp. 1-34 | DOI | MR | Zbl

[5] Green, R. Some properties of Specht modules for the wreath product of symmetric groups, Ph. D. Thesis, University of Kent (2019)

[6] Green, R. Cellular structure of wreath product algebras, J. Pure Appl. Algebra, Volume 224 (2020) no. 2, pp. 819 -835 | DOI | MR | Zbl

[7] James, G. D. The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978

[8] James, G. D.; Kerber, A. The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981

[9] James, G. D.; Peel, M. H. Specht series for skew representations of symmetric groups, J. Algebra, Volume 56 (1979) no. 2, pp. 343-364 | DOI | MR | Zbl

[10] Stanley, R. P. Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, 1999 | DOI

[11] Tsuchioka, S. A modular branching rule for wreath products, Combinatorial representation theory and related topics (RIMS Kôkyûroku Bessatsu, B8), Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 21-35 | MR | Zbl

[12] Wildon, M. A model for the double cosets of Young subgroups http://www.ma.rhul.ac.uk/~uvah099/Maths/doubleRevised2.pdf

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