ALGEBRAIC COMBINATORICS

no. 4
Specht module branching rules for wreath products of symmetric groups
Green, Reuben1
1 Pembroke College St Aldate’s Oxford OX1 1DW
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 609-628.

We review a class of modules for the wreath product S m ≀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 ≀S n and S m ≀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.

Received:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2022__5_4_609_0,
     author = {Green, Reuben},
     title = {Specht module branching rules for wreath products of symmetric groups},
     journal = {Algebraic Combinatorics},
     pages = {609--628},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {4},
     year = {2022},
     doi = {10.5802/alco.223},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.223/}
}
TY  - JOUR
AU  - Green, Reuben
TI  - Specht module branching rules for wreath products of symmetric groups
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 609
EP  - 628
VL  - 5
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.223/
DO  - 10.5802/alco.223
LA  - en
ID  - ALCO_2022__5_4_609_0
ER  - 
%0 Journal Article
%A Green, Reuben
%T Specht module branching rules for wreath products of symmetric groups
%J Algebraic Combinatorics
%D 2022
%P 609-628
%V 5
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.223/
%R 10.5802/alco.223
%G en
%F ALCO_2022__5_4_609_0
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

Cited by Sources: