ALGEBRAIC COMBINATORICS

no. 1
Irreducible representations of the symmetric groups from slash homologies of p-complexes
Chan, Aaron1; Wong, William1
1 Graduate School of Mathematics Nagoya University Furocho, Chikusaku, Nagoya, Japan
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 125-144.

In the 40s, Mayer introduced a construction of (simplicial) p-complex by using the unsigned boundary map and taking coefficients of chains modulo p. We look at such a p-complex associated to an (n-1)-simplex; in which case, this is also a p-complex of representations of the symmetric group of rank n — specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology — a homology theory introduced by Khovanov and Qi — of such a p-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called p-standard tableaux.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.153
Classification: 20C30
Keywords: Modular representation, symmetric group, permutation module, p-complex, slash cohomology, p-standard tableau.
Chan, Aaron 1; Wong, William 1

1 Graduate School of Mathematics Nagoya University Furocho, Chikusaku, Nagoya, Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2021__4_1_125_0,
     author = {Chan, Aaron and Wong, William},
     title = {Irreducible representations of the symmetric groups from slash homologies of $p$-complexes},
     journal = {Algebraic Combinatorics},
     pages = {125--144},
     publisher = {MathOA foundation},
     volume = {4},
     number = {1},
     year = {2021},
     doi = {10.5802/alco.153},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.153/}
}
TY  - JOUR
TI  - Irreducible representations of the symmetric groups from slash homologies of $p$-complexes
JO  - Algebraic Combinatorics
PY  - 2021
DA  - 2021///
SP  - 125
EP  - 144
VL  - 4
IS  - 1
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.153/
UR  - https://doi.org/10.5802/alco.153
DO  - 10.5802/alco.153
LA  - en
ID  - ALCO_2021__4_1_125_0
ER  - 
%0 Journal Article
%T Irreducible representations of the symmetric groups from slash homologies of $p$-complexes
%J Algebraic Combinatorics
%D 2021
%P 125-144
%V 4
%N 1
%I MathOA foundation
%U https://doi.org/10.5802/alco.153
%R 10.5802/alco.153
%G en
%F ALCO_2021__4_1_125_0
Chan, Aaron; Wong, William. Irreducible representations of the symmetric groups from slash homologies of $p$-complexes. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 125-144. doi : 10.5802/alco.153. https://alco.centre-mersenne.org/articles/10.5802/alco.153/

[1] Bell, Steven; Jones, Philip; Siemons, Johannes On modular homology in the Boolean algebra. II, J. Algebra, Volume 199 (1998) no. 2, pp. 556-580 | DOI | MR | Zbl

[2] de Boeck, Maarten; Evseev, Anton; Lyle, Sinéad; Speyer, Liron On bases of some simple modules of symmetric groups and Hecke algebras, Transform. Groups, Volume 23 (2018) no. 3, pp. 631-669 | DOI | MR

[3] Elias, Ben; Qi, You Categorifying Hecke algebras at prime roots of unity, part I (2020) (https://arxiv.org/abs/2005.03128)

[4] Erdmann, Karin Tensor products and dimensions of simple modules for symmetric groups, Manuscripta Math., Volume 88 (1995) no. 3, pp. 357-386 | DOI | MR | Zbl

[5] James, Gordon D. The irreducible representations of the symmetric groups, Bull. London Math. Soc., Volume 8 (1976) no. 3, pp. 229-232 | DOI | MR | Zbl

[6] James, Gordon D. A characteristic-free approach to the representation theory of 𝔖 n , J. Algebra, Volume 46 (1977) no. 2, pp. 430-450 | DOI | MR

[7] James, Gordon D. The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978, v+156 pages | MR

[8] Khovanov, Mikhail; Qi, You An approach to categorification of some small quantum groups, Quantum Topol., Volume 6 (2015) no. 2, pp. 185-311 | DOI | MR | Zbl

[9] Kleshchev, Alexander Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. (2), Volume 54 (1996) no. 1, pp. 25-38 | DOI | MR | Zbl

[10] Kleshchev, Alexander Completely splittable representations of symmetric groups, J. Algebra, Volume 181 (1996) no. 2, pp. 584-592 | DOI | MR | Zbl

[11] Mathieu, Olivier On the dimension of some modular irreducible representations of the symmetric group, Lett. Math. Phys., Volume 38 (1996) no. 1, pp. 23-32 | DOI | MR | Zbl

[12] Mayer, Walther A new homology theory. I, Ann. of Math. (2), Volume 43 (1942), pp. 370-380 | DOI | MR

[13] Mayer, Walther A new homology theory. II, Ann. of Math. (2), Volume 43 (1942), pp. 594-605 | DOI | MR

[14] Mirmohades, Djalal Homologically optimal categories of sequences lead to N-complexes (2014) (https://arxiv.org/abs/1405.3921)

[15] Mohanty, Sri Gopal Lattice path counting and applications, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1979, xi+185 pages (Probability and Mathematical Statistics) | MR | Zbl

[16] Peel, Michael H. Specht modules and symmetric groups, J. Algebra, Volume 36 (1975) no. 1, pp. 88-97 | DOI | MR | Zbl

[17] Ruff, Oliver Completely splittable representations of symmetric groups and affine Hecke algebras, J. Algebra, Volume 305 (2006) no. 2, pp. 1197-1211 | DOI | MR | Zbl

[18] Sagan, Bruce E. The symmetric group: Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | DOI | MR | Zbl

[19] Wildon, Mark The multistep homology of the simplex and representations of symmetric groups (2018) (https://arxiv.org/abs/1803.00465)

Cited by Sources: