Irreducible representations of the symmetric groups from slash homologies of p-complexes
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: 2020-05-29
Revised: 2020-09-19
Accepted: 2020-09-27
Published online: 2021-02-16
DOI: https://doi.org/10.5802/alco.153
Classification: 20C30
Keywords: Modular representation, symmetric group, permutation module, p-complex, slash cohomology, p-standard tableau.
@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/item/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/item/ALCO_2021__4_1_125_0/

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