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.

Revised: 2020-09-19

Accepted: 2020-09-27

Published online: 2021-02-16

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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