# ALGEBRAIC COMBINATORICS

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 $\left(n-1\right)$-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:
Accepted:
Published online:
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/articles/10.5802/alco.153/}
}
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/

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