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 
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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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