In the 40s, Mayer introduced a construction of (simplicial) -complex by using the unsigned boundary map and taking coefficients of chains modulo . We look at such a -complex associated to an -simplex; in which case, this is also a -complex of representations of the symmetric group of rank — 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 -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 -standard tableaux.
Revised: 2020-09-19
Accepted: 2020-09-27
Published online: 2021-02-16
Classification: 20C30
Keywords: Modular representation, symmetric group, permutation module, -complex, slash cohomology, -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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