English

New columns in decomposition matrices of symmetric groups for every block

Representation Theory 2026-06-27 v1

Abstract

The central unsolved problem in the modular representation theory of symmetric groups is to find the decomposition matrices, which describe how irreducible representations in characteristic zero decompose upon reduction modulo a prime characteristic pp. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose pp-divisible hooks have all even arm lengths. In particular in odd characteristic pp, for every possible block of every possible symmetric group SnS_n, we determine at least one complete column. These columns are multiplicity free and are described by a recently introduced combinatorial statistic of partitions (depending on pp), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules H(2m)H^{(2^m)}.

Keywords

Cite

@article{arxiv.2606.28731,
  title  = {New columns in decomposition matrices of symmetric groups for every block},
  author = {David J. Hemmer and Pavel Turek},
  journal= {arXiv preprint arXiv:2606.28731},
  year   = {2026}
}

Comments

34 pages, 8 figures