New columns in decomposition matrices of symmetric groups for every block

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Researchers have identified new columns in decomposition matrices for symmetric groups labeled by partitions with even arm lengths for $p$-divisible hooks, solving at least one column for every block in odd characteristic $p$. These multiplicity-free columns are characterized using the recently introduced "odd sequence" statistic of partitions. This finding also determines the indecomposable summands of Foulkes modules $H^{(2^m)}$.

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Exploring new columns in decomposition matrices of symmetric groups could lead to interesting discussions about modular representation theory and combinatorial statistics.

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arXiv:2606.28731v2 Announce Type: replace 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 $p$. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose $p$-divisible hooks have all even arm lengths. In particular in odd characteristic $p$, for every possible block of every possible symmetric group $S_n$, 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 $p$), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules $H^{(2^m)}$.