On the modular representation theory of the partition algebra
We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine exactly for which values of the parameter the partition algebras are semisimple over an arbitrary field. Furthermore, we show that the blocks of the partition algebra over an arbitrary field categorify weight spaces of an action of the quantum groups \(\mathcal {U}_q(\widehat{\mathfrak {sl}_p})\) and \(\mathcal {U}_q(\mathfrak {sl}_\infty )\) on an analogue of the Fock space. In particular, we recover the block structure which was recently determined by Bowman et al. In order to do so, we use induction and restriction functors as well as analogues of JucysMurphy elements. The description of decomposition numbers will be in terms of combinatorics of partitions but can also be given a Lie theoretic interpretation in terms of a Weyl group of type \(A\): a simple module \(L(\mu )\) is a composition factor of a cell module \(\Delta (\lambda )\) if and only if \(\lambda \) and \(\mu \) differ by the action of a transposition.
partition algebra decomposition numbers blocks jucysmurphy elements diagram algebras