On the modular representation theory of the partition algebra

Shalile, Armin

Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 245-282.

Résumé

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 $\mathcal U_q(\widehat{\frak{sl}_p})$ and $\mathcal U_q(\frak{sl}_\infty)$ on an analogue of the Fock space. In particular, we recover the block structure which was recently determined by C. Bowman, M. De Visscher and O. King ["The block structure of the partition algebras in characteristic $p\geq 0$", Preprint, arXiv:1402.5157]. In order to do so, we use induction and restriction functors as well as analogues of Jucys-Murphy 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.

Détail

Classification : 05E10, 16G20, 20C30
Keywords: partition algebra, diagram algebras, decomposition numbers, blocks, Jucys-Murphy elements