a b s t r a c tThe Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.In particular, there is an ordering of simple and cell modules such that D B r (δ) is lower triangular with diagonal entries equal to 1. Remark 1.10. When F is a field of characteristic 0, then the decomposition matrix of the Brauer algebra was determined by Martin, see [16]. If the characteristic of F is 0 or larger than p, then an algorithm to compute the decomposition matrix of the Brauer algebra can also be found in [23]
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 for which exact 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 Uqp x slpq and Uqpsl8q 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 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 Lpµq is a composition factor of a cell module ∆pλq if and only if λ and µ differ by the action of a transposition.
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