Alcove geometry and a translation principle for the Brauer algebra

There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra can be reduced to a study of the block containing the weight 0. We define parabolic Kazhdan–Lusztig polynomials for the Brauer algebra and show in certain low rank examples that they determine standard module decomposition numbers and filtrations.