Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra

Abstract: We determine the decomposition numbers for the Brauer and walled Brauer algebra in characteristic zero in terms of certain polynomials associated to cap and curl diagrams (recovering a result of Martin in the Brauer case). We consider a second family of polynomials associated to such diagrams, and use these to determine projective resolutions of the standard modules. We then relate these two families of polynomials to Kazhdan-Lusztig theory via the work of Lascoux-Sch\"utzenberger and Boe, inspired by work of Brundan and Stroppel in the cap diagram case.