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

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.