Cellular bases for the Brauer and Birman–Murakami–Wenzl algebras

An explicit combinatorial construction is given for cellular bases (in the sense of Graham and Lehrer) for the Birman–Murakami–Wenzl and Brauer algebra. We provide cell modules for the Birman–Murakami–Wenzl and Brauer algebras with bases index by certain bitableaux, generalising the Murphy basis for the Specht modules of the Iwahori–Hecke algebra of the symmetric group. The bases for the cell modules given here are constructed non-diagrammatically and hence are relatively amenable to computation.