Let \(n\) be a natural number, \(r, s\) and \(m\) non-negative integers and let \(R\) be a commutative ring. For \(x\in R\), the walled Brauer algebra \(B_{r, s}(x)\) is the subalgebra of the Brauer algebra \(B_{r+s}(x)\) spanned by certain diagrams, the walled Brauer diagrams. Let \(V\) be a free \(R\)-module of rank \(n\) and let \(V^\ast=\mathrm{Hom}_R(V, R)\). Then, the walled Brauer algebra for the parameter \(x=n\) acts on the mixed tensor space \(V^{\otimes r}\otimes V^{\ast\otimes s}\) and satisfies Schur-Weyl duality together with the action of the universal enveloping algebra \(U\) of the general linear algebra. This situation is very similar to (in fact a generalization of) the classical Schur-Weyl duality where both the group algebra \(R\mathfrak{S}_m\) of the symmetric group and \(U\) act on the ordinary tensor space \(V^{\otimes m}\). The group algebra of the symmetric group is a cellular algebra in the sense of J. J. Graham and G. I. Lehrer [Invent. Math. 123, No. 1, 1–34 (1996; Zbl 0853.20029)] with a cellular basis (the Murphy basis, see [G. E. Murphy, J. Algebra 173, No. 1, 97–121 (1995; Zbl 0829.20022)]) which has remarkable properties and one may ask, if the walled Brauer algebra has a cellular basis with similar properties.
The authors in the paper under review construct a cellular basis of the walled Brauer algebra which has similar properties as the Murphy basis of the group algebra of the symmetric group. In particular, the restriction of a cell module to a certain subalgebra can be easily described via this basis. Furthermore, the mixed tensor space possesses a filtration by cell modules – although not by cell modules of the walled Brauer algebra itself, but by cell modules of its image in the algebra of endomorphisms of mixed tensor space.