Abstract. A construction of bases for cell modules of the Birman-Murakami-Wenzl (or B-M-W) algebra B n (q, r) by lifting bases for cell modules of B n−1 (q, r) is given. By iterating this procedure, we produce cellular bases for B-M-W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys-Murphy elements from the representation theory of the Iwahori-Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B-M-W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori-Hecke algebra of the symmetric group.

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.

We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin. 26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive formula and the original definition of Jucys-Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys-Murphy elements given in this paper are used to construct the seminormal representations for the partition algebras in a separate paper.In §2 we recall the presentation of the partition algebras given by Halverson and Ram [HR] and East [Ea]. In §3 we state a recursion defining a family of operators (L i , L i+ 1 2 : i = 0, 1, . . .) (1.2) and establish that the operators (1.2) form a commuting family with properties analogous to the Jucys-Murphy elements that arise in the representation theory of the symmetric group. Simultaneously, we establish some basic commutativity results for certain operators denotedwhich arose in the recursive definition of the Jucys-Murphy elements (1.2). In §4 we show that the elements of (1.3) are involutions which are related to the Coxeter generators for the symmetric group in a precise way. Using the relation between the involutions (1.3) and the Coxeter generators for the symmetric group, and the properties established in §3, we derive a new presentation for the partition algebras. In §5 we give formulae for the actions of the Jucys-Murphy elements (1.2) and the involutions (1.3) on tensor space. Using Schur-Weyl duality, we demonstrate that the recursive definition of Jucys-Murphy elements given in §3 is equivalent to the definition of Jucys-Murphy elements given by Halverson and Ram [HR].

Abstract. Using a new presentation for partition algebras (J. Algebraic Combin. 37(3):401-454, 2013), we derive explicit combinatorial formulae for the seminormal representations of the partition algebras. These results generalise to the partition algebras the classical formulae given by Young for the symmetric group.