John Enyang Nagoya University On the Semisimplicity of the Brauer and Birman-Murakami-Wenzl algebras Friday 13th August, 12:05-12:55pm, Carslaw 175. The Birman-Murakami-Wenzl (or B-M-W) algebras arise as centraliser algebras for the action of the Drinfeld-Jimbo quantum groups of type B, C and D on their respective natural representations. The B-M-W algebras are known to be cellular in the sense of Graham and Lehrer. We give an explicit construction of new cellular bases for the B-M-W algebras, bases which are indexed by paths in the Bratteli diagram associated with the B-M-W algebras and with respect to which the Jucys-Murphy elements in the B-M-W algebra act upper triangularly; we also give explicit formulae, in terms of paths in the Bratteli diagram associated with the B-M-W algebras, for the generalised eigenvalues arising from the action of the Jucys-Murphy operators on the cell (or Specht) modules of the B-M-W algebras. Our construction is motivated by the Specht module theory for the representations of the Iwahori-Hecke algebra of the symmetric group. The Jucys-Murphy elements act upper triangularly with respect to the Murphy bases for the Specht modules of the Iwahori-Hecke algebra of the symmetric group, allowing one to distinguish between the Specht modules and to state explicitly in terms of standard tableaux, the necessary and sufficient conditions for the Iwahori-Hecke algebra of the symmetric group to be semisimple. The Jucys-Murphy operators play a somewhat weaker role in the representation theory of the B-M-W algebras. Here, the action of the Jucys-Murphy operators allow us to state explicitly, in terms of paths in the associated Bratteli diagrams, sufficient but not necessary conditions for the B-M-W algebras to be semi-simple. We give examples showing that even if parameters are chosen such that a B-M-W algebra is semisimple, the Jucys-Murphy elements may fail to distinguish between distinct cell modules of the given algebra. Finally we show how our construction may be applied to the representation theory of Brauer's centraliser algebras.