| Classical Schur-Weyl duality is concerned with the commuting actions of the complex general linear group and the symmetric group on a tensor power of the natural representation of the complex general linear group. In this setting, the action of the symmetric group generates the full centralizer algebra for the action of the complex general linear group on the tensor space. In the analogous situation where the general linear group is replaced by the a symplectic or orthogonal group, the centralizer algebras are generated by the action of the Brauer algebras, defined by Richard Brauer in the 1930's.; More recently, the commuting actions of the quantum general linear group and the Iwahori-Hecke algebra of the symmetric group on tensor space have been studied. Here, the analogues of the Brauer algebras will be the Birman-Murakami-Wenzl algebras (B-M-W algebras). The action of the B-M-W algebras will generate the full centralizer algebras for the actions of the quantum orthogonal and symplectic groups on tensor powers of their respective natural representations.; This dissertation contains an explicit construction of the representations of the B-M-W algebras and of another class of algebras defined by Leduc which generalize the walled Brauer algebras of Benkart et al. These representations are constructed using a “good basis” of the algebra in question; more precisely by means of a cellular basis in the sense of Graham and Lehrer. This thesis gives an explicit combinatorial construction of such bases. |
