Gradings on the Brauer algebras and double affine BMW algebras

Abstract

In this thesis we study algebras that appear in di erent generalisations of the well-known Schur-Weyl duality. This classical result gives a remarkable connection between the irreducible finite-dimensional representations of the general linear and symmetric groups. In this work we are interested in the algebras appearing in similar correspondences for the orthogonal/symplectic groups and their quantum analogues.

In the first part of the thesis, we work with Brauer algebras. These algebras were first introduced by Brauer to obtain a better understanding of the irreducible finite-dimensional representations of the orthogonal /symplectic groups. Afterwards, it was shown that Brauer algebras are also important examples of graded cellular algebras. Moreover, two completely di erent approaches were considered to give a grading on the Brauer algebras. Our goal is to show that the two constructions give the same grading on the Brauer algebras. Namely, we give an explicit graded isomorphism between two constructions.

In the second part of thesis, we discuss generalizations of the quantised version of the previous construction, where the Brauer algebras are replaced by BMW algebras and symplectic/orthogonal groups are replaced by their quantum groups. In particular, to study specific representations of D-modules on the quantum group corresponding to the sympletic groups, we introduce double affine BMW algebras. Furthermore, we give some representations of these algebras and give its combinatorial description.