Abstract of thesis

The quantum general linear group, as defined by Dipper and Donkin, is a certain non-commutative deformation of the coordinate algebra of the corresponding classical group. In this thesis we study representations of this quantum group, and in particular develop an infinitesimal theory mimicking that of the classical case.

Our first main result generalises a theorem of Erdmann. This determines, using infinitesimal methods, precisely when a non-split extension exists between two Weyl modules for SL(2, k), in prime characteristic. We extend this result to quantum GL(2, k). As a corollary of this result we see that, when such extensions exist, they are unique up to isomorphism. We determine the structure of these in certain small cases, and consider a conjecture as to the structure of a much larger class of such.

Our particular quantum group was designed to provide a means of studying the q-Schur algebra of Dipper and James, and the rest of the thesis is concerned with this. In the classical case, the blocks of the Schur algebra have been determined by Donkin, and we verify that the appropriate modification of this result holds in the quantum case (with the same proof). This requires us to prove a quantum version of the strong linkage principle.

Doty, Nakano and Peters have defined an infinitesimal version of the Schur algebra, and we next consider a quantum analogue of this. After developing some of the basic representation theory of this algebra, we prove an infinitesimal version of Kostka duality. Finally, we determine the blocks of the infinitesimal Schur algebras corresponding to GL(2, k), and conclude by verifying that a similar result also holds in the quantum case.

Anton Cox (A.G.Cox@city.ac.uk)