Integral forms for Weyl modules of GL(2,Q)

• Main Author: Stevens, Perdita Emma
• Published: University of Warwick 1992
• Subjects: 510; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.334002

In this thesis we determine the integral forms in Weyl modules for GL(2,Q). We work with the Schur algebra exclusively; we do not use the Lie algebra of GL(2,Q). In Chapter 1 we give the necessary background. We begin to simplify the problem, using the known reduction of it to the problem of finding those integral forms which lie between certain limits -Y and V. Together with localisation at each prime p, this enables us to restrict our attention to the structure of X/Vp. We show that we can deduce the integral structure of any Weyl module from that of Weyl modules with highest weight (r, 0) for an integer r. We describe a duality which arises on X/Vp. In Chapter 2 we prove a rather surprising number-theoretic result which allows us to simplify the problem further. In Chapter 3 we arrive at a very simple characterisation of the integral forms, namely that they can be represented as those integer labellings of a particular graph, the scoreable set lattice, which satisfy a certain criterion. We exploit this to prove various general results about the structure of X/Vp. We show how it is possible, using our methods, to describe the structure of X/Vp in arbitrarily complicated cases in terms of simpler structures. In Chapter 4, we discuss the relevance of our work to the theory of modular Weyl modules, and we explain how our work relates to that of others.