Title:

On the A1structure and quiver for the Weyl extension algebra of GL2(Fp)

This research belongs to the field of Representation Theory and tries to solve questions through homological algebraic methods. This project deals with the study of symmetries of the plane and aims at measuring how much a mathematical object of importance for that study fails to satisfy the property of not needing bracketing when multiplying three elements together, which is called associativity. More precisely, we study the rational representations of GL2(Fp), the general linear group of order 2 over an algebraically closed field of prime characteristic p. Representations are a means to understand group or algebra elements as linear transformations on a vector space of a given dimension, and it is possible to 'build' representations from smaller ones, e.g. the set of socalled standard representations. The way to glue these building blocks together is governed by the algebra of extensions between standard representations. In a series of papers culminating with [MT13], Miemietz and Turner described precisely the algebra structure of that extension algebra. It is the homology of a differentialgraded algebra and this project aims at estimating how nonassociative it is by computing its A1algebra structure. For any p, we give the quiver of that extension algebra, and for p = 2, we show that there exists a subalgebra of the extension algebra which admits a trivial A1algebra structure, and what's more, in a somewhat peculiar way. We also give its quiver and discuss some of its properties.
