Dimensions of Ext-groups of Weyl modules for GL_2

TL;DR

This paper analyzes the extension algebra of Weyl modules for GL_2 over fields of positive characteristic, providing bounds on Ext-group dimensions and an algorithm for their computation.

Contribution

It offers explicit bounds and an algorithm for calculating Ext-group dimensions, building on prior explicit descriptions of the extension algebra.

Findings

Bounds on the growth of Ext-group dimensions

An algorithm to compute Ext-group dimensions

Explicit basis for the extension algebra

Abstract

Let F be an algebraically closed field of positive characteristic p. The third author and Will Turner gave an explicit description of the extension algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit basis. We examine this basis and use it to give upper and lower bounds for the growth behaviour of the dimensions of Ext-groups. Moreover, we implement an algorithm to determine the dimension of any given Ext-group.