On the Ext groups between Weyl modules for GL_n

TL;DR

This paper computes extension groups between specific Weyl modules for GL_n, advancing understanding of their modular representation theory and recovering recent related results.

Contribution

It provides a complete determination of Ext groups between Weyl modules with weights differing by a root and between exterior powers and Weyl modules.

Findings

Calculated Ext groups between neighboring Weyl modules for GL_n

Determined Ext^1 between exterior powers and Weyl modules

Recovers a recent result of Andersen for GL_n

Abstract

This paper studies extension groups between certain Weyl modules for the algebraic group GL_n over the integers. Main results include: (1) A complete determination of Ext groups between Weyl modules whose highest weights differ by a single root and (2) Determination of Ext^1 between an exterior power of the defining representation and any Weyl module. The significance of these results for modular representation theory of GL_n is discussed in several Remarks. Notably the first result leads to a calculation of Ext groups between neighboring Weyl modules for GL_n and also recovers the GL_n case of a recent result of Andersen. Some generalities about Ext groups between Weyl modules and a brief overview of known results about these groups are also included.