On integral $\mathrm{Ext^2}$ between certain Weyl modules of $\mathrm{GLn}$

TL;DR

This paper explicitly computes the second extension groups between certain Weyl modules of the general linear group over integers, providing new insights into their integral structure and relationships.

Contribution

It determines the integral Ext^2 groups between specific Weyl modules of GL_n, a problem previously unresolved in the integral setting.

Findings

Explicit formulas for Ext^2 between the modules

Extension groups depend on the parameters of the partitions

Results apply to integral Schur algebras and general linear groups

Abstract

Consider partitions of the form $\lambda=(a,1^b)$ and $\mu=(a+1,b-1)$,\\ where $a+1>b-1$. In this paper, we determine the extension groups $\mathrm{Ext}_A^2(K_{\lambda}F,K_{\mu}F)$, where $F$ is a free $\mathbb{Z}-$module of finite rank $n$, $K_{\lambda}F$ and $K_{\mu}F$ are the Weyl modules of the general linear group $GL_n(\mathbb{Z})$ corresponding to $\lambda$ and $\mu$, respectively, $A=S_\mathbb{Z}(n,r)$ is the integral Schur algebra and $r=a+b$.