Almost split sequences for polynomial $G_r T$-modules and polynomial parts of Auslander-Reiten components

TL;DR

This paper investigates the structure of polynomial modules and Auslander-Reiten components for algebraic groups, extending classical results to new categories and identifying specific quivers for $ ext{GL}_2$ and Borel subgroups.

Contribution

It introduces analogues of infinitesimal Schur algebras for reductive groups and describes their Auslander-Reiten theory, including explicit quiver descriptions for $ ext{GL}_2$ and Borel subgroups.

Findings

Components with complexity 1 contain finitely many polynomial modules.

Explicit description of the polynomial part of the Auslander-Reiten quiver for $ ext{GL}_2$.

Extension of results to modules of higher Frobenius kernels for Borel subgroups.

Abstract

In 1996, Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via $G_r T$-modules to representations of the algebraic group $G = \mathrm{GL}_n$. We study analogues of these algebras and their Auslander-Reiten theory for reductive algebraic groups $G$ and Borel subgroups $B$ by considering the categories of polynomial representations of $G_r T$ and $B_r T$ as full subcategories of $\mathrm{mod} \thinspace G_r T$ and $\mathrm{mod}\thinspace B_r T$, respectively. We show that every component $\Theta$ of the stable Auslander-Reiten quiver $\Gamma_s(G_r T)$ of $\mathrm{mod}\thinspace G_r T$ whose constituents have complexity 1 contains only finitely many polynomial modules. For $G = \mathrm{GL}_2$, $r = 1$ and $T \subseteq G$ the torus of diagonal matrices, we identify the polynomial part of the stable Auslander-Reiten quiver of $G_r T$ and use this to determine the Auslander-Reiten quiver of the infinitesimal Schur algebras in this situation. For the Borel subgroup $B$ of lower triangular matrices of $\mathrm{GL}_2$, the category of $B_r T$-modules is related to representations of elementary abelian groups of rank $r$. In this case, we can extend our results about modules of complexity $1$ to modules of higher Frobenius kernels arising as outer tensor products.