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

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.