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 $\mathrm{mod}{G}_{r}T$ and $\mathrm{mod}{B}_{r}T$, respectively. We show that every component $\Theta$ of the stable Auslander-Reiten quiver ${\Gamma }_s(G_{r}T)$ of $\mathrm{mod}{G}_{r}T$ 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.

32 pages

References in corpus (1)

• On simple polynomial $G_{r}T$-modules