Almost split sequences for polynomial GrT-modules and polynomial parts of Auslander-Reiten components

In [8], Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via GrT-modules to representations of the algebraic group G=GLn. 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 GrT and BrT as full subcategories of modGrT and modBrT, respectively. We show that every component Θ of the stable Auslander-Reiten quiver Γs(GrT) of modGrT whose constituents have complexity 1 contains only finitely many polynomial modules. For G=GL2,r=1 and T⊆G the torus of diagonal matrices, we identify the polynomial part of the stable Auslander-Reiten quiver of GrT 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 GL2, the category of BrT-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.