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

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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.