2016/08/01 by Christian Drenkhahn, Drenkhahn, Christian
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1608.00429
32 pages
arxiv created 2016/09/12 · arxiv updated 2016/09/13
In 1996, Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via Gr T-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 Gr T and Br T as full subcategories of mod \thinspace Gr T and mod\thinspace Br T, respectively. We show that every component Θ of the stable Auslander-Reiten quiver Γs(Gr T) of mod\thinspace Gr T 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 Gr 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 GL2, the category of Br 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.