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Almost split sequences for polynomial G r T-modules and polynomial parts of Auslander-Reiten components
| Content Provider | Semantic Scholar |
|---|---|
| Author | Drenkhahn, Christian |
| Copyright Year | 2018 |
| Abstract | 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. 2010 Mathematics Subject Classification: 16G70, 20M32, 14L15 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://export.arxiv.org/pdf/1608.00429 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
