Polynomial representations of GrT,simple polynomial modules andpolynomial parts of Auslander-Reitencomponents

In this thesis, we use the general framework given by Doty to consider polynomial representations of $G_r T$ for $G$ either a reductive group or a Borel subgroup of such a group and view the category of polynomial $G_r T$-modules as a full subcategory of the category of $X(T)$-graded $G_r$-modules. It is our aim to extend results on $\mathrm{GL}_n$ to this case and to study the interplay of the Auslander-Reiten theory of $\m G_r T$ and of the infinitesimal Schur algebras by considering the polynomial parts of components of the stable Auslander-Reiten quiver $\Gamma_s(G_r T)$ of the category of finite-dimensional $G_r T$-modules. We give sufficient conditions on $G$ making a classification of simple polynomial $G_r T$-modules similar to the case $G = \mathrm{GL}_n$ possible and apply this to recover the corresponding result for $\mathrm{GL}_n$ with a different proof, extending it to symplectic similitude groups, Levi subgroups of $\mathrm{GL}_n$ and, in a weaker form, to odd orthogonal similitude groups. For components $\Theta \subseteq \Gamma_s(G_r T)$ of complexity $1$, we show that $\Theta$ only contains finitely many polynomial $G_r T$-modules. If $G$ is the Borel subgroup of upper triangular matrices of $\mathrm{GL}_2$ and $T \subseteq G$ the torus of diagonal matrices, we extend this result to modules arising as outer tensor $M \otimes_k N$, where $M$ has complexity $1$. In this situation, $G_r T$-modules correspond to graded modules over an elementary abelian $p$-group of rank $r$. As an application, we show that the components of simple modules contain only finitely many polynomial modules in this case. For $G = \mathrm{GL}_2$, $r = 1$, we use Premets classification of indecomposable modules for the restricted enveloping algebra $U_0(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ to fully determine the polynomial parts of the components of $\Gamma_s(G_1 T)$ and the Auslander-Reiten quiver of the corresponding infinitesimal Schur algebras.


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