arXiv subjects

Rolf Farnsteiner

Publications and source records attributed to Rolf Farnsteiner.

12 recordsLinked to original sources

arXiv2024

Representations of Kronecker quivers and Steiner bundles on Grassmannians

Let $\mathbb{k}$ be an algebraically closed field. Connections between representations of the generalized Kronecker quivers $K_r$ and vector bundles on $\mathbb{P}^{r-1}$ have been known for quite some time. This article is concerned with a particular aspect of this correspondence, involving more generally Steiner bundles on Grassmannians $\mathrm{Gr}_d(\mathbb{k}^r)$ and certain full subcategories $\mathrm{rep}_{\mathrm{proj}}(K_r,d)$ of relative projective $K_r$-representations. Building on a categorical equivalence first explicitly established by Jardim and Prata, we employ representation-theoretic techniques provided by Auslander-Reiten theory and reflection functors to organize indecomposable Steiner bundles in a manner that facilitates the study of bundles enjoying certain properties such as uniformity and homogeneity. Conversely, computational results on Steiner bundles motivate investigations in $\mathrm{rep}_{\mathrm{proj}}(K_r,d)$, which elicit the conceptual sources of some recent work on the subject. From a purely representation-theoretic vantage point, our paper initiates the investigation of certain full subcategories of the, for $r\!\ge\!3$, wild category of $K_r$-representations. These may be characterized as being right Hom-orthogonal to certain algebraic families of elementary test modules.

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arXiv2023

Varieties of subalgebras and endotrivial modules

Let $(\mathfrak{g},[p])$ be a finite dimensional restricted Lie algebra over a perfect field $\mathbbm{k}$ of characteristic $p\!\ge \!3$. By combining methods from recent work of Benson-Carlson \cite{BC20} with those of \cite{CF21,Fa17} we obtain a description of the endotrivial $(\mathfrak{g},[p])$-modules in case $\mathfrak{g}$ is supersolvable.

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arXiv2021

Varieties of Elementary Abelian Lie Algebras and Degrees of Modules

Let $(\mathfrak{g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak{g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots,p\!-\!1\}$), we study the projective variety $\mathbb{E}(2,\mathfrak{g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge\!5$, then the topological space $\mathbb{E}(2,\mathfrak{g}/C(\mathfrak{g}))$, associated to the factor algebra of $\mathfrak{g}$ by its center $C(\mathfrak{g})$, is shown to be connected. We give applications concerning categories of $(\mathfrak{g},[p])$-modules of constant $j$-rank and certain invariants, called $j$-degrees.

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arXiv2018

Commuting varieties for nilpotent radicals

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin-Röhrle concerning the commuting variety of Lie(U) for char(k)=0 to fields, whose characteristic is good for G

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arXiv2016

Finite group schemes of $p$-rank $\leq1$

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the $p$-rank $\mathsf{rk}_p(\mathcal{G})$ of $\mathcal{G}$ and determine the structure of those group schemes of $p$-rank $1$, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height $1$, which correspond to restricted Lie algebras. Our results show that group schemes of $p$-rank $\leq 1$ are closely related to those being of finite or domestic representation type.

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arXiv2014

Representations of finite group schemes and morphisms of projective varieties

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $°^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$ to $M$ that take values in Grassmannians of $M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant $j$-rank have a well-defined degree ranging between $0$ and $j\rk^j(M)$, where $\rk^j(M)$ is the generic $j$-rank of $M$. The extreme values are attained when the module $M$ has the equal images property or the equal kernels property. We establish a formula linking the $j$-degrees of $M$ and its dual $M^\ast$. For a self-dual module $M$ of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back $α^\ast(M)$ of $M$ along a $p$-point $α: k[X]/(X^p) \lra k\cG$.

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arXiv2012

Extensions of tame algebras and finite group schemes of domestic representation type

Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k- algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of finite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary polyhedral group schemes.

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arXiv2012

Weyl groups for non-classical restricted Lie algebras and the Chevalley restriction theorem

Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.

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arXiv2011

Categories of modules given by varieties of p-nilpotent operators

For a finite group scheme G over an algebraically closed field k of characteristic p>0 we study G-modules M, which are defined in terms of properties of their pull-backs along p-points of G. We show that the corresponding subcategories strongly depend on the structure of G. The second part of the paper discusses recent work by Carlson-Friedlander-Suslin concerning the subcategory of equal images modules from the vantage point of Auslander-Reiten theory.

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arXiv2009

Jordan Types for Indecomposable Modules of Finite Group Schemes

In this article we study the interplay between algebro-geometric notions related to $π$-points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $π$-points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on $π$-points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial modules.

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arXiv2009

Complexity, Periodicity and One-Parameter Subgroups

We use the variety of one-parameter subgroups to define a numerical invariant for a representation of an infinitesimal group scheme. For an indecomposable module M of complexity 1, this number is related to the period of M.

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arXiv2009

Support Varieties, AR-Components, and Good Filtrations

In continuation of work begun in \cite{FR}, we study in this article those Auslander--Reiten components of the algebras $\Dist(G_r)$ that contain simple modules or baby Verma modules, where $\Dist(G_r)$ is the algebra of distributions of the $r$-th Frobenius kernel of the smooth reductive group $G$, defined over an algebraically closed field of positive characteristic.

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