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29 March 2024
 
  » arxiv » 1410.3166

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Irreducible components of varieties of representations I. The local case
Birge Huisgen-Zimmermann ;
Date 13 Oct 2014
AbstractLet $Lambda$ be a local truncated path algebra over an algebraically closed field $K$, i.e., $Lambda$ is a quotient of a path algebra $KQ$ by the paths of length $L+1$, where $Q$ is the quiver with a single vertex and a finite number of loops and $L$ is a positive integer. For any $d>0$, we determine the irreducible components of the varieties that parametrize the $d$-dimensional representations of $Lambda$, namely, the components of the classical affine variety ${ mf{Rep}}_{d}(Lambda)$ and -- equivalently -- those of the projective parametrizing variety ${ m GRASS}_d(Lambda)$. Our method is to corner the components by way of a twin pair of upper semicontinuous maps from ${ mf{Rep}}_{d}(Lambda)$ to a poset consisting of sequences of semisimple modules.
An excerpt of the main result is as follows. Given a sequence ${f S} = ({f S}_0, ..., {f S}_L)$ of semisimple modules with $dim igoplus_{0 le l le L} {f S}_l = d$, let ${ mf{Rep}}, {f S}$ be the subvariety of ${ mf{Rep}}_{d}(Lambda)$ consisting of the points that parametrize the modules with radical layering ${f S}$. (The radical layering of a $Lambda$-module $M$ is the sequence $igl(J^l M / J^{l+1} Migr)_{0 le l le L}$, where $J$ is the Jacobson radical of $Lambda$.) Suppose the quiver $Q$ has $r ge 2$ loops. If $d le L+1$, the variety ${ mf{Rep}}_{d}(Lambda)$ is irreducible. If, on the other hand, $d > L+1$, then the irreducible components of ${ mf{Rep}}_{d}(Lambda)$ are the closures of the subvarieties ${ mf{Rep}}, {f S}$ for those sequences ${f S}$ which satisfy the inequalities $dim {f S}_l le r dim {f S}_{l+1}$ and $dim {f S}_{l+1} le r dim {f S}_l$ for $0 le l < L$. As a byproduct, the main result provides generic information on the modules corresponding to the irreducible components of the parametrizing varieties.
Source arXiv, 1410.3166
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