#### Vol. 12, No. 2, 2018

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Closures in varieties of representations and irreducible components

### Kenneth R. Goodearl and Birge Huisgen-Zimmermann

Vol. 12 (2018), No. 2, 379–410
##### Abstract

For any truncated path algebra $\Lambda$ of a quiver, we classify, by way of representation-theoretic invariants, the irreducible components of the parametrizing varieties ${\mathbf{Rep}}_{d}\left(\Lambda \right)$ of the $\Lambda$-modules with fixed dimension vector $d$. In this situation, the components of ${\mathbf{Rep}}_{d}\left(\Lambda \right)$ are always among the closures $\overline{\mathbf{Rep}\phantom{\rule{0.3em}{0ex}}\mathbb{S}}$, where $\mathbb{S}$ traces the semisimple sequences with dimension vector $d$, and hence the key to the classification problem lies in a characterization of these closures.

Our first result concerning closures actually addresses arbitrary basic finite-dimensional algebras over an algebraically closed field. In the general case, it corners the closures $\overline{\mathbf{Rep}\phantom{\rule{0.3em}{0ex}}\mathbb{S}}$ by means of module filtrations “governed by $\mathbb{S}$”; when $\Lambda$ is truncated, it pins down the $\overline{\mathbf{Rep}\phantom{\rule{0.3em}{0ex}}\mathbb{S}}$ completely.

The analysis of the varieties $\overline{\mathbf{Rep}\phantom{\rule{0.3em}{0ex}}\mathbb{S}}$ leads to a novel upper semicontinuous module invariant which provides an effective tool towards the detection of components of ${\mathbf{Rep}}_{d}\left(\Lambda \right)$ in general. It detects all components when $\Lambda$ is truncated.

##### Keywords
varieties of representations, irreducible components, generic properties of representations
##### Mathematical Subject Classification 2010
Primary: 16G10
Secondary: 14M15, 14M99, 16G20