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On the number of subrepresentations of a general quiver representation. (English) Zbl 1146.16007
Let \(Q\) be a quiver with no oriented cycles. For dimension vectors \(\alpha\) and \(\beta\), define \(N(\beta,\alpha)\) as the number of \(\beta\)-dimensional subrepresentations of a general \(\alpha\)-dimensional representation of \(Q\). If \(\langle\beta,\alpha-\beta\rangle=0\) (here \(\langle\cdot,\cdot\rangle\) denotes the Ringel bilinear form), then \(N(\beta,\alpha)\) is finite. Denote by \(M(\beta,\alpha)\) the dimension of the space of semi-invariant polynomial functions with weight \(\langle\beta,\cdot\rangle\) on the space of \((\alpha-\beta)\)-dimensional representations of \(Q\) (note that any non-zero semi-invariant on \(\text{Rep}(Q,\alpha-\beta)\) has weight of such special form).
The main result of this paper is that \(M(\beta,\alpha)=N(\beta,\alpha)\) when \(\langle\beta,\alpha-\beta\rangle=0\). The proof is that the number \(M(\beta,\alpha)\) can be expressed via a Littlewood-Richardson calculation, which is then compared by the authors with the expression of \(N(\beta,\alpha)\) given by W. Crawley-Boevey [Bull. Lond. Math. Soc. 28, No. 4, 363-366 (1996; Zbl 0863.16008)] using intersection theory.
Applying results of A. Schofield [J. Lond. Math. Soc., II. Ser. 43, No. 3, 383-395 (1991; Zbl 0779.16005)], a basis of the corresponding space of semi-invariants is obtained. The result is generalized to covariants as follows: the cohomology class of the variety of \(\beta\)-dimensional subrepresentations of an \(\alpha\)-dimensional representation in general position can be expressed in terms of multiplicities in isotypic components of the coordinate ring of \(\text{Rep}(Q,\beta-\alpha)\).

16G20 Representations of quivers and partially ordered sets
13A50 Actions of groups on commutative rings; invariant theory
14M15 Grassmannians, Schubert varieties, flag manifolds
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