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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 $$\text{Rep}(Q,\beta-\alpha)$$.

##### MSC:
 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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