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Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Authors: Harm Derksen and Jerzy Weyman
Journal: J. Amer. Math. Soc. 13 (2000), 467-479
MSC (2000): Primary 13A50; Secondary 14L24, 14L30, 16G20, 20G05
Published electronically: March 13, 2000
MathSciNet review: 1758750
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Let $Q$ be a quiver without oriented cycles. For a dimension vector $\beta$ let $\operatorname{Rep}(Q, \beta)$ be the set of representations of $Q$ with dimension vector $\beta$. The group $\operatorname{GL}(Q, \beta)$ acts on $\operatorname{Rep}(Q, \beta)$. In this paper we show that the ring of semi-invariants $\operatorname{SI} (Q,\beta)$ is spanned by special semi-invariants $c^V$ associated to representations $V$ of $Q$. From this we show that the set of weights appearing in $\operatorname{SI}(Q,\beta)$ is saturated. In the case of triple flag quiver this reduces to the results of Knutson and Tao on the saturation of the set of triples of partitions for which the Littlewood-Richardson coefficient is nonzero.

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Additional Information

Harm Derksen
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02151

Jerzy Weyman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Quiver representations, semi-invariants, Littlewood-Richardson coefficients, Klyachko cone, saturation
Received by editor(s): July 20, 1999
Published electronically: March 13, 2000
Additional Notes: The second author was supported by NSF, grant DMS 9700884 and KBN No. PO3A 012 14.
Article copyright: © Copyright 2000 American Mathematical Society

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