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

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- by Harm Derksen and Jerzy Weyman
- J. Amer. Math. Soc.
**13**(2000), 467-479 - DOI: https://doi.org/10.1090/S0894-0347-00-00331-3
- Published electronically: March 13, 2000
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## Abstract:

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.## References

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## Bibliographic Information

**Harm Derksen**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02151
- Email: hderksen@math.mit.edu
**Jerzy Weyman**- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
- MR Author ID: 182230
- ORCID: 0000-0003-1923-0060
- Email: weyman@neu.edu
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**13**(2000), 467-479 - MSC (2000): Primary 13A50; Secondary 14L24, 14L30, 16G20, 20G05
- DOI: https://doi.org/10.1090/S0894-0347-00-00331-3
- MathSciNet review: 1758750