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
- Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, DOI 10.1016/0001-8708(82)90039-1
- C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), no. 3, 330–354. MR 422314, DOI 10.1016/S0001-8708(76)80003-5
- Séminaire Bourbaki. Vol. 1997/98, Société Mathématique de France, Paris, 1998 (French). Exposés 835–849; Astérisque No. 252 (1998). MR 1685659
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. MR 677715, DOI 10.1016/0021-8693(82)90105-3
- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
- A. Klyachko, Stable vector bundles and Hermitian operators, IGM, University of Marne-la-Vallee, preprint (1994).
- Allen Knutson and Terence Tao, The honeycomb model of $\textrm {GL}_n(\textbf {C})$ tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055–1090. MR 1671451, DOI 10.1090/S0894-0347-99-00299-4
- Claus Michael Ringel, Representations of $K$-species and bimodules, J. Algebra 41 (1976), no. 2, 269–302. MR 422350, DOI 10.1016/0021-8693(76)90184-8
- Aidan Schofield, Semi-invariants of quivers, J. London Math. Soc. (2) 43 (1991), no. 3, 385–395. MR 1113382, DOI 10.1112/jlms/s2-43.3.385
- Aidan Schofield, General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), no. 1, 46–64. MR 1162487, DOI 10.1112/plms/s3-65.1.46
- A. Schofield, M. van den Bergh, Semi-invariants of quivers for arbitrary dimension vectors, preprint, math.RA/9907174.
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