×

zbMATH — the first resource for mathematics

A remark on quiver varieties and Weyl groups. (English) Zbl 1143.14309
Summary: We define an action of the Weyl group on the quiver varieties \(M_{m,\lambda}(d,v)\) with generic \((m,\lambda)\). To do it we describe a set of generators of the projective ring of a quiver variety. We also prove connectness for the smooth quiver variety \(M(d,v)\) and normality for \(M_0(d,v)\) in the case of a quiver of finite type and \(d-v\) a regular weight.

MSC:
14L30 Group actions on varieties or schemes (quotients)
16G20 Representations of quivers and partially ordered sets
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] L. Le Bruyn - C. Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585-598. Zbl0693.16018 MR958897 · Zbl 0693.16018 · doi:10.2307/2001477
[2] W. Crawley-Boevey, Geometry of the moment map for representations of quivers, preprint available at http://www.amsta.leeds.ac.uk/ pmtwc. Zbl1037.16007 MR1834739 · Zbl 1037.16007 · doi:10.1023/A:1017558904030
[3] H. Derksen - J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc. 13 (2000), 467-479. Zbl0993.16011 MR1758750 · Zbl 0993.16011 · doi:10.1090/S0894-0347-00-00331-3
[4] G. Kempf - L. Ness, The length of vectors in representation spaces, In: “Algebraic geometry”. Proc. Summer Meeting Univ. Copenhagen 1978, Vol. 732 of LNM, Springer, 1979, pp. 233-243. Zbl0407.22012 MR555701 · Zbl 0407.22012
[5] G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), 141-182. Zbl0915.17008 MR1623674 · Zbl 0915.17008 · doi:10.1006/aima.1998.1729
[6] G. Lusztig, Quiver varieties and Weyl group actions, Ann. Inst. Fourier (Grenoble) 50 (2000), 461-489. Zbl0958.20036 MR1775358 · Zbl 0958.20036 · doi:10.5802/aif.1762 · numdam:AIF_2000__50_2_461_0 · eudml:75426
[7] A. Maffei, A remark on quiver varieties and Weyl groups, preprint available at http://xxx.lanl.gov. · Zbl 1143.14309
[8] A. Maffei, “Quiver varieties”, PhD thesis, Universit√† di Roma “La Sapienza”, 1999. · Zbl 1095.16008
[9] L. Migliorini, Stability of homogeneous vector bundles, Boll. Un. Mat. Ital. 7-B (1996), 963-990. Zbl0885.14024 MR1430162 · Zbl 0885.14024
[10] D. Mumford - J. Fogarty - F. Kirwan, “Geometric invariant theory”, Ergebn. der Math., Vol. 34, Springer, third edition, 1994. Zbl0797.14004 MR1304906 · Zbl 0797.14004 · doi:10.1007/978-3-642-57916-5
[11] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365-416. Zbl0826.17026 MR1302318 · Zbl 0826.17026 · doi:10.1215/S0012-7094-94-07613-8
[12] H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), 515-560. Zbl0970.17017 MR1604167 · Zbl 0970.17017 · doi:10.1215/S0012-7094-98-09120-7
[13] P. Newstead, “Introduction to moduli problems and orbit spaces”, Tata Lectures, Vol. 51, Springer, 1978. Zbl0411.14003 MR546290 · Zbl 0411.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.