Rational smoothness of varieties of representations for quivers of Dynkin type
[Lissité rationnelle des variétés de représentations pour les carquois de type Dynkin]
Caldero, Philippe ; Schiffler, Ralf
Annales de l'Institut Fourier, Tome 54 (2004), p. 295-315 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

On étudie les clôtures au sens de Zariski des orbites de représentations des carquois de type A, D ou E. A l’aide de la base canonique de Lusztig, on caractérise les clotures d’orbites rationnellement lisses et l’on prouve que ces variétés sont lisses si et seulement si elle sont rationnellement lisses.

We study the Zariski closures of orbits of representations of quivers of type A, D ou E. With the help of Lusztig’s canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth.

Publié le : 2004-01-01
DOI : https://doi.org/10.5802/aif.2019
Classification:  17B37,  16G20,  14B05
Mots clés: groupes quantiques, representations de carquois, singularites, base canonique 
@article{AIF_2004__54_2_295_0,
     author = {Caldero, Philippe and Schiffler, Ralf},
     title = {Rational smoothness of varieties of representations for quivers of Dynkin type},
     journal = {Annales de l'Institut Fourier},
     volume = {54},
     year = {2004},
     pages = {295-315},
     doi = {10.5802/aif.2019},
     zbl = {02123568},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2004__54_2_295_0}
}

Caldero, Philippe; Schiffler, Ralf. Rational smoothness of varieties of representations for quivers of Dynkin type. Annales de l'Institut Fourier, Tome 54 (2004) pp. 295-315. doi : 10.5802/aif.2019. http://gdmltest.u-ga.fr/item/AIF_2004__54_2_295_0/

[BL] S. Billey; V. Lakshmibai Singular loci of schubert varieties, Birkhäuser, Boston-Basel-Berlin, Progress in Math, Tome 182 (2000) | MR 1782635 | Zbl 0959.14032

[BM] W. Borho; R. Macpherson Partial resolutions of nilpotent varieties, Analyse et topologie sur les espaces singuliers II (Astérisque) Tome 101-102 (1983), pp. 23-74 | MR 737927 | Zbl 0576.14046

[Bon95] K. Bongartz Degenerations for representations of tame quivers, Annales Scientifiques de L'école Normale Supérieure (IV), Tome 28 (1995), pp. 647-668 | Numdam | MR 1341664 | Zbl 0844.16007

[Bri98] M. Brion Equivariant cohomology and equivariant intersection theory, Representation Theory and Algebraic geometry, Kluwer Acad. Publ (1998), pp. 1-37 | MR 1649623 | Zbl 0946.14008

[BS] R. Bédard; R. Schiffler Rational smoothness of varieties of representations for quivers of type A, Represent. Theory, Tome 7 (2003), pp. 481-548 | MR 2017066 | Zbl 1060.17005

[Cal] Ph. Caldero A multiplicative property of quantum flag minors, Representation Theory, Tome 7 (2003), pp. 164-176 | MR 1973370 | Zbl 1030.17009

[Car94] J. Carrell; W. Haboush And B. Parshall Eds The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, Algebraic groups and their generalizations, American Mathematical Society, Providence (1994), pp. 53-62 | MR 1278700 | Zbl 0818.14020

[Dan96] V.I. Danilov; I.R. Shafarevich Eds Cohomology of algebraic varieties, ch. Algebraic geometry II (Encyclopedia of Math. Sciences) (1996), pp. 1-125 | MR 1392957 | Zbl 0832.14009

[Deo85] V. Deodhar Local Poincaré duality and nonsingularities of Schubert varieties, Comm. Alg, Tome 13 (1985), pp. 1379-1388 | MR 788771 | Zbl 0579.14046

[Gre95] J.A. Green Hall algebras, hereditary algebras and quantum groups, Invent. Math, Tome 120 (1995), pp. 361-377 | MR 1329046 | Zbl 0836.16021

[Kas91] M. Kashiwara On crystal bases of the q-analogue of the universal enveloping algebra, Duke Math. J, Tome 63 (1991), pp. 465-516 | MR 1115118 | Zbl 0739.17005

[Lus90a] G. Lusztig Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc, Tome 3 (1990), pp. 447-498 | MR 1035415 | Zbl 0703.17008

[Lus90b] G. Lusztig Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc, Tome 3 (1990), pp. 257-296 | MR 1013053 | Zbl 0695.16006

[Lus93] G. Lusztig Introduction to quantum groups, Birkhäuser, Progress in Mathematics, Tome 110 (1993) | MR 1227098 | Zbl 0788.17010

[Nör] R. Nörenberg From elementary calculations to Hall polynomials (preprint) | MR 1987346 | Zbl 1060.16018

[Rei99] M. Reineke Multiplicative properties of dual canonical bases of quantum groups, Journal of Algebra, Tome 211 (1999), pp. 134-149 | MR 1656575 | Zbl 0917.17008

[Rie94] C. Riedtmann Lie algebras generated by indecomposables, Journal of algebra, Tome 170 (1994), pp. 526-546 | MR 1302854 | Zbl 0841.16018

[Rin90] C. M. Ringel Hall algebras, Banach Center Publ, Tome 26 (1990), pp. 433-447 | MR 1171248 | Zbl 0778.16004

[Rin93] C. M. Ringel Hall algebras revisited (Israel Math. Conf. Proc) Tome 7 (1993), pp. 171-176 | MR 1261907 | Zbl 0852.17009