no. 2
Rational smoothness of varieties of representations for quivers of Dynkin type
[Lissité rationnelle des variétés de représentations pour les carquois de type Dynkin]
Caldero, Philippe1 ; Schiffler, Ralf
1 Université Claude Bernard Lyon I, Département de Mathématiques, 69622 Villeurbanne (France), Carleton University, School of mathematics and statistics, 1125 Colonel By drive, room 4302 Herzberg building, Ottawa, Ontario K1S 5B6 (Canada)
Annales de l'Institut Fourier, Tome 54 (2004) no. 2, pp. 295-315.

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.

DOI : 10.5802/aif.2019
Classification : 17B37, 16G20, 14B05
Keywords: quantum groups, representations of quivers, singularities, canonical basis
Mot clés : groupes quantiques, representations de carquois, singularites, base canonique

Caldero, Philippe 1 ; Schiffler, Ralf 

1 Université Claude Bernard Lyon I, Département de Mathématiques, 69622 Villeurbanne (France), Carleton University, School of mathematics and statistics, 1125 Colonel By drive, room 4302 Herzberg building, Ottawa, Ontario K1S 5B6 (Canada) 
@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},
     pages = {295--315},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {2},
     year = {2004},
     doi = {10.5802/aif.2019},
     zbl = {02123568},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2019/}
}
TY  - JOUR
AU  - Caldero, Philippe
AU  - Schiffler, Ralf
TI  - Rational smoothness of varieties of representations for quivers of Dynkin type
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 295
EP  - 315
VL  - 54
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2019/
DO  - 10.5802/aif.2019
LA  - en
ID  - AIF_2004__54_2_295_0
ER  - 
%0 Journal Article
%A Caldero, Philippe
%A Schiffler, Ralf
%T Rational smoothness of varieties of representations for quivers of Dynkin type
%J Annales de l'Institut Fourier
%D 2004
%P 295-315
%V 54
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2019/
%R 10.5802/aif.2019
%G en
%F 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) no. 2, pp. 295-315. doi : 10.5802/aif.2019. https://aif.centre-mersenne.org/articles/10.5802/aif.2019/

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

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

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

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

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

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

[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 (1994), pp. 53-62 | MR | Zbl

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :