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Philippe Caldero; Ralf Schiffler
Rational smoothness of varieties of representations for quivers of Dynkin type
(Lissité rationnelle des variétés de représentations pour les carquois de type Dynkin)
Annales de l'institut Fourier, 54 no. 2 (2004), p. 295-315
Article PDF | Reviews Zbl 02123568 | 1 citation in Cedram
Class. Math.: 17B37, 16G20, 14B05
Keywords: quantum groups, representations of quivers, singularities, canonical basis
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.
[BL] S. Billey & V. Lakshmibai, Singular loci of schubert varieties, Progress in Math 182, Birkhäuser, 2000 MR 1782635 | Zbl 0959.14032
[Bon95] K. Bongartz, “Degenerations for representations of tame quivers”, Annales Scientifiques de L'école Normale Supérieure (IV) 28 (1995), p. 647-668
Numdam | MR 1341664 | Zbl 0844.16007
[BM] W. Borho & R. MacPherson, Partial resolutions of nilpotent varieties, Astérisque, 1983, p. 23-74 MR 737927 | Zbl 0576.14046
[BS] R. Bédard & R. Schiffler, “Rational smoothness of varieties of representations for quivers of type $\uppercase{a}$”, Represent. Theory 7 (2003), p. 481-548 MR 2017066 | Zbl 1060.17005
[Bri98] M. Brion, Equivariant cohomology and equivariant intersection theory, Kluwer Acad. Publ, 1998, p. 1-37 MR 1649623 | Zbl 0946.14008
[Cal] Ph. Caldero, “A multiplicative property of quantum flag minors”, Representation Theory 7 (2003), p. 164-176 MR 1973370 | Zbl 1030.17009
[Car94] J. Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, American Mathematical Society, 1994, p. 53-62 MR 1278700 | Zbl 0818.14020
[Dan96] V.I. Danilov, Cohomology of algebraic varieties, Encyclopedia of Math. Sciences, 1996, p. 1-125 MR 1392957 | Zbl 0832.14009
[Deo85] V. Deodhar, “Local Poincaré duality and nonsingularities of Schubert varieties”, Comm. Alg 13 (1985), p. 1379-1388 MR 788771 | Zbl 0579.14046
[Gre95] J.A. Green, “Hall algebras, hereditary algebras and quantum groups”, Invent. Math 120 (1995), p. 361-377
Article | MR 1329046 | Zbl 0836.16021
[Kas91] M. Kashiwara, “On crystal bases of the $q$-analogue of the universal enveloping algebra”, Duke Math. J 63 (1991), p. 465-516
Article | MR 1115118 | Zbl 0739.17005
[Lus90a] G. Lusztig, “Canonical bases arising from quantized enveloping algebras”, J. Amer. Math. Soc 3 (1990), p. 447-498 MR 1035415 | Zbl 0703.17008
[Lus90b] G. Lusztig, “Finite dimensional Hopf algebras arising from quantized universal enveloping algebras”, J. Amer. Math. Soc 3 (1990), p. 257-296 MR 1013053 | Zbl 0695.16006
[Lus93] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkhäuser, 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 211 (1999), p. 134-149 MR 1656575 | Zbl 0917.17008
[Rie94] C. Riedtmann, “Lie algebras generated by indecomposables”, Journal of algebra 170 (1994), p. 526-546 MR 1302854 | Zbl 0841.16018
[Rin90] C. M. Ringel, “Hall algebras”, Banach Center Publ 26 (1990), p. 433-447 MR 1171248 | Zbl 0778.16004
[Rin93] C. M. Ringel, Hall algebras revisited, Israel Math. Conf. Proc, 1993, p. 171-176 MR 1261907 | Zbl 0852.17009
Philippe Caldero; Ralf Schiffler
Rational smoothness of varieties of representations for quivers of Dynkin type
(Lissité rationnelle des variétés de représentations pour les carquois de type Dynkin)
Annales de l'institut Fourier, 54 no. 2 (2004), p. 295-315
Article PDF | Reviews Zbl 02123568 | 1 citation in Cedram
Class. Math.: 17B37, 16G20, 14B05
Keywords: quantum groups, representations of quivers, singularities, canonical basis
Résumé - Abstract
Bibliography
[Bon95] K. Bongartz, “Degenerations for representations of tame quivers”, Annales Scientifiques de L'école Normale Supérieure (IV) 28 (1995), p. 647-668
Numdam | MR 1341664 | Zbl 0844.16007
[BM] W. Borho & R. MacPherson, Partial resolutions of nilpotent varieties, Astérisque, 1983, p. 23-74 MR 737927 | Zbl 0576.14046
[BS] R. Bédard & R. Schiffler, “Rational smoothness of varieties of representations for quivers of type $\uppercase{a}$”, Represent. Theory 7 (2003), p. 481-548 MR 2017066 | Zbl 1060.17005
[Bri98] M. Brion, Equivariant cohomology and equivariant intersection theory, Kluwer Acad. Publ, 1998, p. 1-37 MR 1649623 | Zbl 0946.14008
[Cal] Ph. Caldero, “A multiplicative property of quantum flag minors”, Representation Theory 7 (2003), p. 164-176 MR 1973370 | Zbl 1030.17009
[Car94] J. Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, American Mathematical Society, 1994, p. 53-62 MR 1278700 | Zbl 0818.14020
[Dan96] V.I. Danilov, Cohomology of algebraic varieties, Encyclopedia of Math. Sciences, 1996, p. 1-125 MR 1392957 | Zbl 0832.14009
[Deo85] V. Deodhar, “Local Poincaré duality and nonsingularities of Schubert varieties”, Comm. Alg 13 (1985), p. 1379-1388 MR 788771 | Zbl 0579.14046
[Gre95] J.A. Green, “Hall algebras, hereditary algebras and quantum groups”, Invent. Math 120 (1995), p. 361-377
Article | MR 1329046 | Zbl 0836.16021
[Kas91] M. Kashiwara, “On crystal bases of the $q$-analogue of the universal enveloping algebra”, Duke Math. J 63 (1991), p. 465-516
Article | MR 1115118 | Zbl 0739.17005
[Lus90a] G. Lusztig, “Canonical bases arising from quantized enveloping algebras”, J. Amer. Math. Soc 3 (1990), p. 447-498 MR 1035415 | Zbl 0703.17008
[Lus90b] G. Lusztig, “Finite dimensional Hopf algebras arising from quantized universal enveloping algebras”, J. Amer. Math. Soc 3 (1990), p. 257-296 MR 1013053 | Zbl 0695.16006
[Lus93] G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkhäuser, 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 211 (1999), p. 134-149 MR 1656575 | Zbl 0917.17008
[Rie94] C. Riedtmann, “Lie algebras generated by indecomposables”, Journal of algebra 170 (1994), p. 526-546 MR 1302854 | Zbl 0841.16018
[Rin90] C. M. Ringel, “Hall algebras”, Banach Center Publ 26 (1990), p. 433-447 MR 1171248 | Zbl 0778.16004
[Rin93] C. M. Ringel, Hall algebras revisited, Israel Math. Conf. Proc, 1993, p. 171-176 MR 1261907 | Zbl 0852.17009
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310