Topology of character varieties and representations of quivers

Tamás Hausel; Emmanuel Letellier; Fernando Rodriguez Villegas

doi:10.1016/j.crma.2010.01.025

Lie Algebras/Algebraic Geometry

Topology of character varieties and representations of quivers
[Topologie des variétés de caractères et représentations de carquois]

Présenté par : Gérard Laumon

Tamás Hausel ¹ ; Emmanuel Letellier ² ; Fernando Rodriguez Villegas ³

¹ University of Oxford, UK
² LMNO, département de mathématiques, Université de Caen, BP 5186, 14032 Caen, France
³ Department of Mathematics, University of Texas at Austin, Texas, USA

Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 131-135.

Résumé
Abstract

Dans Hausel et al. (2008) [10] nous avons énoncé une conjecture qui généralise la formule de Cauchy pour les polynômes de Macdonald. Cette formule contient l'information sur les polynômes de Hodge mixtes des variétés de représentations du groupe fondamental d'une surface de Riemann épointée de genre g. Nous avons montré plusieurs résultats qui appuient notre conjecture. Ici nous présentons de nouveaux résultats qui sont des conséquences de ceux de Hausel et al. (2008) [10].

In Hausel et al. (2008) [10] we presented a conjecture generalizing the Cauchy formula for Macdonald polynomial. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a punctured Riemann surface of genus g. We proved several results which support this conjecture. Here we announce new results which are consequences of those in Hausel et al. (2008) [10].

Reçu le : 2009-05-21
Accepté le : 2010-01-21
Publié le : 2010-02-01

DOI : 10.1016/j.crma.2010.01.025

Affiliations des auteurs :

Tamás Hausel ¹ ; Emmanuel Letellier ² ; Fernando Rodriguez Villegas ³

¹ University of Oxford, UK
² LMNO, département de mathématiques, Université de Caen, BP 5186, 14032 Caen, France
³ Department of Mathematics, University of Texas at Austin, Texas, USA

@article{CRMATH_2010__348_3-4_131_0,
     author = {Tam\'as Hausel and Emmanuel Letellier and Fernando Rodriguez Villegas},
     title = {Topology of character varieties and representations of quivers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {131--135},
     publisher = {Elsevier},
     volume = {348},
     number = {3-4},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.025},
     language = {en},
}

TY  - JOUR
AU  - Tamás Hausel
AU  - Emmanuel Letellier
AU  - Fernando Rodriguez Villegas
TI  - Topology of character varieties and representations of quivers
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 131
EP  - 135
VL  - 348
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2010.01.025
LA  - en
ID  - CRMATH_2010__348_3-4_131_0
ER  -

%0 Journal Article
%A Tamás Hausel
%A Emmanuel Letellier
%A Fernando Rodriguez Villegas
%T Topology of character varieties and representations of quivers
%J Comptes Rendus. Mathématique
%D 2010
%P 131-135
%V 348
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2010.01.025
%G en
%F CRMATH_2010__348_3-4_131_0

Tamás Hausel; Emmanuel Letellier; Fernando Rodriguez Villegas. Topology of character varieties and representations of quivers. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 131-135. doi : 10.1016/j.crma.2010.01.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.025/

Bibliographie
Cité par

[1] S. Bloch; A. Okounkov The character of the infinite wedge representation, Adv. Math., Volume 149 (2000) no. 1, pp. 1-60

[2] W. Crawley-Boevey Geometry of the moment map for representations of quivers, Comput. Math., Volume 126 (2001), pp. 257-293

[3] W. Crawley-Boevey Normality of Mardsen–Weinstein reductions for representations of quivers, Math. Ann., Volume 325 (2003), pp. 55-79

[4] W. Crawley-Boevey; M. Van den Bergh Absolutely indecomposable representations and Kac–Moody Lie algebras, Invent. Math., Volume 155 (2004) no. 3, pp. 537-559 (with an appendix by Hiraku Nakajima)

[5] P. Deligne Théorie de Hodge II, Inst. Hautes Etudes Sci. Publ. Math., Volume 40 (1971), pp. 5-47

[6] A.M. Garsia; M. Haiman A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin., Volume 5 (1996) no. 3, pp. 191-244

[7] L. Göttshce Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces, Comm. Math. Phys., Volume 206 (1999), pp. 105-136

[8] L. Göttsche; W. Soergel Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann., Volume 296 (1993), pp. 235-245

[9] T. Hausel, Arithmetic harmonic analysis, Macdonald polynomials and the topology of the Riemann–Hilbert monodromy map, with an appendix by E. Letellier, in preparation

[10] T. Hausel; E. Letellier; F. Rodriguez-Villegas Arithmetic harmonic analysis on character and quiver varieties, 2008 | arXiv

[11] T. Hausel, E. Letellier, F. Rodriguez-Villegas, Arithmetic harmonic analysis on character and quiver varieties II, in preparation

[12] V. Kac Root systems, representations of quivers and invariant theory, Lecture Notes in Math., vol. 996, Springer-Verlag, 1982, pp. 74-108

Cité par Sources :

Commentaires - Politique