Localization in quiver moduli spaces

Weist, Thorsten

doi:10.1090/S1088-4165-2013-00436-3

Localization in quiver moduli spaces
HTML articles powered by AMS MathViewer

by Thorsten Weist PDF

Represent. Theory 17 (2013), 382-425 Request permission

Abstract:

Torus fixed points of quiver moduli spaces are given by stable representations of the universal (abelian) covering quiver. As far as the Kronecker quiver is concerned they can be described by stable representations of certain bipartite quivers coming along with a stable colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. For certain roots it is even possible to construct all torus fixed points.

References

I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065
A. Białynicki-Birula, On fixed point schemes of actions of multiplicative and additive groups, Topology 12 (1973), 99–103. MR 313261, DOI 10.1016/0040-9383(73)90024-4
K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
R. Martínez-Villa and J. A. de la Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), no. 3, 277–292. MR 724038, DOI 10.1016/0022-4049(83)90062-2
Frederik Denef, Quantum quivers and Hall/hole halos, J. High Energy Phys. 10 (2002), 023, 42. MR 1952307, DOI 10.1088/1126-6708/2002/10/023
M. Douglas, Oral communication to Markus Reineke from 02.08.2004.
J.-M. Drezet, Cohomologie des variétés de modules de hauteur nulle, Math. Ann. 281 (1988), no. 1, 43–85 (French). MR 944602, DOI 10.1007/BF01449215
Michael Drmota, Combinatorics and asymptotics on trees, Cubo 6 (2004), no. 2, 105–136 (English, with English and Spanish summaries). MR 2092045
P. Gabriel, The universal cover of a representation-finite algebra, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 68–105. MR 654725
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248. MR 364254, DOI 10.1007/BF01357141
Frank Harary and Edgar M. Palmer, Graphical enumeration, Academic Press, New York-London, 1973. MR 0357214
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. MR 677715, DOI 10.1016/0021-8693(82)90105-3
A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
W. Lück, Algebraische Topologie. Vieweg, Wiesbaden 2005.
William S. Massey, Singular homology theory, Graduate Texts in Mathematics, vol. 70, Springer-Verlag, New York-Berlin, 1980. MR 569059, DOI 10.1007/978-1-4684-9231-6
A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canadian J. Math. 30 (1978), no. 5, 997–1015. MR 506256, DOI 10.4153/CJM-1978-085-0
Shigeru Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. MR 2004218, DOI 10.1017/CBO9781316257074
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
Markus Reineke, Localization in quiver moduli, J. Reine Angew. Math. 631 (2009), 59–83. MR 2542217, DOI 10.1515/CRELLE.2009.041
Markus Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), no. 2, 349–368. MR 1974891, DOI 10.1007/s00222-002-0273-4
Markus Reineke, Moduli of representations of quivers, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 589–637. MR 2484736, DOI 10.4171/062-1/14
Aidan Schofield, General representations of quivers, Proc. London Math. Soc. (3) 65 (1992), no. 1, 46–64. MR 1162487, DOI 10.1112/plms/s3-65.1.46
R. Sedgewick, P. Flajolet, An Introduction to the Analysis of Algorithms, Addison Wesley, 1996.
T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, Birkhäuser, Boston, Mass., 1981. MR 632835
T. Weist, Asymptotische Eulercharakteristik von Kroneckermodulräumen. Diplomarbeit, Westfälische-Wilhelms-Universität Münster, 2005.
Thorsten Weist, Tree modules of the generalised Kronecker quiver, J. Algebra 323 (2010), no. 4, 1107–1138. MR 2578596, DOI 10.1016/j.jalgebra.2009.11.033