Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T00:15:34.259Z Has data issue: false hasContentIssue false
  • English
  • Français

Bending Flows for Sums of Rank One Matrices

Published online by Cambridge University Press:  20 November 2018

Hermann Flaschka  and
John Millson
Hermann Flaschka
Affiliation:
Department of Mathematics, The University of Arizona, Tucson, AZ 85721 e-mail: flaschka@math.arizona.edu
John Millson
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742 e-mail: jjm@math.umd.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study certain symplectic quotients of $n$-fold products of complex projective $m$-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group.

Keywords

53D20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 1 , 01 February 2005 , pp. 114 - 158
Copyright
Copyright © Canadian Mathematical Society 2005

References

[AHH] Adams, M. R., Harnad, J. and Hurtubise, J., Dual moment maps into loop algebras. Lett. Math. Phys. 20 (1990), 299–308.Google Scholar
[BeSch] Berceanu, S. and Schlichenmaier, M., Coherent states, embeddings, polar divisors and Cauchy formulas. J. Geom. Phys. 34 (2000), 336–368.Google Scholar
[BrJa] Bröcker, Th. and Jänich, K., Introduction to Differential Topology. Cambridge University Press, Cambridge, 1982.Google Scholar
[DO] Dolgachev, I. and Ortland, D., Point Sets in Projective Space and Theta Functions. Astérisque 165, (1988).Google Scholar
[FH] Fulton, W. and Harris, J., Representation Theory, A First Course. Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991.Google Scholar
[GGMS] Gel’fand, I. M., Goresky, R. M., R. D. MacPherson and Serganova, V. V., Combinatorial geometries, convex polyhedra and Schubert cells. Adv. Math. 63 (1987), 301–316.Google Scholar
[Go] Goldman, W. M., Complex Hyperbolic Geometry. Oxford Mathematical Monographs, The Clarendon Press, New York, 1999.Google Scholar
[GZ86] Gel’fand, I. and Zelevinsky, A., Multiplicities and proper bases for gln. In: Group Theoretical Methods in Physics, Proceedings of the Third Yurmala Seminar, (Markov, M. A., Mank’o, V. I., Dodonov, V. V., eds.) VNU Science Press, Utrecht, The Netherlands, 1986, pp. 147–159.Google Scholar
[Goldin] Goldin, R. F., The cohomology ring of weight varieties and polygon spaces. Adv. Math. 160 (2001), 175–204.Google Scholar
[GS83] Guillemin, V. and Sternberg, S., The Gel’fand-Cetlin system and quantization of the complex flag manifolds. J. Funct. Anal. 52 (1983), 106–128.Google Scholar
[HM] Hangan, Th. and Masala, G., A geometric interpretation of the shape invariant for geodesic triangles in complex projective spaces. Geom. Dedicata 49 (1994), 129–134.Google Scholar
[HK97] Hausmann, J.-C. and Knutson, A., Polygons spaces and Grassmannians. Enseign. Math. 43 (1997), 173–198.Google Scholar
[HL94] Heinzner, P. and Loose, F., Reduction of complex Hamiltonian G-spaces. Geom. Funct. Anal. 4 (1994), 288–297.Google Scholar
[Hi] Hitchin, N. J., Integrable systems. Twistors, loop groups and Riemann surfaces. Oxford Graduate Texts in Mathematics 4, The Clarendon Press, Oxford, 1999.Google Scholar
[Hu] Hu, Y., The geometry and topology of quotient varieties of torus actions. DukeMath. J. 68 (1992), 151–184.Google Scholar
[KM96] Kapovich, M. and Millson, J., The symplectic geometry of polygons in Euclidean space. J. Diff. Geom. 44 (1996), 479–513.Google Scholar
[KM01] Kapovich, M. and Millson, J., Quantization of bending deformations of polygons i. E3, hypergeometric integrals and the Gassner representation. Canad. Math. Bull. 44 (2001), 36–60.Google Scholar
[Kato] Kato, T., Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, New York, 1966.Google Scholar
[KKS78] Kazhdan, D., Kostant, B. and Sternberg, S., Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure Appl. Math. 31 (1978), 481–508.Google Scholar
[Ki] Kirwan, F. C., Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes 31, Princeton University Press, Princeton, NJ. 1984.Google Scholar
[Kly92] Klyachko, A., Spatial polygons and stable configurations of points on the projective line. In: Algebraic Geometry and its Applications, Proceedings of the 8th Algebraic Geometry Conference, Yaroslavl’ 1992, (Tikhomirov, A. and Tyurin, A., eds.), Vieweg, Braunschweig, 1994, pp. 67–84.Google Scholar
[Kly98] Klyachko, A., Stable bundles, representation theory and Hermitean operators. Selecta Math. 4 (1998), 419–445.Google Scholar
[MR] Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry. 2nd Edition, Springer-Verlag, New York, 1999.Google Scholar
[MFK] Mumford, D., Fogarty, J., and Kirwan, F., Geometric Invariant Theory, 3rd ed. Ergebnisse der Mathematik und ihre Grenzgebiete 34, Springer Verlag, Berlin, 1991.Google Scholar
[Ne84] Ness, L., A stratification of the null cone via the moment map. Amer. J. Math. 106 (1984), 1281–1329.Google Scholar
[Sj95] Sjamaar, R. Holomorphic slices, symplectic reduction and multiplicities of representations. Annals of Math. 141 (1995), 87–129.Google Scholar
[Ze] Želobenko, D. P., Compact Lie Groups and Their Representations, Translations of Mathematical Monographs 40, American Mathematical Society, Providence, R.I., 1973.Google Scholar