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On Orbifold Criteria for Symplectic Toric Quotients
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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We introduce the notion of regular symplectomorphism and graded regular symplectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly symplectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicialness of a torus representation to Gaussian elimination.
Keywords: singular symplectic reduction; invariant theory; orbifold. 
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     author = {Carla Farsi and Hans-Christian Herbig and Christopher Seaton},
     title = {On {Orbifold} {Criteria} for {Symplectic} {Toric} {Quotients}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a31/}
}
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Carla Farsi; Hans-Christian Herbig; Christopher Seaton. On Orbifold Criteria for Symplectic Toric Quotients. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a31/

[1] Barthel G., Brasselet J. P., Fieseler K. H., Kaup L., “Combinatorial intersection cohomology for fans”, Tohoku Math. J., 54 (2002), 1–41, arXiv: math.AG/0002181 | DOI | MR | Zbl

[2] Bosio F., Meersseman L., “Real quadrics in $C^n$, complex manifolds and convex polytopes”, Acta Math., 197 (2006), 53–127, arXiv: math.GT/0405075 | DOI | MR | Zbl

[3] Brøndsted A., An introduction to convex polytopes, Graduate Texts in Mathematics, 90, Springer-Verlag, New York, 1983 | MR

[4] Cox D. A., Little J. B., Schenck H. K., Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011 | MR | Zbl

[5] Coxeter H. S. M., Regular complex polytopes, 2nd ed., Cambridge University Press, Cambridge, 1991 | MR | Zbl

[6] Cushman R., Sjamaar R., “On singular reduction of Hamiltonian spaces”, Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, 1991, 114–128 | MR

[7] Cushman R., Śniatycki J., “Differential structure of orbit spaces”, Canad. J. Math., 53 (2001), 715–755 | DOI | MR

[8] Du Val P., Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964 | MR

[9] Dunbar W. D., Greenwald S. J., McGowan J., Searle C., “Diameters of 3-sphere quotients”, Differential Geom. Appl., 27 (2009), 307–319, arXiv: math.DG/0702680 | DOI | MR | Zbl

[10] Eisenbud D., The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, 229, Springer-Verlag, New York, 2005 | MR | Zbl

[11] Falbel E., Paupert J., “Fundamental domains for finite subgroups in $U(2)$ and configurations of Lagrangians”, Geom. Dedicata, 109 (2004), 221–238 | DOI | MR | Zbl

[12] Fulton W., Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993 | MR | Zbl

[13] Gessel I. M., “Generating functions and generalized Dedekind sums”, Electron. J. Combin., 4:2 (1997), 11, 17 pp. | MR

[14] Gotay M. J., Bos L., “Singular angular momentum mappings”, J. Differential Geom., 24 (1986), 181–203 | MR | Zbl

[15] Haefliger A., “Groupoïdes d'holonomie et classifiants”, Astérisque, 116, 1984, 70–97 | MR | Zbl

[16] Hatcher A., Vogtmann K., “Rational homology of ${\rm Aut}(F_n)$”, Math. Res. Lett., 5 (1998), 759–780 | MR | Zbl

[17] Herbig H. C., Iyengar S. B., Pflaum M. J., “On the existence of star products on quotient spaces of linear Hamiltonian torus actions”, Lett. Math. Phys., 89 (2009), 101–113, arXiv: 0811.2152 | DOI | MR | Zbl

[18] Lerman E., Montgomery R., Sjamaar R., “Examples of singular reduction”, Symplectic Geometry, London Math. Soc. Lecture Note Ser., 192, Cambridge University Press, Cambridge, 1993, 127–155 | MR | Zbl

[19] Marsden J., Weinstein A., “Reduction of symplectic manifolds with symmetry”, Rep. Math. Phys., 5 (1974), 121–130 | DOI | MR | Zbl

[20] Mather J. N., “Differentiable invariants”, Topology, 16 (1977), 145–155 | DOI | MR | Zbl

[21] Meyer K. R., “Symmetries and integrals in mechanics”, Dynamical Systems, Proc. Sympos. (Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 259–272 | MR

[22] Molien T., “Über die Invarianten der linearen Substitutionsgruppen”, Sitzungsber. der Königl. Preuss. Akad. d. Wiss., v. 2, 1897, 1152–1156 | Zbl

[23] Multarzyński P., Żekanowski Z., “On general Hamiltonian dynamical systems in differential spaces”, Demonstratio Math., 24 (1991), 539–555 | MR | Zbl

[24] Navarro González J. A., Sancho de Salas J. B., $C^\infty$-differentiable spaces, Lecture Notes in Mathematics, 1824, Springer-Verlag, Berlin, 2003 | DOI | MR | Zbl

[25] Pflaum M. J., Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics, 1768, Springer-Verlag, Berlin, 2001 | MR | Zbl

[26] Schwarz G. W., “Smooth functions invariant under the action of a compact Lie group”, Topology, 14 (1975), 63–68 | DOI | MR | Zbl

[27] Schwarz G. W., “The topology of algebraic quotients”, Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988), Progr. Math., 80, Birkhäuser Boston, Boston, MA, 1989, 135–151 | MR

[28] Sikorski R., Wstȩp do geometrii różniczkowej, Biblioteka Matematyczna, 42, Państwowe Wydawnictwo Naukowe, Warsaw, 1972 | MR | Zbl

[29] Sjamaar R., Lerman E., “Stratified symplectic spaces and reduction”, Ann. of Math. (2), 134 (1991), 375–422 | DOI | MR | Zbl

[30] Spanier E. H., Algebraic topology, Springer-Verlag, New York, 1981 | MR

[31] Sturmfels B., Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993 | MR