64
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank 1 Lie groups

, , , , &
Pages 1000-1027 | Received 10 Jun 2022, Accepted 30 Aug 2023, Published online: 18 Sep 2023

References

  • Aparicio Monforte, A., Kauers, M. (2013). Formal Laurent series in several variables. Expo. Math. 31(4):350–367. DOI: 10.1016/j.exmath.2013.01.004.
  • Arms, J. M., Gotay, M. J., Jennings, G. (1990). Geometric and algebraic reduction for singular momentum maps. Adv. Math. 79(1):43–103. DOI: 10.1016/0001-8708(90)90058-U.
  • Bao, J., Hanany, A., He, Y.-H., Hirst, E. (2022). Some open questions in quiver gauge theory. Proyecciones (Antofagasta, On line). 41(2): 355–386. DOI: 10.22199/issn.0717-6279-5274.
  • Bedratyuk, L. (2010). The Poincare series for the algebra of covariants of a binary form. Int. J. Algebra. 4(25):1201–1207.
  • Bedratyuk, L. (2010). Weitzenböck derivations and classical invariant theory: I. Poincaré series. Serdica Math. J. 36(2):99–120.
  • Bedratyuk, L. (2011). Bivariate Poincaré series for the algebra of covariants of a binary form. ISRN Algebra. Article 312789, 11. DOI: 10.5402/2011/312789.
  • Bedratyuk, L. (2011). Poincaré series of the multigraded algebras of SL2-invariants. Ukrainian Math. J. 63(6):880–890. DOI: 10.1007/s11253-011-0550-8.
  • Bedratyuk, L., Bedratyuk, L. (2011). Multivariate Poincaré series for algebras of SL2-invariants. C. R. Acad. Bulgare Sci. 64(6):807–814.
  • Bedratyuk, L., Ilash, N. (2015). The degree of the algebra of covariants of a binary form. J. Commut. Algebra. 7(4):459–472. DOI: 10.1216/JCA-2015-7-4-459.
  • Bedratyuk, L., Xin, G. (2011). MacMahon partition analysis and the Poincaré series of the algebras of invariants of ternary and quaternary forms. Linear Multilinear Algebra. 59(7):789–799. DOI: 10.1080/03081087.2010.536763.
  • Brion, M. (1982). Invariants de plusieurs formes binaires. Bull. Soc. Math. France. 110(4):429–445. DOI: 10.24033/bsmf.1971.
  • Broer, B. (1990). On the generating functions associated to a system of binary forms. Indag. Math. (N.S.). 1(1):15–25. DOI: 10.1016/0019-3577(90)90029-M.
  • Broer, B. (1994). Classification of Cohen-Macaulay modules of covariants for systems of binary forms. Proc. Amer. Math. Soc. 120(1):37–45. DOI: 10.2307/2160164.
  • Broer, B. (1994). Hilbert series for modules of covariants. In: Haboush, W. J., Parshall, B. J., eds. Algebraic Groups and Their Generalizations: Classical Methods (University Park, PA, 1991), vol. 56 of Proc. Sympos. Pure Math. Providence, RI: American Mathematical Society, pp. 321–331. DOI: 10.1090/pspum/056.1.
  • Broer, B. (1994). A new method for calculating Hilbert series. J. Algebra. 168(1): 43–70. DOI: 10.1006/jabr.1994.1220.
  • Cape, J., Herbig, H.-C., Seaton, C. (2016). Symplectic reduction at zero angular momentum. J. Geom. Mech. 8(1):13–34. DOI: 10.3934/jgm.2016.8.13.
  • Collins, T. C., Székelyhidi, G. (2018). K-semistability for irregular Sasakian manifolds. J. Differential Geom. 109(1):81–109. DOI: 10.4310/jdg/1525399217.
  • Collins, T. C., Székelyhidi, G. (2019). Sasaki-Einstein metrics and K-stability. Geom. Topol. 23(3):1339–1413. DOI: 10.2140/gt.2019.23.1339.
  • Cowie, L. E., Herbig, H.-C., Herden, D., Seaton, C. (2019). The Hilbert series and a-invariant of circle invariants. J. Pure Appl. Algebra. 223:395–421. DOI: 10.1016/j.jpaa.2018.03.017.
  • de Carvalho Cayres Pinto, P., Herbig, H.-C., Herden, D., Seaton, C. (2020). The Hilbert series of SL2-invariants. Commun. Contemp. Math. 22(7): Article 1950017, 38. DOI: 10.1142/S0219199719500172.
  • Derksen, H., Kemper, G. (2015). Computational Invariant Theory. Encyclopaedia of Mathematical Sciences, Vol. 130, enlarged ed. Heidelberg, Germany: Springer. With two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov, Invariant Theory and Algebraic Transformation Groups, VIII. DOI: 10.1007/978-3-662-48422-7.
  • Farsi, C., Herbig, H.-C., Seaton, C. (2013). On orbifold criteria for symplectic toric quotients. SIGMA Symmetry Integrability Geom. Methods Appl. 9: Paper 032. DOI: 10.3842/SIGMA.2013.032.
  • Feng, B., Hanany, A., He, Y.-H. (2007). Counting gauge invariants: the plethystic program. J. High Energy Phys. 2007(3):090. DOI: 10.1088/1126-6708/2007/03/090.
  • Forger, M. (1998). Invariant polynomials and Molien functions. J. Math. Phys. 39(2):1107–1141. DOI: 10.1063/1.532373.
  • Gessel, I. M. (1997). Generating functions and generalized Dedekind sums. Electron. J. Combin. 4(2) (The Wilf Festschrift volume): Paper #R11. DOI: 10.37236/1326.
  • Hanany, A., Mekareeya, N., Razamat, S. S. (2013). Hilbert series for moduli spaces of two instantons. J. High Energy Phys. 1:070. DOI: 10.1007/JHEP01(2013)070.
  • Herbig, H.-C., Herden, D., Seaton, C. (2015). On compositions with x2/(1−x). Proc. Amer. Math. Soc. 143:4583–4596. DOI: 10.1090/proc/12806.
  • Herbig, H.-C., Herden, D., Seaton, C. (2020). Hilbert series associated to symplectic quotients by SU2. Internat. J. Algebra Comput. 30(7):1323–1357. DOI: 10.1142/S0218196720500435.
  • Herbig, H.-C., Herden, D., Seaton, C. (2021). The Laurent coefficients of the Hilbert series of a Gorenstein algebra. Exp. Math. 30(1):56–75. DOI: 10.1080/10586458.2018.1492473.
  • Herbig, H.-C., Herden, D., Seaton, C. (2023). Hilbert series of symplectic quotients by the 2-torus. Collect. Math. 74(2): 415–442. DOI: 10.1007/s13348-022-00357-6.
  • Herbig, H.-C., Iyengar, S. B., Pflaum, M. J. (2009). On the existence of star products on quotient spaces of linear Hamiltonian torus actions. Lett. Math. Phys. 89(2):101–113. DOI: 10.1007/s11005-009-0331-6.
  • Herbig, H.-C., Lawler, E., Seaton, C. (2020). Constructing symplectomorphisms between symplectic torus quotients. Beitr. Algebra Geom. 61(4):581–604. DOI: 10.1007/s13366-020-00486-8.
  • Herbig, H.-C., Schwarz, G. W. (2013). The Koszul complex of a moment map. J. Symplectic Geom. 11(3):497–508. DOI: 10.4310/JSG.2013.v11.n3.a9.
  • Herbig, H.-C., Schwarz, G. W., Seaton, C. (2015). When is a symplectic quotient an orbifold? Adv. Math. 280:208–224. DOI: 10.1016/j.aim.2015.04.016.
  • Herbig, H.-C., Schwarz, G. W., Seaton, C. (2020). Symplectic quotients have symplectic singularities. Compos. Math. 156(3):613–646. DOI: 10.1112/S0010437X19007784.
  • Herbig, H.-C., Seaton, C. (2014). The Hilbert series of a linear symplectic circle quotient. Exp. Math. 23(1):46–65. DOI: 10.1080/10586458.2013.863745.
  • Herbig, H.-C., Seaton, C. (2015). An impossibility theorem for linear symplectic circle quotients. Rep. Math. Phys. 75:303–331. DOI: 10.1016/S0034-4877(15)00019-1.
  • Hilbert, D. (1893). Ueber die vollen Invariantensysteme. Math. Ann. 42(3):313–373. DOI: 10.1007/BF01444162.
  • Ilash, N. (2015). The Poincaré series for the algebras of joint invariants and covariants of n linear forms. C. R. Acad. Bulgare Sci. 68(6):715–724.
  • Ilash, N. (2017). Poincaré series for the algebras of joint invariants and covariants of n quadratic forms. Carpathian Math. Publ. 9(1):57–62. DOI: 10.15330/cmp.9.1.57-62.
  • Knightly, A., Li, C. (2006). Traces of Hecke Operators, Mathematical Surveys and Monographs, Vol. 133. Providence, RI: American Mathematical Society. DOI: 10.1090/surv/133.
  • Littelmann, P., Procesi, C. (1990). On the Poincaré series of the invariants of binary forms. J. Algebra 133(2):490–499. DOI: 10.1016/0021-8693(90)90284-U.
  • Macdonald, I. G. (2008). Symmetric Functions and Hall Polynomials. Oxford Classic Texts in the Physical Sciences, 2nd ed. New York, NY: The Clarendon Press, Oxford University Press. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, reprint of the 2008 paperback edition.
  • Palm, M. R. (2012). Explicit GL(2) Trace Formulas and Uniform, Mixed Weyl Laws. PhD dissertation. Georg-August-Universität Göttingen, Göttingen, Germany.
  • Popov, V. L. (1992). Groups, Generators, Syzygies, and Orbits in Invariant Theory. Translations of Mathematical Monographs, Vol. 100. (Martsinkovsky, A., trans.) Providence, RI: American Mathematical Society. DOI: 10.1090/mmono/100.
  • Popov, V. L., Vinberg, È. B. (1994). Invariant Theory. Algebraic Geometry IV. Encyclopaedia of Mathematical Sciences, Vol. 55. Berlin: Springer-Verlag.
  • Sagan, B. E. (2001). The Symmetric Group. Graduate Texts in Mathematics, Vol. 203, 2nd ed. New York, NY: Springer-Verlag. DOI: 10.1007/978-1-4757-6804-6.
  • Schwarz, G. W. (1980). Lifting smooth homotopies of orbit spaces. Inst. Hautes Études Sci. Publ. Math. 51:37–135. DOI: 10.1007/BF02684776.
  • Stanley, R. P. (1978). Hilbert functions of graded algebras. Adv. Math. 28(1):57–83. DOI: 10.1016/0001-8708(78)90045-2.
  • Stanley, R. P. (1979). Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.). 1(3):475–511. DOI: 10.1090/S0273-0979-1979-14597-X.
  • Stanley, R. P. (1996). Combinatorics and Commutative Algebra. Progress in Mathematics, Vol. 41, 2nd ed. Boston, MA: Birkhäuser.
  • Sturmfels, B. (1993). Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation. Vienna, Austria: Springer-Verlag. DOI: 10.1007/978-3-7091-4368-1.
  • Mathematica Edition, Version 12.1. (2020). Champaign, Illinois: Wolfram Research, Inc.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.