Abstract
Let (N,G), where \(N\unlhd G\leq \text {SL}_{n}(\mathbb {C})\), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V ) = ⊕k≥ 0Sk(V ) by giving explicit formulas of the Poincaré series for the induced modules and the restriction modules. In particular, this provides a uniform formula of the Poincaré series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincaré series in terms of Tchebychev polynomials in the sense that one needs only the dimensions of the subgroups and their group-types to completely determine the Poincaré series.
Similar content being viewed by others
References
Benkart, G.: Poincaré series for tensor invariants and the McKay correspondence. Adv. Math. 290, 236–259 (2016)
Butin, F., Perets, G.S.: Branching law for finite subgroups of \(\mathrm {{{SL}}}_{3}\mathbb {C}\) and McKay correspondence. J. Group Theory 17, 191–251 (2014)
Damianou, P.A.: A beautiful sine formula. Amer. Math. Monthly 121, 120–135 (2014)
Ebeling, W.: Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity. Manuscripta Math. 107, 271–282 (2002)
Ebeling, W.: A McKay correspondence for the Poincaré series of some finite subgroups of \(\text {{{SL}}}_{3}(\mathbb {C})\). J. Singul. 18, 397–408 (2018)
Fulton, W., Harris, J.: Representation Theory, A First Course, p 129. Springer, New York (1991)
Gonzalez-Sprinberg, G., Verdier, J.-L.: Construction géométrique de la correspondence de McKay. Ann. Sci. École Norm. Sup. (4) 16, 409–449 (1983)
Hanany, A., He, Y.-H.: A monograph on the classification of the discrete subgroups of SU(4). J. High Energy Phys. 2, 12 (2001)
Hu, X., Jing, N., Cai, W.: Generalized McKay quivers of rank three. Acta Math. Sin. (Engl. Ser.) 29, 1351–1368 (2013)
Jing, N., Wang, D., Zhang, H.: Poincaré series, exponents of affine Lie algebras, and McKay-Slodowy correspondence. J. Algebra 546, 135–162 (2020)
Knörrer, H.: Group representations and the resolution of rational double points. Finite Groups-Coming of Age (Montreal, Que., 1982), 175–222, Contemp Math, vol. 45. Amer. Math. Soc., Providence (1985)
Korányi, A.: Spectral properties of the Cartan matrices. Acta Sci. Math. (Szeged) 57, 587–592 (1993)
Kostant, B.: On finite subgroups of SU(2), simple Lie algebras, and the McKay correspondence. Proc. Nat. Acad. Sci. U.S.A. 81, 5275–5277 (1984)
Kostant, B.: The McKay correspondence, the Coxeter element and representation theory. The mathematical heritage of Élie Cartan (Lyon 1984). Astérisque Numéro Hors Série, pp. 209–255 (1985)
Kostant, B.: The Coxeter Element and the Branching Law for the Finite Subgroups of SU(2). The Coxeter Legacy, pp 63–70. Amer. Math. Soc., Providence (2006)
McKay, J.: Graphs, Singularities, and Finite Groups. The Santa Cruz Conference on Finite Groups, (Univ. California, Santa Cruz, Calif., 1979), pp. 183–186, Proc. Sympos Pure Math., vol. 37. Amer. Math. Soc., Providence (1980)
Rivlin, T.J.: Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. Pure and Applied Mathematics (New York), 2nd edn. Wiley, New York (1990)
Slodowy, P.: Simple Singularities and Simple Algebraic Groups. Lecture Notes in Math, vol. 815. Springer, Berlin (1980)
Steinberg, R.: Finite subgroups of SU2, Dynkin diagrams and affine Coxeter elements. Pacific J. Math. 118, 587–598 (1985)
Stekolshchik, R.: Notes on Coxeter Transformations and the McKay Correspondence. Springer Monographs in Mathematics. Springer, Berlin (2008)
Suter, R.: Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations. Manuscripta Math. 122, 1–21 (2007)
Yau, S.S.-T., Yu, Y.: Gorenstein quotient singularities in dimension three. Mem. Amer. Math. Soc. 505, 105 (1993)
Acknowledgments
N. Jing would like to thank the partial support of Simons Foundation grant no. 523868 and NSFC grant no. 11531004. H. Zhang would like to thank the support of NSFC grant no 11871325.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Alistair Savage
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Jing, N., Wang, D. & Zhang, H. Poincaré Series of Relative Symmetric Invariants for \(\text {SL}_{n}(\mathbb {C})\). Algebr Represent Theor 24, 601–623 (2021). https://doi.org/10.1007/s10468-020-09962-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-09962-0