Illinois Journal of Mathematics

Invariant CR mappings between hyperquadrics

Dusty Grundmeier, Kemen Linsuain, and Brendan Whitaker

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We analyze a canonical construction of group-invariant CR Mappings between hyperquadrics due to D’Angelo. Given source hyperquadric of $Q(1,1)$, we determine the signature of the target hyperquadric for all finite subgroups of $SU(1,1)$. We also extend combinatorial results proven by Loehr, Warrington, and Wilf on determinants of sparse circulant determinants. We apply these results to study CR mappings invariant under finite subgroups of $U(1,1)$.

Article information

Illinois J. Math., Volume 62, Number 1-4 (2018), 321-340.

Received: 15 March 2018
Revised: 31 July 2018
First available in Project Euclid: 13 March 2019

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Zentralblatt MATH identifier

Primary: 32V20: Analysis on CR manifolds 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 32H35: Proper mappings, finiteness theorems


Grundmeier, Dusty; Linsuain, Kemen; Whitaker, Brendan. Invariant CR mappings between hyperquadrics. Illinois J. Math. 62 (2018), no. 1-4, 321--340. doi:10.1215/ijm/1552442665.

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  • M. Salah Baouendi, P. Ebenfelt and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999.
  • J. P. D'Angelo, Invariant holomorphic mappings, J. Geom. Anal. 6 (1996), no. 2, 163–179. \bid,doi=10.1007/BF02921598
  • J. P. D'Angelo, The combinatorics of certain group-invariant mappings, Complex Var. Elliptic Equ. 58 (2013), no. 5, 621–634. \bid,doi=10.1080/17476933.2011.599117
  • J. P. D'Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993.
  • J. P. D'Angelo and D. A. Lichtblau, Spherical space forms, CR mappings, and proper maps between balls, J. Geom. Anal. 2 (1992), no. 5, 391–415. \bid,doi=10.1007/BF02921298
  • J. P. D'Angelo and M. Xiao, Symmetries in CR complexity theory, Adv. Math. 313 (2017), 590–627. \bid,doi=10.1016/j.aim.2017.04.014
  • J. P. D'Angelo and M. Xiao, Symmetries and regularity for holomorphic maps between balls, to appear in Math. Res. Lett.
  • F. Forstnerič, Proper holomorphic mappings: A survey, Several complex variables (Stockholm, 1987/1988), Math. Notes, vol. 38, Princeton University Press, Princeton, NJ, 1993, pp. 297–363.
  • F. Forstnerič, Proper holomorphic maps from balls, Duke Math. J. 53 (1986), no. 2, 427–441. \bid,doi=10.1215/S0012-7094-86-05326-3
  • F. Forstnerič, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), no. 1, 31–61. \bid,doi=10.1007/BF01394144
  • D. Grundmeier, Signature pairs for group-invariant Hermitian polynomials, Internat. J. Math. 22 (2011), no. 3, 311–343. \bid,doi=10.1142/S0129167X11006775
  • D. Grundmeier, Group-invariant CR mappings, ProQuest LLC, Ann Arbor, MI, 2011. Ph.D. thesis, University of Illinois at Urbana-Champaign.
  • D. Grundmeier, J. Lebl and L. Vivas, Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics, Math. Ann. 358 (2014), no. 3–4, 1059–1089. \bid,doi=10.1007/s00208-013-0989-z
  • N. A. Loehr, G. S. Warrington and H. S. Wilf, The combinatorics of a three-line circulant determinant, Israel J. Math. 143 (2004), 141–156. \bid,doi=10.1007/BF02803496
  • B. Simon, Orthogonal polynomials on the unit circle. Part 2: Spectral theory, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005.
  • R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inf. Theory 2 (2006), no. 3, 155–239.