Unitarily invariant valuations and Tutte’s sequence

Bernig, Andreas

doi:10.1090/proc/15264

Unitarily invariant valuations and Tutte’s sequence
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Proc. Amer. Math. Soc. 149 (2021), 829-841 Request permission

Abstract:

We prove Fu’s power series conjecture which relates the algebra of isometry invariant valuations on complex space forms to a formal power series from combinatorics which was introduced by Tutte. The $n$th coefficient of this series is the number of triangulations of a triangle with $3n$ internal edges; or the number of intervals in Tamari’s lattice $Y_n$.

References

Judit Abardia, Eduardo Gallego, and Gil Solanes, The Gauss-Bonnet theorem and Crofton-type formulas in complex space forms, Israel J. Math. 187 (2012), 287–315. MR 2891707, DOI 10.1007/s11856-011-0083-8
S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244–272. MR 1837364, DOI 10.1007/PL00001675
Semyon Alesker, Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Differential Geom. 63 (2003), no. 1, 63–95. MR 2015260
S. Alesker, The multiplicative structure on continuous polynomial valuations, Geom. Funct. Anal. 14 (2004), no. 1, 1–26. MR 2053598, DOI 10.1007/s00039-004-0450-2
Semyon Alesker, Theory of valuations on manifolds: a survey, Geom. Funct. Anal. 17 (2007), no. 4, 1321–1341. MR 2373020, DOI 10.1007/s00039-007-0631-x
S. Alesker, Theory of valuations on manifolds. IV. New properties of the multiplicative structure, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1910, Springer, Berlin, 2007, pp. 1–44. MR 2347038, DOI 10.1007/978-3-540-72053-9_{1}
Olivier Bernardi and Nicolas Bonichon, Intervals in Catalan lattices and realizers of triangulations, J. Combin. Theory Ser. A 116 (2009), no. 1, 55–75. MR 2469248, DOI 10.1016/j.jcta.2008.05.005
Andreas Bernig, Algebraic integral geometry, Global differential geometry, Springer Proc. Math., vol. 17, Springer, Heidelberg, 2012, pp. 107–145. MR 3289841, DOI 10.1007/978-3-642-22842-1_{5}
Andreas Bernig and Joseph H. G. Fu, Convolution of convex valuations, Geom. Dedicata 123 (2006), 153–169. MR 2299731, DOI 10.1007/s10711-006-9115-7
Andreas Bernig and Joseph H. G. Fu, Hermitian integral geometry, Ann. of Math. (2) 173 (2011), no. 2, 907–945. MR 2776365, DOI 10.4007/annals.2011.173.2.7
Andreas Bernig, Joseph H. G. Fu, and Gil Solanes, Integral geometry of complex space forms, Geom. Funct. Anal. 24 (2014), no. 2, 403–492. MR 3192033, DOI 10.1007/s00039-014-0251-1
Andreas Bernig, Joseph H. G. Fu, and Gil Solanes, Dual curvature measures in Hermitian integral geometry, Analytic aspects of convexity, Springer INdAM Ser., vol. 25, Springer, Cham, 2018, pp. 1–17. MR 3753099
F. Chapoton, Sur le nombre d’intervalles dans les treillis de Tamari, Sém. Lothar. Combin. 55 (2005/07), Art. B55f, 18 (French, with English and French summaries). MR 2264942
Robert Cori and Gilles Schaeffer, Description trees and Tutte formulas, Theoret. Comput. Sci. 292 (2003), no. 1, 165–183. Selected papers in honor of Jean Berstel. MR 1964632, DOI 10.1016/S0304-3975(01)00221-3
S. C. Coutinho, A primer of algebraic $D$-modules, London Mathematical Society Student Texts, vol. 33, Cambridge University Press, Cambridge, 1995. MR 1356713, DOI 10.1017/CBO9780511623653
Joseph H. G. Fu, Algebraic integral geometry, Integral geometry and valuations, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2014, pp. 47–112. MR 3330875
Anna-Laura Sattelberger and Bernd Sturmfels, D-Modules and Holonomic Functions, Preprint arXiv:1910.01395.
N.J.A. Sloane, The on-line encyclopedia of integer sequences, https://oeis.org/, 2006.
W. T. Tutte, A census of planar triangulations, Canadian J. Math. 14 (1962), 21–38. MR 130841, DOI 10.4153/CJM-1962-002-9
Thomas Wannerer, Integral geometry of unitary area measures, Adv. Math. 263 (2014), 1–44. MR 3239133, DOI 10.1016/j.aim.2014.06.005
Thomas Wannerer, The module of unitarily invariant area measures, J. Differential Geom. 96 (2014), no. 1, 141–182. MR 3161388
Doron Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), no. 3, 321–368. MR 1090884, DOI 10.1016/0377-0427(90)90042-X