Algebraic & Geometric Topology

Hofer–Zehnder capacity and Bruhat graph

Alexander Caviedes Castro

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Abstract

We find bounds for the Hofer–Zehnder capacity of spherically monotone coadjoint orbits of compact Lie groups with respect to the Kostant–Kirillov–Souriau symplectic form in terms of the combinatorics of their Bruhat graphs. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal in that case to the diameter of a weighted Cayley graph.

Article information

Source
Algebr. Geom. Topol., Volume 20, Number 2 (2020), 565-600.

Dates
Received: 2 February 2017
Revised: 4 December 2018
Accepted: 19 April 2019
First available in Project Euclid: 30 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.agt/1588212070

Digital Object Identifier
doi:10.2140/agt.2020.20.565

Mathematical Reviews number (MathSciNet)
MR4092307

Zentralblatt MATH identifier
07195372

Subjects
Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 57R17: Symplectic and contact topology
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]

Keywords
symplectic capacities coadjoint orbits Bruhat graph Hofer–Zehnder capacity

Citation

Caviedes Castro, Alexander. Hofer–Zehnder capacity and Bruhat graph. Algebr. Geom. Topol. 20 (2020), no. 2, 565--600. doi:10.2140/agt.2020.20.565. https://projecteuclid.org/euclid.agt/1588212070


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References

  • K Behrend, Gromov–Witten invariants in algebraic geometry, Invent. Math. 127 (1997) 601–617
  • I N Bernstein, I M Gelfand, S I Gelfand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973) 3–26 In Russian; translated in Russian Math. Surveys 28 (1973) 1–26 and reprinted in “Representation theory”, London Math. Soc. Lecture Note Ser. 69, Cambridge Univ. Press (1982) 115–140
  • A Caviedes Castro, Upper bound for the Gromov width of flag manifolds, J. Symplectic Geom. 13 (2015) 745–764
  • I Ekeland, H Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z. 200 (1989) 355–378
  • I Ekeland, H Hofer, Symplectic topology and Hamiltonian dynamics, II, Math. Z. 203 (1990) 553–567
  • F Farnoud, O Milenkovic, Sorting of permutations by cost-constrained transpositions, IEEE Trans. Inform. Theory 58 (2012) 3–23
  • A Floer, H Hofer, C Viterbo, The Weinstein conjecture in $P\times {\mathbb C}^l$, Math. Z. 203 (1990) 469–482
  • W Fulton, R Pandharipande, Notes on stable maps and quantum cohomology, from “Algebraic geometry” (J Kollár, R Lazarsfeld, D R Morrison, editors), Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI (1997) 45–96
  • W Fulton, C Woodward, On the quantum product of Schubert classes, J. Algebraic Geom. 13 (2004) 641–661
  • M Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
  • V Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$–spaces, Progr. Math. 122, Birkhäuser, Boston (1994)
  • V Guillemin, E Lerman, S Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge Univ. Press (1996)
  • H Hofer, C Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992) 583–622
  • H Hofer, E Zehnder, A new capacity for symplectic manifolds, from “Analysis, et cetera” (P H Rabinowitz, E Zehnder, editors), Academic, Boston (1990) 405–427
  • T Hwang, D Y Suh, Symplectic capacities from Hamiltonian circle actions, J. Symplectic Geom. 15 (2017) 785–802
  • A A Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, Amer. Math. Soc., Providence, RI (2004)
  • G Liu, G Tian, Weinstein conjecture and GW–invariants, Commun. Contemp. Math. 2 (2000) 405–459
  • A Loi, R Mossa, F Zuddas, Symplectic capacities of Hermitian symmetric spaces of compact and noncompact type, J. Symplectic Geom. 13 (2015) 1049–1073
  • G Lu, Gromov–Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006) 1–63
  • G Lu, Symplectic capacities of toric manifolds and related results, Nagoya Math. J. 181 (2006) 149–184
  • D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, 2nd edition, American Mathematical Society Colloquium Publications 52, Amer. Math. Soc., Providence, RI (2012)
  • D McDuff, S Tolman, Topological properties of Hamiltonian circle actions, Int. Math. Res. Pap. (2006) art. id. 72826
  • M Pabiniak, Gromov width of non-regular coadjoint orbits of $U(n)$, $\mathrm{SO}(2n)$ and $\mathrm{SO}(2n+1)$, Math. Res. Lett. 21 (2014) 187–205
  • A Postnikov, Quantum Bruhat graph and Schubert polynomials, Proc. Amer. Math. Soc. 133 (2005) 699–709
  • Y Rinott, Multivariate majorization and rearrangement inequalities with some applications to probability and statistics, Israel J. Math. 15 (1973) 60–77
  • B Siebert, Algebraic and symplectic Gromov–Witten invariants coincide, Ann. Inst. Fourier (Grenoble) 49 (1999) 1743–1795
  • M Usher, Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms, Geom. Topol. 15 (2011) 1313–1417
  • A Vince, A rearrangement inequality and the permutahedron, Amer. Math. Monthly 97 (1990) 319–323