Abstract
We show that the orthogonal separation coordinates on the sphere S n are naturally parametrised by the real version of the Deligne–Mumford–Knudsen moduli space \({\bar{M}_{0,n+2}({\mathbb{R}})}\) of stable curves of genus zero with n + 2 marked points. We use the combinatorics of Stasheff polytopes tessellating \({\bar{M}_{0,n+2}({\mathbb{R}})}\) to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates, as well as of Stäckel systems from the mosaic operad structure on \({\bar{M}_{0,n+2}({\mathbb{R}})}\).
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References
Aguirre L., Felder G., Veselov A.P.: Gaudin subalgebras and stable rational curves. Composit. Math. 147(5), 1463–1478 (2011)
Benenti, S.: Orthogonal separable dynamical systems. Differential geometry and its applications (Opava, 1992), pp. 163–184, Math. Publ., 1, Silesian Univ. Opava, Opava (1993)
Bolsinov A.V., Matveev V.S.: Geometrical interpretation of Benenti systems. J. Geom. Phys. 44(4), 489–506 (2003)
Boyer C.P., Kalnins E.G., Winternitz P.: Separation of variables for the Hamilton–Jacobi equation on complex projective spaces. SIAM J. Math. Anal. 16(1), 93–109 (1985)
Crampin M.: Conformal Killing tensors with vanishing torsion and the separation of variables in the Hamilton–Jacobi equation. Differ. Geom. Appl. 18(1), 87–102 (2003)
Davis M., Januszkiewicz T., Scott R.: Nonpositive curvature of blowups. Selecta Math. 4(4), 491–547 (1998)
Devadoss S.L.: Tesselations of moduli spaces and the mosaic operad. Contemp. Math. 239, 91–114 (1999)
Devadoss, S.L., Read, R.C.: Cellular structures determined by polygons and trees. Ann. Combin. 5, 71–98 (2011)
Devadoss S.: A realization of graph associahedra. Discrete Math. 309(1), 271–276 (2009)
Eisenhart L.P.: Separable systems of Stäckel. Ann. Math. 35, 284–305 (1934)
Etingof P., Henriques A., Kamnitzer J., Rains E.: The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. Ann. Math. 171, 731–777 (2010)
Jacobi, C.G.J.: Vorlesungen über Dynamik. In: C.G.J. Jacobi’s Gesammelte Werke (in German). G. Reimer, Berlin (1881)
Kalnins E.G., Miller W. Jr.: Separation of variables on n-dimensional Riemannian manifolds. I. The n-sphere S n and Euclidean n-space R n . J. Math. Phys. 27, 1721–1736 (1986)
Kalnins E.G., Miller W., Reid G.J.: Separation of variables for Riemannian spaces of constant curvature. I. Orthogonal separable coordinates for S nC and E nC . Proc. R. Soc. Lond. Ser. A 394(1806), 183–206(1984)
Kapranov M.M.: The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation. J. Pure Appl. Algebra 85(2), 119–142 (1993)
Knudsen F.F.: The projectivity of the moduli space of stable curves. II. The stacks M g,n . Math. Scand. 52,161–199 (1983)
Lamé G.: Sur les surfaces isothermes dans les corps homogènes en équilibre de température. J. Math. Pures Appl. 2, 147–188 (1837)
Lee C.: The associahedron and triangulations of the n-gon. Eur. J. Combin. 10(6), 551–560 (1989)
Levi-Civita T.: Sulla integrazione della equazione de Hamilton Jacobi per separazione de variabili. Math. Ann. 59, 383–397 (1904)
Matveev, V.S., Mounoud, P.: Gallot–Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications. Ann. Global Anal. Geom. 38(3), 259–271 (2010)
McLenaghan R.G., Milson R., Smirnov R.G.: Killing tensors as irreducible representations of the general linear group. C. R. Acad. Sci. Paris 339, 621–624 (2004)
Müller-Hoissen, F., Pallo, J.M., Stasheff, J. (eds.): Associahedra, Tamari Lattices and Related Structures. Tamari Memorial Festschrift. Progress in Mathematics, vol. 299. Birkhäuser, Basel (2012)
Neumann C.: De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur. J. Reine Angew. Math. 56, 46–63 (1859)
Olevskiĭ M.N.: Triorthogonal systems in spaces of constant curvature in which the equation Δ2 u + λ u = 0 admits separation of variables. Math. Sbornik 27, 379–426 (1950)
Schöbel K.: Algebraic integrability conditions for Killing tensors on constant sectional curvature manifolds. J. Geom. Phys. 62(5), 1013–1037 (2012)
Schöbel K.: The variety of integrable Killing tensors on the 3-sphere. SIGMA 10, 080 (2014)
Stäckel, P.: Die Integration der Hamilton–Jacobischen Differentialgleichung mittelst Separation der Variabeln. Habilitationsschrift Universität Halle (1891)
Stasheff J.: Homotopy associativity of H-spaces. I, II. Trans. Am. Math. Soc. 108, 275–312 (1963)
Vershik, A.M., Okounkov, A.Yu.: A new approach to representation theory of symmetric groups. II. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), 307, 57–98 (2004) [English transl.: J. Math. Sci. 131, 5471–5494 (2005)]
Vilenkin, N.Ya.: Special functions and group representation theory. Nauka Publ. Comp., Moscow (1965) (in Russian)
Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie Groups and Special Functions. Class I Representations, Special Functions, and Integral Transforms, vol. 2. Kluwer Academic Publishers, Dordrecht (1993)
OEIS Foundation Inc., Sequence A001190, The On-Line Encyclopedia of Integer Sequences (2011). http://oeis.org/A001190
OEIS Foundation Inc., Sequence A032132, The On-Line Encyclopedia of Integer Sequences (2011). http://oeis.org/A032132
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Schöbel, K., Veselov, A.P. Separation Coordinates, Moduli Spaces and Stasheff Polytopes. Commun. Math. Phys. 337, 1255–1274 (2015). https://doi.org/10.1007/s00220-015-2332-x
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DOI: https://doi.org/10.1007/s00220-015-2332-x