caa a22 4500 475738446 CHVBK 20180406123453.0 cr unu---uuuuu 170329e20000701xx s 000 0 eng 10.1007/PL00005790 doi (NATIONALLICENCE)springer-10.1007/PL00005790 Monodromy of certain Painlevé-VI transcendents and reflection groups [Elektronische Daten] [B. Dubrovin, M. Mazzocco] Abstract. : We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ=1/2 and 2α=(2μ-1)2 with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Springer-Verlag Berlin Heidelberg, 2000 Dubrovin B. International School of Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy¶(e-mail: dubrovin@sissa.it, mazzocco@sissa.it), Italy aut Mazzocco M. International School of Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy¶(e-mail: dubrovin@sissa.it, mazzocco@sissa.it), Italy aut Inventiones mathematicae Springer Berlin Heidelberg 141/1(2000-07-01), 55-147 0020-9910 141:1<55 2000 141 222 https://doi.org/10.1007/PL00005790 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/PL00005790 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Dubrovin B. International School of Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy¶(e-mail: dubrovin@sissa.it, mazzocco@sissa.it), Italy aut NATIONALLICENCE 700 1- Mazzocco M. International School of Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy¶(e-mail: dubrovin@sissa.it, mazzocco@sissa.it), Italy aut NATIONALLICENCE 773 0- Inventiones mathematicae Springer Berlin Heidelberg 141/1(2000-07-01), 55-147 0020-9910 141:1<55 2000 141 222 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer