The fifty-two icosahedral solutions to Painlevé VI

Abstract

The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F 4 Weyl group action and many properties of the solutions will be given.There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known P VI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev.Of the remaining 28 solutions we will find 20 explicitly using the method of [5] (via Jimbo's asymptotic formula). Amongst those constructed there is one solution that is "generic" in that its parameters lie on none of the affine F 4 hyperplanes, one that is equivalent to the Dubrovin-Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.

Conformal field theory of Painlevé VI

“…This approach, characteristic for classical integrable systems in general, turns out to be quite successful. In particular, it represents the key element of the solution of PVI connection problem [18], as well as of the construction [5,11] and classification [25] of algebraic solutions. Logarithmic derivative of Painlevé VI tau function…”

Algebraic solutions of the sixth Painlevé equation

“…C, p. 4. Solution 32, 18 branches, θ = (4/7, 4/7, 4/7, 1/3): [3], first obtained explicitly in [5], p. 9. (The formula (6) for v in [5], p. 8 is incorrect and should be replaced with v 2 = −2(j + 1)(5j 2 − 2j + 13).…”

“…The center Z of B 3 is generated by (β z β x ) 3 and therefore it acts on M trivially. This leads to an action of the modular group Γ ∼ = B 3 /Z on M, with…”

An ergodic study of Painlevé VI

“…The logarithm of this rate will give the entropy of the Poincaré return map γ * . Recently several authors [4,5,9,13,14,25,26] have been interested in finding algebraic solutions, which must have only finitely many branches under the analytic continuations along all loops in Z. On the other hand, in this article we will be concerned with those solutions which are finitely many-valued along a fixed single loop.…”