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From Klein to Painlevé via Fourier, Laplace and Jimbo. (English) Zbl 1070.34123

Implementing the idea of B. Dubrovin and M. Mazzocco [Invent. Math. 141, 55–147 (2000; Zbl 0960.34075)], the author explicitly constructs a new algebraic solution to the classical sixth Painlevé equation. This solution is related to the Klein reflection group and is not equivalent to any other solution coming from a finite subgroup of $S{L}_{2}\left(ℂ\right)$.

The study is based on the known fact that the braid group orbit in the monodromy group of the Fuchsian system associated to equation PVI is finite if and only if the corresponding solution of PVI is finite branching. E.g., the braid group orbit in the finite monodromy group is always finite. Next basic fact is that the equation PVI admits a linear $3×3$ Fuchsian representation with rank 1 residue matrices at the finite singularities $0,1,t$, so that the relevant monodromy matrices are conjugate to complex pseudo-reflections in three dimensions [the author, Ann. Inst. Fourier 53, 1009–1022 (2003; Zbl 1081.34086)]. The author observes that the previously found algebraic solutions are related to various real reflection groups which do not exhaust the set of all possible reflection groups. The simplest of the nonreal reflection groups, the Klein reflection group, has order 336 and is used by the author as an illustrative example.

Starting with a known triple of generators of the Klein reflection group, the author interpret them as generators of the monodromy group for an above mentioned $3×3$ Fuchsian system with rank 1 residue matrices. Then he gives detailed explanations how to construct a scalar gauge transformation to reduce this system to the classical $2×2$ Fuchsian case keeping the invariants of the monodromy group under control. Having the conventional $2×2$ Fuchsian equation with the monodromy group known up to an overall conjugation, the author applies the machinery developed by Dubrovin and Mazzocco and finds the polynomial in two variables $F\left(t,y\right)=0$ defining the desired algebraic solution (its rational parameterization is found using an appropriate computer algebra package). Finally, he shows how to reconstruct a $3×3$ system from the found solution.

A significant piece of work is devoted to the proof that the Klein algebraic solution found is not equivalent to any other solution corresponding to a finite subgroup of $S{L}_{2}\left(ℂ\right)$. The author first shows that the monodromy data corresponding to the solution generate an infinite subgroup of $S{U}_{2}$, and then he proves that a solution equivalent to the Klein solution under Okamoto’s affine ${F}_{4}$ symmetry group action also corresponds to the monodromy data generating an infinite subgroup of $S{L}_{2}\left(ℂ\right)$.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34M40 Stokes phenomena and connection problems (ODE in the complex domain) 20F55 Reflection groups; Coxeter groups
##### Keywords:
Painlevé equation; reflection group; algebraic solution