Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Abstract

Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2×2 system to obtain RS 2 4 -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations.An important observation is that the same rational covering R(x) can be used in several RS-pullback transformations. Application of different RS-transformations to respectively different matrix differential equations gives different algebraic Painlevé VI solutions. For example, [26] demonstrates usage of the same degree 10 covering to pullback three different hypergeometric equations (with the local monodromy differences 1/2, 1/3, k/7 with k = 1, 2 or 3) and obtain three algebraic Painlevé VI solutions unrelated by fractional-linear or Okamoto transformations.In our other concrete examples, we start with matrix hypergeometric equations with the icosahedral monodromy group. The pullbacked Fuchsian equations have the icosahedral monodromy group as well. Corresponding Painlevé VI solutions are called icosahedral [3]. There are 52 types of icosahedral Painlevé VI solutions in total [3], up to branching representation of the icosahedral monodromy group (or equivalently, up to Okamoto transformations). We recompute icosahedral solutions of Boalch types 26, 27, 31, 32. Second order ordinary Fuchsian equations (or 2 × 2 first order matrix Fuchsian equations) with a finite monodromy group are always pullbacks of a standard hypergeometric equation with the same monodromy group, as asserted by celebrated Klein's theorem [19]. In particular, existence of pull-back transformations for the four icosahedral examples follows from Klein's theorem. R. Fuchs [12] soon considered extension of Klein's theorem to algebraic solutions of Painlevé equations. Recently, Ohyama and Okumura [22] showed that algebraic solutions of Painlevé equations from the first to the fifth do arise from pullback transformations of confluent hypergeometric equations, affirming the formulation of R. Fuchs. The pullback method for computing algebraic Painlevé VI solutions was previously suggested in [18], [2], [17], [7]. The alternative representation-theoretic approach is due to Dubrovin-Mazzocco [9]. Recently, it was used [20] to complete classification of algebraic Painlevé VI solutions.The article is organized as follows. Section 2 presents two almost Belyi coverings we employ. The coverings have degree 8 and 12; they were previously used in [18]. Section 3 demonstrates two examples of full RS-pullback transformations, both with respect to the degree 8 covering. In Section 4 we formulate basic algebraic facts useful in computations of RS-pullback transformations....

Flat Structure on the Space of Isomonodromic Deformations

“…Kitaev [38,39], A.V. Kitaev and R. Vidūnas [62,63], K. Iwasaki [28]. In his review article [13], Boalch classified all the known algebraic solutions to the Painlevé VI based on types of the monodromy groups of the associated 2 × 2 or 3 × 3 linear differential equations.…”

Flat Structure on the Space of Isomonodromic Deformations

“…In this section we show some examples of potential vector fields in three variables which correspond to algebraic solutions to the Painlevé VI equation. Algebraic solutions to the Painlevé VI equation were studied and constructed by many authors including N. J. Hitchin [24,25], B. Dubrovin [17], B. Dubrovin -M. Mazzocco [19], P. Boalch [6,7,9,10,11], A. V. Kitaev [37,38], A. V. Kitaev -R. Vidūnas [39,61], K. Iwasaki [27]. The classification of algebraic solutions to the Painlevé VI equation was achieved by Lisovyy and Tykhyy [42].…”

Differential relations for almost Belyi maps

“…The pull-backs of E(1/2, 1/3, 2/7) have the same 4 + 1 singularities, plus a new apparent singularity at x = ∞. Schlessinger transformations neutralizing this singularity give algebraic solutions of P V I (2/7, 2/7, 4/7, 2/7), P V I (2/7, 2/7, 1/3, 2/7), as demonstrated in [32]. Similarly, the pull-backs of E(1/2, 1/3, 3/7) have the same 4 + 1 singularities, plus a new singularity at x = ∞ with the monodromy difference 3.…”

“…Similarly, the pull-backs of E(1/2, 1/3, 3/7) have the same 4 + 1 singularities, plus a new singularity at x = ∞ with the monodromy difference 3. Neutralizing Schlessinger transformations lead to algebraic solutions of P V I (3/7, 3/7, 6/7, 4/7) and P V I (3/7, 3/7, 1/3, 4/7), as shown in [32].…”