Painlevé equations and the middle convolution

Abstract

We use the middle convolution to obtain some old and new algebraic solutions of the Painlevé VI equations.

Katz's Middle Convolution Algorithm

“…And, Crawley-Boevey views the construction, again in algebraic terms, as something about root systems. Boalch considered a particular example of middle convolution in a non-rigid case [15], and the link with Katz's construction was made in [47]. In [45] following [86,Chap.…”

“…Boalch considered a particular example of middle convolution in a non-rigid case [15], and the link with Katz's construction was made in [47]. In [45] following [86,Chap. 5.1] it is shown that the explicit matrix definition of M C has a geometric or cohomological interpretation as a higher direct image-this is the point of view we adopt here.…”

“…5.1] it is shown that the explicit matrix definition of M C has a geometric or cohomological interpretation as a higher direct image-this is the point of view we adopt here. The braid-style calculations of group cohomology necessary to get the local form of monodromy out of this geometric definition were done in [45] but using the Pochammer basis for the group cohomology classes, rather than a standard basis as we shall use here. In spite of the numerous references on this subject, we go through the details, where possible keeping simplifying assumptions for our expository purpose.…”

Constructive Solutions to the Riemann–Hilbert Problem and Middle Convolution

“…Dettweiler and Reiter's algebraic analogue [7,8] of Katz' middle convolution is a certain transformation of Fuchsian systems which preserves an index of rigidity. Middle convolution is related to the Euler transformation of the Fuchsian system.…”

“…We shall first review definitions of the Fuchsian system and its monodromy representation, formulate the Riemann-Hilbert problem and present some of the known methods of its constructive solutions (other methods and further references can be found in, for example, a survey paper [14]). Then we shall briefly describe an algorithm of middle convolution following [7,8] and, finally, we shall present our new approach (the general scheme) to constructive solutions of the Riemann-Hilbert problem via middle convolution with two illustrative examples.…”