Middle convolutions of KZ-type equations and single-elimination tournaments
Toshio Oshima
TL;DR
This work develops a microlocal framework for Knizhnik–Zamolodchikov–type equations by extending the generalized Riemann scheme to capture local residue-spectral data under singularity resolution and middle convolution. It formalizes the spectra of KZ-type systems via maximal commuting families of residue matrices and proves explicit transformation rules for these spectra under middle convolution, expressed through combinatorial data encoded as single-elimination tournaments. The approach unifies desingularization, spectral decompositions, and mc transformations, illustrating the theory with $n=4$ examples and connecting to Appell–type hypergeometric functions. The results provide a principled bridge between spectral geometry of KZ-type equations and combinatorial tournament structures, with implications for rigidity, accessory parameters, and multi-variable hypergeometric functions.
Abstract
We introduce an extension of the generalized Riemann scheme for Fuchsian ordinary differential equations in the case of KZ-type equations. This extension describes the local structure of equations obtained by resolving the singularities of KZ-type equations. We present the transformation of this extension under middle convolutions. As a consequence, we derive the corresponding transformation of the eigenvalues and multiplicities of the residue matrices of KZ-type equations under middle convolutions. We interpret the result in terms of the combinatorics of single-elimination tournaments.