A unified description of the asymmetric q-P_{v} and d-P_{iv} equations and their Schlesinger transformations

Abstract

We present a geometric description, based on the affine Weyl group E_{6}^{(1)}, of two discrete analogues of the Painlev\'e VI equation, known as the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to describe in a unified way the evolution of the mapping along the independent variable and along the various parameters (the latter evolution being the one induced by the Schlesinger transformations). It turns out that both discrete Painlev\'e equations exhibit the property of self-duality: the same equation governs the evolution along any direction in the space of E_{6}^{(1)}.