Universal Characters and q-Painlevé Systems

Abstract

We propose an integrable system of q-difference equations of which the universal characters satisfy and regard it as a q-analogue of the UC hierarchy; see [10]. Via a similarity reduction of this integrable system, rational solutions of the q-Painlevé systems are constructed in terms of the universal characters.

q-Painlevé VI Equation Arising from q-UC Hierarchy

“…This hierarchy is considered as an integrable system characterized by the universal character (see [9]) which is a generalization of Schur polynomial attached to a pair of partitions. Also a q-difference analogue of the hierarchy (q-UC hierarchy) was studied in [26]. In this paper we prove, by using tau functions, that q-P VI coincides with a certain similarity reduction of the q-UC hierarchy.…”

“…If a 0 a 1 = a 4 a 5 , then ℓ leaves the variable η = a 2 /a 3 invariant; see (2.3). In this case, (2.14) is equivalent to the bilinear equations of the fifth q-Painlevé equation (q-P V ); see [7,12,26]. We cite the result [22] by Takenawa; he showed that the defining variety of q-P V is actually included in that of q-P VI .…”

“…In this section we briefly review the notion of the universal character and the q-UC hierarchy; see [9,23,26].…”

“…Definition 3.1 (see [26]). The following system of q-difference equations is called the q-UC hierarchy:…”

“…We refer to the result [26] where the higher order q-Painlevé equations also turn out to be certain similarity reductions of the q-UC hierarchy; cf. [24].…”