Universal character and q-difference Painlevé equations

Abstract

The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we introduce an integrable system of q-difference lattice equations satisfied by the universal character, and call it the lattice q-UC hierarchy. We regard it as generalizing both q-KP and q-UC hierarchies. Suitable similarity and periodic reductions of the hierarchy yield the q-difference Painlevé equations of types (A 2g+1 + A 1 ) (1) (g ≥ 1), D(1) , and E(1) 6 . As its consequence, a class of algebraic solutions of the q-Painlevé equations is rapidly obtained by means of the universal character. In particular, we demonstrate explicitly the reduction procedure for the case of type E (1) 6 via the framework of τ -functions based on the geometry of certain rational surfaces.

Cluster integrable systems, q-Painlevé equations and their quantization

“…And if we add π, σ we get an extended Weyl group. This group is the symmetry group of this Painlevé equation, the formulas coincide with ones in [56, 6 , in accordance to (2.3) (and slightly different from q in [56]). …”

From Kp/Uc Hierarchies to Painlevé Equations

“…We refer the reader to [51] where a more general class of functional relations of the UC hierarchy including the above is established. See also [49,Appendix] regarding (6.9). We mention that the first two equations and the last are originated from (6.6) and (6.5), respectively.…”

On an Integrable System of q-Difference Equations Satisfied by the Universal Characters: Its Lax Formalism and an Application to q-Painlevé Equations

“…The q-LUC was introduced in [12] and it is, as seen below, originated from the quadratic relations satisfied by the universal characters. For convenience, we shall extend the universal character, (2.1), to be defined for a pair of arbitrary sequences of integers [λ, µ].…”

“…For instance, the whole set of homogeneous polynomial solutions of the UC hierarchy is in one-toone correspondence with the set of the universal characters. Also in [12] a q-difference analogue of the hierarchy was studied in connection with the q-Painlevé equations; it is an integrable system of q-difference equations originated from certain quadratic relations among the universal characters and was named the lattice q-UC hierarchy (q-LUC), since whose dependent variables (τ-functions) are arranged on a two-dimensional lattice Z 2 ; cf. [11,14].…”

“…Observing the universal characters be consistent with the similarity reduction, we can immediately construct particular solutions of the dynamics in terms of the universal characters; cf. [12].…”