Related papers: q-Painlev\'e systems arising from q-KP hierarchy
The $q$-Painlev\'e equation of type $E^{(1)}_6$ is obtained by Pad\'e method. Special solutions in determinant formula to the $q$-Painlev\'e equation is presented. A relation between Pad\'e method and B\"acklund transformation of type…
We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear $2\times 2$ system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the…
We derive a $q$-analogue of the matrix sixth Painlev\'e system via a connection-preserving deformation of a certain Fuchsian linear $q$-difference system. In specifying the linear $q$-difference system, we utilize the correspondence between…
It is known that discrete Painlev\'e equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlev\'e equations in which the symmetries become clearly visible. We know how to obtain…
We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and…
The generalized Drinfel'd-Sokolov hierarchies studied by de Groot-Hollowood-Miramontes are extended from the viewpoint of Sato-Wilson dressing method. In the A_1^(1) case, we obtain the hierarchy that include the derivative nonlinear…
We study the underlying relationship between Painleve equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity…
We show that the Garnier system in n-variables has affine Weyl group symmetry of type $B^{(1)}_{n+3}$. We also formulate the $\tau$ functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice $Q(C_{n+3})$ and…
We find four kinds of six-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order…
A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U_q with a representation…
A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.
In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlev\'e equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlev\'e…
We present a unified description of birational representation of Weyl groups associated with T-shaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of tau-functions is given…
Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, which has meromorphic continuation to…
The geometric approach for Painlev\'e and quasi-Painlev\'e differential equations in the complex plane is applied to non-autonomous Hamiltonian systems, quartic in the dependent variables. By computing their defining manifolds (analogue of…
We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups.…
In this paper, we find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $D_3^{(2)}$. We also find a four-parameter family of 2-coupled $D_3^{(2)}$-systems in dimension eight with…
We present a linear $q$-difference equation of rank $3$, which admits the affine Weyl group symmetry of type $E_8^{(1)}$. We further compare this equation with Moriyama-Yamada's quantum curve which has $W(E_8^{(1)})$-symmetry. The symmetry…
We propose a Q-system for the $A_m^{(1)}$ quantum integrable spin chain. We also find compact determinant expressions for all the Q-functions, both for the rational and trigonometric cases.
A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and…