Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae . For all these classes of polynomials a 2x2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toeplitz determinants of the moments of the pseudo-measure. In particular the results imply that any (shifted) Toeplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.Comment: 35 pages, 1 figur

Cluster algebras and Poisson geometry

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.Comment: minor change

Cluster algebras in scattering amplitudes with special 2D kinematics

We study the cluster algebra of the kinematic configuration space $Conf_n(\mathbb{P}^3)$ of a n-particle scattering amplitude restricted to the special 2D kinematics. We found that the n-points two loop MHV remainder function found in special 2D kinematics depend on a selection of \XX-coordinates that are part of a special structure of the cluster algebra related to snake triangulations of polygons. This structure forms a necklace of hypercubes beads in the corresponding Stasheff polytope. Furthermore in $n = 12$, the cluster algebra and the selection of \XX-coordinates in special 2D kinematics replicates the cluster algebra and the selection of \XX-coordinates of $n=6$ two loop MHV amplitude in 4D kinematics.Comment: 22 page

Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville's equation and the map plays the role of a B\"acklund transformation.Comment: 17 pages, 7 figures. Corrected typos and updated reference detail

Solitons and Almost-Intertwining Matrices

We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some rank one matrix T. A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such triples. The tau-functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars-Schneider particle system

Plethora of cluster structures on GLnGL_n

We continue the study of multiple cluster structures in the rings of regular functions on $GL_n$, $SL_n$ and $\operatorname{Mat}_n$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group $\mathcal G$ corresponds to a cluster structure in $\mathcal O(\mathcal G)$. Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of $A_n$ type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$ type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $SL_n$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $SL_n$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications

Cluster structures on simple complex Lie groups and the Belavin-Drinfeld classification

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SLn, n < 5, and for any G in the case of the standard Poisson- Lie structure