arXiv subjects

Joe Pallister

Publications and source records attributed to Joe Pallister.

4 recordsLinked to original sources

arXiv2021

Period 2 quivers and their T- and Y-systems

We give a general definition for period $2$ quivers before making some prudent assumptions to reduce the number of possibilities. Finding these quivers requires solving a complicated system of equations between the number of arrows in the quiver. We find all solutions of these systems for quivers with $N=3,4,5$ vertices and give some solutions for $N=6$. We then consider the $T$- and $Y$-systems associated with these quivers, some of which exhibit periodic quantities which lead to reductions; in a few cases we obtain the Somos-4 and Somos-5 recurrences.

math-ph↗

arXiv2021

$\tilde{A}$ and $\tilde{D}$ type cluster algebras: Triangulated surfaces and friezes

By viewing $\tilde{A}$ and $\tilde{D}$ type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze pattern (or bipartite belt) or (ii) the periodic quantities previously found for the cluster map associated with these frieze patterns. We show that these cluster variables form friezes which are precisely the ones found in [1] by applying the cluster character to the associated cluster category.

math.RA↗

arXiv2020

Linear relations and integrability for cluster algebras from affine quivers

We consider frieze sequences corresponding to sequences of cluster mutations for affine D and E type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

math.DS↗

arXiv2020

Linear relations for Laurent polynomials and lattice equations

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalize the Heideman-Hogan recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. The latter are obtained from linear relations with periodic coefficients, which were found recently by Kamiya et al. from travelling wave reductions of a linearizable lattice equation on a 6-point stencil. We introduce another linearizable lattice equation on the same stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.

nlin.SI↗