University of Minnesota Combinatorics Seminar: 2021-2022

Date: 10/22/2021

Speaker: Nicholas Ovenhouse

Double Dimers and the Super Ptolemy Relation
Given a quadrilateral inscribed in a circle, Ptolemy's theorem relates the lengths of the diagonals to the lengths of the sides. Given an inscribed polygon, the Ptolemy relation can be used to express the length of any diagonal as a Laurent polynomial in the lengths of the diagonals in some fixed triangulation. This is a realization of type-A cluster algebras, where the cluster variables are the lengths of the diagonals. There are formulas (due to Musiker and Schiffler) for these Laurent polynomials in terms of dimer covers (perfect matchings) of a certain graph. Penner and Zeitlin have recently defined a super algebra which generalizes this situation by introducing new non-commuting variables. They also defined a super version of Ptolemy's relation. In joint work with Musiker and Zhang, we gave a formula for the corresponding elements of the super algebra in terms of double dimer covers on the same graph.

Double Dimers and the Super Ptolemy Relation

Given a quadrilateral inscribed in a circle, Ptolemy's theorem relates the lengths of the diagonals to the lengths of the sides. Given an inscribed polygon, the Ptolemy relation can be used to express the length of any diagonal as a Laurent polynomial in the lengths of the diagonals in some fixed triangulation. This is a realization of type-A cluster algebras, where the cluster variables are the lengths of the diagonals. There are formulas (due to Musiker and Schiffler) for these Laurent polynomials in terms of dimer covers (perfect matchings) of a certain graph.

Penner and Zeitlin have recently defined a super algebra which generalizes this situation by introducing new non-commuting variables. They also defined a super version of Ptolemy's relation. In joint work with Musiker and Zhang, we gave a formula for the corresponding elements of the super algebra in terms of double dimer covers on the same graph.