Variations on Baur--Marsh's determinant

TL;DR

This paper explores new determinant formulas related to cluster variables in type A and D cluster algebras, including interpretations as Cayley--Menger determinants and explicit product formulas.

Contribution

It introduces two variations of Baur--Marsh's determinant, including determinants of squared cluster variables and a new formula for type D cluster algebras from surfaces.

Findings

Determinant of squared cluster variables relates to Cayley--Menger determinant.

Explicit product formula for determinants in type D cluster algebra.

Extension of Baur--Marsh's original determinant computation.

Abstract

Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. In this article we wish to describe two variations. On the one hand, we compute determinants of matrices assembled from the squares of the cluster variables in Baur--Marsh's matrix. One such determinant admits an interpretation as a Cayley--Menger determinant. On the other hand, we wish to present a formula for the determinant of a matrix of cluster variables in a cluster algebra of type D. This cluster algebra is associated with a marked oriented surface. As in Baur--Marsh's setup the matrix is indexed by the marked points of the surface and an entry is given by the cluster variable corresponding to an arc between two marked points. Our formula asserts that the determinant may again be written as a product of cluster variables.