Exploring the Varchenko Determinant of Partial Cubes

TL;DR

This paper generalizes the Varchenko determinant to partial cubes, exploring when these determinants have a clean factorization and identifying classes with well-behaved structures, thus advancing understanding of their algebraic properties.

Contribution

It extends the concept of the Varchenko determinant from oriented matroid complexes to partial cubes, highlighting cases with and without clean factorizations.

Findings

Identified partial cubes with determinants lacking clean factorization.

Found classes of partial cubes with well-structured Varchenko determinants.

Open questions for characterizing partial cubes with desirable determinant properties.

Abstract

The Varchenko matrix is known to have a well-structured determinant for complexes of oriented matroids (COMs). COMs can be characterized as partial cubes that do not have certain forbidden pc-minors. In this work, we generalize the Varchenko matrix and its determinant to partial cubes. We identify examples of partial cubes whose Varchenko determinants lack a clean factorization, as well as those that exhibit such a structure. These findings open the door for further research into the properties and potential characterizations of partial cubes with well-behaved Varchenko determinants.