Quiver combinatorics for higher-dimensional triangulations

TL;DR

This paper explores the combinatorics of quivers from higher-dimensional cyclic polytope triangulations, linking them to cluster theory, and introduces criteria for bistellar flips and quiver mutations in higher dimensions.

Contribution

It generalizes the relationship between polygon triangulations and type A quivers to higher dimensions, and develops a quiver-theoretic framework for bistellar flips and mutations.

Findings

Triangulations with no interior (d+1)-simplices correspond to cut quivers of type A.

Connectedness of certain higher-dimensional triangulations via bistellar flips.

A quiver criterion for performing bistellar flips in higher dimensions.

Abstract

We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that a $2d$-dimensional triangulation has no interior $(d + 1)$-simplices if and only if its quiver is a cut quiver of type $A$, in the sense of Iyama and Oppermann. This is a higher-dimensional generalisation of the fact that triangulations of polygons with no interior triangles correspond to orientations of an $A_{n}$ Dynkin diagram. An application of this first result is that the set of triangulations of a $2d$-dimensional cyclic polytope with no interior $(d + 1)$-simplices is connected via bistellar flips -- the higher-dimensional analogue of flipping a diagonal inside a quadrilateral. In dimensions higher than 2, bistellar flips cannot be performed at all locations in a triangulation. Our second result gives a quiver-theoretic criterion for performing bistellar flips on a triangulation of a $2d$-dimensional cyclic polytope. This provides a visual tool for studying mutability of higher-dimensional triangulations and points towards what a theory of higher-dimensional quiver mutation could look like. Indeed, we apply this result to give a rule for mutating cut quivers at vertices which are not necessarily sinks or sources.