Quiver combinatorics for higherdimensional triangulations
Abstract
We investigate the combinatorics of quivers that arise from triangulations of evendimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higherdimensional 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 higherdimensional 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 higherdimensional 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 quivertheoretic criterion for performing bistellar flips on a triangulation of a $2d$dimensional cyclic polytope. This provides a visual tool for studying mutability of higherdimensional triangulations and points towards what a theory of higherdimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.09189
 arXiv:
 arXiv:2112.09189
 Bibcode:
 2021arXiv211209189W
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Representation Theory;
 Primary: 52B05;
 Secondary: 05E10;
 52B11
 EPrint:
 26 pages, 9 figures