Cluster Expansions from Punctured Orbifolds

TL;DR

This paper introduces multiple combinatorial formulas for generalized cluster algebras from punctured orbifolds, unifying and extending previous surface and unpunctured orbifold results.

Contribution

It provides new combinatorial expansion formulas in various forms and demonstrates their equivalence, broadening the scope of cluster algebra combinatorics.

Findings

Multiple combinatorial expansion formulas are established.

Formulas are shown to be equivalent.

Work generalizes previous results on surfaces and unpunctured orbifolds.

Abstract

We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their equivalence. This work generalizes and unifies existing work on combinatorial expansion formulas from surfaces and unpunctured orbifolds.