Greg Muller

Locally acyclic cluster algebras

Abstract This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A ‘locally acyclic cluster algebra’ is a cluster algebra which admits a finite cover (in a geometric sense) by acyclic cluster algebras. Many important results about acyclic cluster algebras extend to locally acyclic cluster algebras (such as being finitely generated, integrally closed, and equaling their upper cluster algebra), as well as a result which is new even for acyclic cluster algebras (regularity over Q when the exchange matrix has full rank). Several techniques are developed for determining whether a cluster algebra is locally acyclic. Cluster algebras of marked surfaces with at least two boundary marked points are shown to be locally acyclic, providing a large class of examples of cluster algebras which are locally acyclic but not acyclic. Some specific examples are worked out in detail.

A = U for Locally Acyclic Cluster Algebras ?

This note presents a self-contained proof that acyclic and locally acyclic cluster algebras coincide with their upper cluster algebras.

Singularities of locally acyclic cluster algebras

We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped.

The twist for positroid varieties

The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikovs boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plucker coordinates.

Computing Upper Cluster Algebras

This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated. We include several examples of presentations produced by these methods.

The greedy basis equals the theta basis

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.

Character algebras of decorated SL2.C/-local systems

Let S be a connected and locally 1‐connected space, and let M S . A decorated SL2.C/‐local system is an SL2.C/‐local system on S , together with a chosen element of the stalk at each component of M. We study the decorated SL2.C/‐character algebra of .S;M/: the algebra of polynomial invariants of decorated SL2.C/‐local systems on .S;M/. The character algebra is presented explicitly. The character algebra is shown to correspond to the C ‐algebra spanned by collections of oriented curves in S modulo local topological rules. As an intermediate step, we obtain an invariant-theory result of independent interest: a presentation of the algebra of SL2.C/‐invariant functions on End.V/ m V n , where V is the tautological representation of SL2.C/. 13A50, 14D20, 57M27, 57M07