An affine almost positive roots model

Abstract

We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset Φc of the root system and a compatibility degree on Φc, given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fanc(Φ). Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fanc(Φ) induced by real roots to the g-vector fan of the associated cluster algebra. We show that Φc is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.

A combinatorial approach to scattering diagrams

“…It is common (for example in the theory of Kac-Moody Lie algebras as in [10]) to place roots and weights in the same vector Algebraic Combinatorics, Vol. 3 #3 (2020) space, place co-roots and co-weights in the dual space, and let the natural pairing play the role that we have given to K. The approach here agrees with our approach in earlier papers, including [27,29,30,28,31] and eliminates the need to enlarge the vector spaces. Most importantly, the present approach lines up perfectly with the definition of scattering diagrams in [9].…”

Affine cluster monomials are generalized minors

“…Let be the maximum of and . Write for the piecewise linear map from the root lattice to the weight lattice; this map, introduced in [ReSt17], is the one used to pass from -vectors to -vectors in the proof of Lemma 2.4 (cf. [RuSt17, Proposition 9]).…”

“…Theorem 1.1 explicitly links these two trichotomies: given the -vector of a cluster variable in , a weight representation for which is extremal is highest-weight, lowest-weight, or neither exactly when the representation of associated to is respectively preprojective, postinjective, or regular [ReSt17, ReSp18]. Thus our results demonstrate that the parallel between these classifications is not a superficial one, but reflects an equivalence between certain structural information about representations of and structural information about representations of .…”

The action of a Coxeter element on an affine root system

“…Here, we improve on [12] by arguing uniformly (rather than type-by-type in the classification of affine root systems) and by clarifying some details for finite orbits. Our immediate motivation is to support an almost-positive roots model [28] for cluster algebras of affine type. (The finite-type model [9,13,14,22,32] uses Theorem 1.1 by way of [5,Exercise V §6.2].…”