Towards a uniform subword complex description of acyclic finite type cluster algebras

Abstract

It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. In this paper, we continue this study by describing the c-and g-vectors, and by providing a conjectured description of the Newton polytopes of the F -polynomials. In particular, we show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F -polynomials, or of the cluster variables, respectively. Date

Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians

“…We employ our algorithms to compute all maximal-dimensional cones in TGr + (3, 8) and compare them to the cluster complex of Gr(3, 8). We verify that [SW05, Conjecture 8.1] holds, which is proven to be true in [BS18]. This serves as a verification of our computations, and as an alternative proof of the conjecture in this specific case.…”

“…It is also conjectured [SW05,Conjecture 8.1] that they encode the combinatorics of cluster algebras of finite type. This was proven recently by Brodsky and Stump for many important cases [BS18].…”

“…The conjecture was proven by Brodsky and Stump [BS18] for cluster algebras of type A and of all types of to rank at most 8, which includes Gr 0 (3, 8). 6.1.…”

Algebraic Weaves and Braid Varieties

“…Let us consider the Mellit chart M, as constructed in Theorem 2.28, and ω M its corresponding 2-form. 12 In this subsection, we will first show that ω M/Tπ is nondegenerate, and thus (holomorphic) symplectic. Then we prove, in Theorem 3.5, that ω induces the holomorphic symplectic structure according to Theorem 1.1.…”

“…Subword Complexes. First, the combinatorics of weaves appears to be closely related to the combinatorics of subword complexes and brick polytopes [12,22,32,42,43,51,59,60,78]. The faces of a subword complex for a braid word γ correspond to all possible subwords of γ such that the Demazure product of their complements equals w 0 .…”