Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by the maximal green sequences of a quiver. We combine the combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.
Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. The two fundamental questions about maximal green sequences are whether a given algebra admits such sequences and, if so, does it admit only finitely many. We study maximal green sequences in the case of string algebras and give sufficient conditions on the algebra that ensure an affirmative answer to these questions.
In this paper, we study the maximal length of maximal green sequences for quivers of type $\widetilde{\mathbf{D}}$ and $\widetilde{\mathbf{E}}$ by using the theory of tilting mutation. We show that the maximal length does not depend on the choice of the orientation, and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.
Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey recent progress on their existence and properties and give a representation-theoretic proof of Greg Muller's theorem stating that full subquivers inherit maximal green sequences. In the appendix, Laurent Demonet describes maximal chains of torsion classes in terms of bricks generalizing a theorem by Igusa.
Maximal green sequences for quantum and Poisson CGL extensions
Northeastern University（东北大学）；International Center for Mathematical Sciences, Institute of Mathematics and Informatics Bulgarian Academy of Sciences（保加利亚科学院数学与计算机科学研究所国际数学科学中心）
We show that the relationship discovered by Oppermann and Thomas between triangulations of cyclic polytopes and the higher Auslander algebras of type $A$, denoted $A_{n}^{d}$, is an incredibly rich one. The \emph{higher Stasheff--Tamari orders} are two orders on triangulations of cyclic polytopes, conjectured to be equivalent, defined in the 1990s by Kapranov and Voevodsky, and Edelman and Reiner. We first show that these orders correspond in even dimensions to natural orders on tilting modules defined by Riedtmann and Schofield and studied by Happel and Unger. This result allows us to show th…
This is a not-to-be-journal-published paper, aimed to serve as reference. It is a summary of the main ideas on the topic appearing in the title, and an opportunity to state correctly the main conjecture in the field.
Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to ask which subsets $C$ can arise as minimal complements for some $W$. We show that in a finite abelian group $G$, every non-empty subset $C$ of size $|C| \leq 2^{2/3}|G|^{1/3}/((3e \log |G|)^{2/3}$ is a minimal complement for some $W$. As a corollary, we deduce that every finite non-empty subset of an infinite abelian group is a minimal complement. We also deri…
Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any Euclidean space. We study this problem in general metric spaces, where a triple $(a,b,c)$ of points in a metric space is considered a $3$-term arithmetic progression if $d(a,b)=d(b,c)=\frac{1}{2}d(a,c)$. In particular, we show that the result of Green and Sisask extends to any Cartan--Hadamard manifold (in particular, to the hyperbolic spaces), but does not…
Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous work has focused on addressing the question: how can we tell when a neural code is convex? Giusti and Itskov identified a local obstruction and proved that convex neural codes have no local obstructions. The converse is true for codes on up to four neurons, but false in general. Nevertheless, we prove this converse holds for codes with up to three maximal codew…
On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups
设$G$是一个有限群.对于$G$上的序列，我们指的是来自$G$的项的有限无序序列，其中允许重复，并且如果它的项可以被排序使得它们的乘积等于$G$的单位元，则我们说它是乘积1序列。大达文波特常数$\mathsf D (G)$是最小乘积1序列的最大长度，即不能分解为两个非平凡乘积1序列的乘积1序列。给出了二面体群和双循环群上长度为$\mathsf D (G)$的所有极小积1序列的明确刻画。基于这些刻画，我们研究了这些群上的积一序列么半群的长度集的并。
