Related papers: From i-boxes to signed words
We consider a finite group acting on a vector space and the corresponding skew group algebra generated by the group and the symmetric algebra of the space. This skew group algebra illuminates the resulting orbifold and serves as a…
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled…
The sign-coherence about $c$-vectors was conjectured by Fomin-Zelevinsky and solved completely by Gross-Hacking-Keel-Kontsevich for integer skew-symmetrizable case. We prove this conjecture associated with $c$-vectors for rank 3 real…
We define combinatorial counterparts to the geometric string vertices of Sen-Zwiebach and Costello-Zwiebach, which are certain closed subsets of the moduli spaces of curves. Our combinatorial vertices contain the same information as the…
The construction of partially compactified cluster algebras on coordinate rings is handled by using codimension 2 arguments on cluster covers. An analog of this in the quantum situation is highly desirable but has not been found yet. In…
Sherman-Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Both constructions involve cluster monomials and normalized Chebyshev polynomials of the…
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…
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…
C.M. Bender and G. V. Dunne showed that linear combinations of words $q^{k}p^{n}q^{n-k}$, where $p$ and $q$ are subject to the relation $qp - pq = \imath$, may be expressed as a polynomial in the symbol $z = \tfrac{1}{2}(qp+pq)$. Relations…
Whereas exceptional sequences have a long history with many well-known connections to combinatorics, signed exceptional sequences are relatively recent. The authors introduced this concept in 2017 [19], although it was retroactively…
We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…
Recently, Kashiwara-Kim-Oh-Park introduced a wide family of monoidal categories of finite-dimensional representations of quantum affine algebras, which provide monoidal categorifications of cluster algebras. In this paper, we prove that,…
We study quantum cluster structures on bosonic extensions of quantum unipotent coordinate rings. For a positive braid group element $b\in \operatorname{Br}^+$, Kashiwara--Kim--Oh--Park introduced a subalgebra $\widehat{\mathcal A}(b)$ and…
Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give a combinatorial framework for phenomena occurring in the context of algebraic groups. Cluster algebras also have links to a wide range of other subjects,…
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…
We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…
We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model…
The cluster automorphism group of a cluster variety was defined by Gekhtman--Shapiro--Vainshtein, and later studied by Lam--Speyer. Braid varieties are interesting affine algebraic varieties indexed by positive braid words. It was proved…
We study generalized cluster algebras introduced by Chekhov and Shapiro. When the coefficients satisfy the normalization and quasi-reciprocity conditions, one can naturally extend the structure theory of seeds in the ordinary cluster…