87 citations · 223 across the 18 of their papers we have counts for
19 papers
Band bases as common triangular bases in cluster algebras from surfaces
Fan Qin, Chao Shen
We consider the skein algebra of an unpunctured marked surface. Thurston previously constructed its band basis topologically. We show that this band basis coincides with the common…
From i-boxes to signed words
Alessandro Contu, Fan Qin, Qiaoling Wei
The combinatorics of i-boxes has recently been introduced by Kashiwara--Kim--Oh--Park in the study of cluster algebras arising from the representation theory of quantum affine alge…
Applications of the freezing operators on cluster algebras
Fan Qin
We apply freezing operators to relate different (quantum) upper cluster algebras. We prove that these operators send localized (quantum) cluster monomials to localized (quantum) cl…
Notes: from dual canonical bases to triangular bases of quantum cluster algebras
Fan Qin
These notes are mainly based on arXiv:2003.13674 and a series of talks given in the workshop CARTEA. For any symmetrizable Kac-Moody algebra and any Weyl group eleme…
Stability scattering diagrams and quiver coverings
Qiyue Chen, Travis Mandel, Fan Qin
Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of a…
A refined multiplication formula for cluster characters
Bernhard Keller, Pierre-Guy Plamondon, Fan Qin
We obtain a multiplication formula for cluster characters on (stably) 2-Calabi-Yau (Frobenius or) triangulated categories. This formula generalizes those known for arbitrary pairs…
Bracelets bases are theta bases
Travis Mandel, Fan Qin
The skein algebra of a marked surface, possibly with punctures, admits the basis of (tagged) bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cl…
The valuation pairing on an upper cluster algebra
Peigen Cao, Bernhard Keller, Fan Qin
It is known that many (upper) cluster algebras are not unique factorization domains. We exhibit the local factorization properties with respect to any given seed : any non-zero…
Twist automorphisms and Poisson structures
Yoshiyuki Kimura, Fan Qin, Qiaoling Wei
We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quant…
Cluster algebras and their bases
Fan Qin
We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topolog…
An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras
Fan Qin
Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of qu…
Dual canonical bases and quantum cluster algebras
Fan Qin
Given any quantum cluster algebra arising from a quantum unipotent subgroup of symmetrizable Kac-Moody type, we verify the quantization conjecture in full generality that the quant…
Bases for upper cluster algebras and tropical points
Fan Qin
It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Pois…
Compare triangular bases of acyclic quantum cluster algebras
Fan Qin
Given a quantum cluster algebra, we show that its triangular bases defined by Berenstein and Zelevinsky and those defined by the author are the same for the seeds associated with a…
Triangular bases in quantum cluster algebras and monoidal categorification conjectures
Fan Qin
We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a…
Quantum groups via cyclic quiver varieties I
Fan Qin
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particul…
-Analog of -Characters, Bases of Quantum Cluster Algebras, and a Correction Technique
Fan Qin
We first study a new family of graded quiver varieties together with a new -deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint…
Graded quiver varieties, quantum cluster algebras and dual canonical basis
Yoshiyuki Kimura, Fan Qin
Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theore…
Quantum Cluster Variables via Serre Polynomials
Fan Qin
For skew-symmetric acyclic quantum cluster algebras, we express the quantum -polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians…