scholarly journals Strong positivity for quantum theta bases of quantum cluster algebras

2021 ◽  
Author(s):  
Ben Davison ◽  
Travis Mandel
Keyword(s):  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Quantum Parameter ◽  
Integer Coefficients ◽  
Positive Elements ◽  
Quantum Cluster ◽  
Symmetric Quantum ◽  
Structure Constants ◽  
Strong Cluster ◽  
Diagram Techniques

AbstractWe construct “quantum theta bases,” extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross–Hacking–Keel–Kontsevich (J Am Math Soc 31(2):497–608, 2018). Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson–Thomas theory of quivers.

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

scholarly journals Orders of torsion units of integral reality-based algebras with rational multiplicities

2018 ◽  
Vol 17 (01) ◽  
pp. 1850015
Author(s):  
Gurmail Singh ◽  
Allen Herman
Keyword(s):  
Algebraic Integer ◽  
Finite Subgroup ◽  
One Dimensional ◽  
Integer Coefficients ◽  
Positive Degree ◽  
Finite Dimensional ◽  
Structure Constants ◽  
Distinguished Basis ◽  
Torsion Units ◽  
Pseudo Inverse

A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over [Formula: see text] whose distinguished basis [Formula: see text] contains [Formula: see text] and is closed under pseudo-inverse. An integral RBA is one whose structure constants in its distinguished basis are integers. If the algebra has a one-dimensional representation taking positive values on [Formula: see text], then we say that the RBA has a positive degree map. These RBAs have a standard feasible trace, and the multiplicities of the irreducible characters in the standard feasible trace are the multiplicities of the RBA. In this paper, we show that for integral RBAs with positive degree map whose multiplicities are rational, any finite subgroup of torsion units whose elements are all of degree [Formula: see text] and have algebraic integer coefficients must have order dividing a certain positive integer determined by the degree map and the multiplicities. The paper concludes with a thorough investigation of the properties of RBAs that force multiplicities to be rational.

scholarly journals Greedy bases in rank 2 quantum cluster algebras

2014 ◽  
Vol 111 (27) ◽  
pp. 9712-9716 ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Dylan Rupel ◽  
Andrei Zelevinsky
Keyword(s):  
Cluster Algebras ◽  
Quantum Cluster ◽  
Rank 2
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