arXiv subjects

Peigen Cao

Publications and source records attributed to Peigen Cao.

At least 19 recordsLinked to original sources

arXiv2026

Partial $F$-invariants and cluster categorifications

The $F$-invariant in cluster algebras is a combinatorial invariant that unifies the $E$-invariant from additive categorification and the $\mathfrak{d}$-invariant from monoidal categorification. In this paper, we study its refinement, the partial $F$-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of $g$-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial $F$-invariants coincide with both the partial $E$-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized $R$-matrices (or partial $\mathfrak{d}$-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of $q$-characters; we verify the conjectural explicit formula for the pole orders between Kirillov--Reshetikhin modules.

math.RT↗

arXiv2026

Linear independence of global monomials on positive spaces

In this paper, we prove that the global monomials on positive spaces are linearly independent. The approach is based on the study of the Newton polytopes of Laurent expansions. Our general results can be applied to the positive spaces arising from cluster algebras and many Laurent phenomenon algebras, which in turn yield the so-called proper Laurent monomial property of Laurent expansions and the linear independence of cluster monomials for such algebras.

math.RT↗

arXiv2026

Quantum cluster algebra realization for stated ${\rm SL}_n$-skein algebras and rotation-invariant bases for polygons

We construct a quantum cluster structure on the skew-field of fractions ${\rm Frac}({\mathscr S}_\omega(\mathfrak{S}))$ of the stated ${\rm SL}_n$-skein algebra ${\mathscr S}_\omega(\mathfrak{S})$, where $\mathfrak{S}$ is a triangulable pb surface without interior punctures. This work complements the construction for the projected stated skein algebra $\widetilde{\mathscr S}_\omega(\mathfrak{S})$ given by the last two authors. Let ${\mathscr S}_\omega^{\rm fr}(\mathfrak{S})$ denote the localization of ${\mathscr S}_\omega(\mathfrak{S})$ at the multiplicative set generated by all frozen variables. Let ${\mathscr A}_\omega^{\rm fr}(\mathfrak{S})$ and ${\mathscr U}_\omega^{\rm fr}(\mathfrak{S})$ (respectively $\overline{\mathscr A}_\omega(\mathfrak{S})$ and $\overline{\mathscr U}_\omega(\mathfrak{S})$) denote the quantum cluster algebra and quantum upper cluster algebra associated to ${\rm Frac}({\mathscr S}_\omega(\mathfrak{S}))$ (respectively ${\rm Frac}(\widetilde{\mathscr S}_\omega(\mathfrak{S}))$). We prove that \[ \widetilde{\mathscr S}_\omega(\mathfrak{S}) = \overline{\mathscr A}_\omega(\mathfrak{S}) = \overline{\mathscr U}_\omega(\mathfrak{S}) \quad \text{and} \quad {\mathscr S}_\omega^{\rm fr}(\mathfrak{S}) = {\mathscr A}_\omega^{\rm fr}(\mathfrak{S}) = {\mathscr U}_\omega^{\rm fr}(\mathfrak{S}) \] whenever $\mathfrak{S}$ is a polygon. As a consequence, when $\mathfrak{S}$ is a polygon, we show that the theta basis of $\overline{\mathscr U}_\omega(\mathfrak{S})$ (respectively ${\mathscr U}_\omega^{\rm fr}(\mathfrak{S})$) yields a rotation-invariant basis of $\overline{\mathscr S}_\omega(\mathfrak{S})$ (respectively ${\mathscr S}_\omega^{\rm fr}(\mathfrak{S})$) with several desirable properties, including positivity and a natural parametrization.

math.QA↗

arXiv2026

Additive categorification of the monoidal $\Lambda$-invariant

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$, Kashiwara-Kim-Oh-Park proved that the Hernandez-Leclerc categories form a monoidal categorification of their Grothendieck rings. Furthermore, these rings are $\Lambda$-cluster algebras, meaning they are equipped with a compatible Poisson structure, constructed via the $\Lambda$-invariant. Under certain natural conditions, where $U_q'(\mathfrak{g})$ is of untwisted simply-laced type, we provide an additive interpretation of the $\Lambda$-invariant within the framework of Higgs categories. More precisely, there is an ice quiver with potential associated with these cluster algebras, and a key ingredient of our work consists in proving that its relative Ginzburg algebra is proper. More generally, if the relative Ginzburg algebra associated with an arbitrary ice quiver with potential is proper, we prove that the corresponding cluster algebra admits the structure of a $\Lambda$-cluster algebra defined in terms of negative extensions in the Higgs category. Moreover, we provide a homological formula to compute the corresponding tropical and $F$-invariants introduced by Cao.

math.RT↗

arXiv2026

Newton polytopes in cluster algebras and $\tau$-tilting theory

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their $F$-polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the $\tau$-rigid modules and the left finite multi-semibricks in $\tau$-tilting theory are uniquely determined by the Newton polytopes of these modules. The key tools used in the proofs are the left Bongartz completion, $F$-invariant and partial $F$-invariant in the context of cluster algebras and $\tau$-tilting theory.

math.RT↗

arXiv2025

F-invariant and E-invariant

$F$-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of $F$-invariant is that it is a coordinate-free invariant, that is, it is mutation invariant under the initial seed mutations. $E$-invariant for a pair of decorated representations of quivers with potentials is introduced by Derksen, Weyman and Zelevinsky, which is also a coordinate-free invariant. The strategies used to show the mutation-invariance of $F$-invariant and $E$-invariant are totally different. In this paper, we give a new proof of the mutation-invariance of $F$-invariant following the strategy used by Derksen, Weyman and Zelevinsky. As a result, we prove that $F$-invariant coincides with $E$-invariant on cluster monomials. We also give a proof of Reading's conjecture, which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of $g$-vectors.

math.RT↗

arXiv2025

Modules determined by their Newton polytopes

In the $\tau$-tilting theory, there exist two classes of foundamental modules: indecomposable $\tau$-rigid modules and left finite bricks. In this paper, we prove the indecomposable $\tau$-rigid modules and the left finite bricks are uniquely determined by their Newton polytopes spanned by the dimensional vectors of their quotient modules. This is a kind of generalization of Gabriel's result that the indecomposable modules over path algebras of Dynkin quivers are uniquely determined by their dimensional vectors.

math.RT↗

arXiv2024

Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel

We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.

math.RA↗

arXiv2024

Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel

math.RT↗

arXiv2023

The valuation pairing on an upper cluster algebra

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 $t$: any non-zero element in a full rank upper cluster algebra can be uniquely written as the product of a cluster monomial in $t$ and another element not divisible by the cluster variables in $t$. Our approach is based on introducing the valuation pairing on an upper cluster algebra: it counts the maximal multiplicity of a cluster variable among the factorizations of any given element. We apply the valuation pairing to obtain many results concerning factoriality, $d$-vectors, $F$-polynomials and the combinatorics of cluster Poisson variables. In particular, we obtain that full rank and primitive upper cluster algebras are factorial; an explanation of $d$-vectors using valuation pairing; a cluster monomial in non-initial cluster variables is determined by its $F$-polynomial; the $F$-polynomials of non-initial cluster variables are irreducible; and the cluster Poisson variables parametrize the exchange pairs of the corresponding upper cluster algebra.

math.RT↗

arXiv2023

Relative left Bongartz completions and their compatibility with mutations

In this paper, we introduce relative left Bongartz completions for a given basic $τ$-rigid pair $(U,Q)$ in the module category of a finite dimensional algebra $A$. They give a family of basic $τ$-tilting pairs containing $(U,Q)$ as a direct summand. We prove that relative left Bongartz completions have nice compatibility with mutations. Using this compatibility we are able to study the existence of maximal green sequences under $τ$-tilting reduction. We also explain our construction and some of the results in the setting of silting theory.

math.RT↗

arXiv2023

F-invariant in cluster algebras

We consider skew-symmetrizable (upper) cluster algebras with a compatible Poisson structure, called $\mathsf{\Lambda}$-(upper) cluster algebras. For any two good elements (e.g., cluster monomials) in a $\mathsf{\Lambda}$-upper cluster algebra, we introduce two invariants, called tropical invariant and $F$-invariant. We prove that (i) the product of two cluster monomials is still a cluster monomial if and only if their $F$-invariant is zero; (ii) if two cluster variables are log-canonical, then they are contained in the same cluster; and (iii) the notion of $F$-invariant for a pair of cluster monomials can be defined for any (upper) cluster algebra, regardless of whether it is a $\mathsf{\Lambda}$-(upper) cluster algebra. When restricting to cluster monomials, we prove that the tropical invariant and $F$-invariant respectively coincide with the $\Lambda$-invariant and twice $\mathfrak{d}$-invariant in the monoidal cluster categorification using various monoidal subcategories of finite-dimensional modules over quantum affine algebras and quiver Hecke algebras; and we prove that the $F$-invariant coincides with the $E$-invariant in the additive cluster categorification using the theory of quivers with potentials. Inspired by $F$-invariant, we introduce the dominant sets for seeds of cluster algebras as a replacement of torsion classes for $\tau$-tilting pairs in $\tau$-tilting theory. With the help of the dominant sets, we prove that the oriented exchange graphs of cluster algebras are acyclic. In particular, this implies that green mutations induce a partial order on the set of seeds (up to seed equivalence) of cluster algebras. We prove that the oriented exchange graphs of cluster algebras coincide with the Hasse quivers of the above posets of seeds.

math.RT↗

arXiv2022

Bongartz completion via $c$-vectors

In the present paper, we first give a characterization for Bongartz completion in $τ$-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using $c$-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra $\mathcal A$ whose vertices consist of seeds of $\mathcal A$ containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a cluster Poisson algebra $\mathcal X_\bullet$, each cluster Poisson seed (up to seed equivalence) of $\mathcal X_\bullet$ is uniquely determined by its negative cluster Poisson variables.

math.RT↗

arXiv2022

On Leclerc's conjectural cluster structures for open Richardson varieties

In 2016, Leclerc constructed conjectural cluster structures on open Richardson varieties using representations of preprojective algebras. A variant with more explicit seeds was obtained by Ménard in his thesis. We show that Ménard's seeds do yield *upper* cluster algebra structures on open Richardson varieties and discuss the problems that remain in order to prove that they are cluster algebra structures.

math.RT↗

arXiv2022

On exchange matrices from string diagrams

Inspired by Fock-Goncharov's amalgamation procedure \cite{Fock-Goncharov-2006}, Shen-Weng introduced string diagrams in \cite{Shen-Weng-2021}, which are very useful to describe many interesting skew-symmetrizable matrices closely related with Lie theory. In this paper, we prove that the skew-symmetrizable matrices from string diagrams are in the smallest class $\mathcal P^\prime$ of skew-symmetrizable matrices containing the $1\times 1$ zero matrix and closed under mutations and source-sink extensions. This result applies to the exchange matrices of cluster algebras from double Bruhat cells, unipotent cells, double Bott-Samelson cells and so on. Our main result can be used to explain why many skew-symmetrizable matrices from Lie theory have reddening sequences. It can be also used to prove some interesting results regarding non-degenerate potentials on many quivers from Lie theory.

math.RT↗

arXiv2019

On some combinatorial properties of generalized cluster algebras

In this paper, we prove some combinatorial results on generalized cluster algebras. To be more precisely, we prove that (i) the seeds of a generalized cluster algebra $\mathcal A(\mathcal S)$ whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of $\mathcal A(\mathcal S)$; (ii) there exists a bijection from the set of cluster variables of a generalized cluster algebra to the set of cluster variables of another generalized cluster algebra, if their initial exchange matrices satisfying a mild condition. Moreover, this bijection preserves the set of clusters of these two generalized cluster algebras. As applications of the second result, we prove some properties of the components of the $d$-vectors of a generalized cluster algebra and we give a characterization for the clusters of a generalized cluster algebra.

math.RA↗

arXiv2019

A conjecture on cluster automorphisms of cluster algebras

A cluster automorphism is a $\mathbb{Z}$-algebra automorphism of a cluster algebra $\mathcal A$ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $\mathcal A$ is just a $\mathbb{Z}$-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.

math.RT↗

arXiv2019

$\mathcal G$-systems

A $\mathcal G$-system is a collection of $\mathbb Z$-bases of $\mathbb Z^n$ with some extra axiomatic conditions. There are two kinds of actions "mutations" and "co-Bongartz completions" naturally acting on a $\mathcal G$-system, which provide the combinatorial structure of a $\mathcal G$-system. It turns out that "co-Bongartz completions" have good compatibility with "mutations". The constructions of "mutations" are known before in different contexts, including cluster tilting theory, silting theory, $τ$-tilting theory, cluster algebras, marked surfaces. We found that in addition to "mutations", there exists another kind of actions "co-Bongartz completions" naturally appearing in these different theories. With the help of "co-Bongartz completions" some good combinatorial results can be easily obtained. In this paper, we give the constructions of "co-Bongartz completions" in different theories. Then we show that $\mathcal G$-systems naturally arise from these theories, and the "mutations" and "co-Bongartz completions" in different theories are compatible with those in $\mathcal G$-systems.

math.RT↗