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Jiarui Fei

Publications and source records attributed to Jiarui Fei.

At least 19 recordsLinked to original sources
arXiv2026

Syzygy Transformations of Cluster Characters and Geometric Twists

We use the full multiplication formula to study syzygy and suspension of cluster characters. In stably 2-Calabi--Yau Frobenius categories, syzygy and cosyzygy induce inverse automorphisms of localized character algebras; in Hom-finite 2-Calabi--Yau triangulated categories, the same uniqueness principle transports all cluster characters under suspension. The latter gives our Auslander--Reiten $F$-polynomial identity. No mutation reachability is required. We extend positroid twist identities to all objects, recover the Grassmannian and unipotent-cell formulas, identify partition-function algebras before boundary localization, and derive tropical covariance when the relevant tropical coordinate map is a homeomorphism.

math.RT↗
arXiv2026

Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions

We study Fulton's universal Schubert polynomials $\mathfrak S_w(c)$ as regular functions on the upper unitriangular group $U_N$. The standard triangular cluster structure associates a Schubert $\sf g$-vector to every permutation. Their convex hull is unimodularly equivalent to $\Delta_1\times\cdots\times\Delta_{N-1}$, and their root-degree fibers are parabolic Bruhat intervals realized by the strata of staircase quiver Grassmannians. The geometric (i.e., generic, canonical, and Mirkovi\'c--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that $\mathfrak S_w(c)$ is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of $\mathfrak S_w(c)$ implies $\Lambda_Q$-rigidity of $Z_{\sf g_w}$. We also classify simultaneously the unit columns of the geometric-to-Schubert transitions, determine support components of the PBW-to-geometric and code-to-Schubert transitions, and exhibit a permutation $w\in S_{10}$ for which the three geometric basis elements are distinct.

math.CO↗
arXiv2026

Cluster Algebras for Bosonic Plethysm

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $V=\Bbbk^\ell$ and $W=\Bbbk^m$, and set \[ \mathcal R_{\ell,m}=\operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}. \] We construct an explicit skew-symmetrizable seed $\Sigma_{\ell,m}$ by restricting and folding the determinantal seed for the flagged $m$-arrow Kronecker quiver. For every $\ell,m\ge2$, we have \[ \mathcal R_{\ell,m}=\mathcal U(\Sigma_{\ell,m}), \] with polynomial frozen coefficients, and $\Sigma_{\ell,m}$ admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone $\mathscr C_{\ell,m}$. Its weight fibers count the multigraded highest-weight multiplicities of $\mathcal R_{\ell,m}$, and the Jacobi--Trudi identity expresses symmetric-square plethysm coefficients as finite alternating sums of these counts. Optimized frozens give an explicit finite system of inequalities for $\mathscr C_{\ell,m}$.

math.RT↗
arXiv2026

A Polyhedral Formula for $n\times2\times2$ Kronecker Coefficients via Cluster Algebras

Let \[ \Bbbk[\Bbbk^3\otimes\Bbbk^2\otimes\Bbbk^2]^{U_3\times U_2\times U_2}. \] We construct an ordinary cluster family with Markov principal part. At $\zeta=-1$, the intersection of its initial Laurent ring with the three adjacent Laurent rings equals $\mathscr U$, and \[ \mathscr U=\mathcal M_u[u_\Delta], \] where $\mathcal M_u$ is the middle algebra. Its theta functions are indexed by a cone with a sixteen-element Hilbert basis. Multiplication by $u_\Delta$ pairs the Hilbert generators of mutable degrees $1$ and $-1$ and reduces each triple-weight space to the slice $\ell=\ell_0$. Counting the lattice points in this slice gives a finite sum with nonnegative summands. Determinant reduction extends the formula to all $n\times2\times2$ Kronecker coefficients.

math.RT↗
arXiv2026

Euler Characteristics of Generic Quiver Grassmannians: Semi-Invariants and Localization

We study the topological Euler characteristic of the quiver Grassmannian for a generic representation. We give a Chern class formula and a finite torus-localization formula. If the quiver $Q$ is acyclic, the formula can be combined with the covariant calculation of Derksen--Schofield--Weyman to obtain a finite integral linear combination of covariant multiplicities, or equivalently of semi-invariant multiplicities for a single flag-extended quiver of $Q$. We also give explicit formulas and examples for generalized Kronecker quivers, and record the same construction for multiplicative characteristic genera.

math.AG↗
arXiv2025

On the Orthogonal Projections

For any rigid presentation $e$, we construct an orthogonal projection functor to ${\rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${\rm rep}(e^\perp)$ and presentations compatible with $e$. For quivers with potentials, we show that ${\rm rep}(e^\perp)$ forms a module category of another quiver with potential. We derive mutation formulas for the $\delta$-vectors of positive and negative complements and the dimension vectors of simple modules in ${\rm rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

math.RT↗
arXiv2025

Schur Rank, Compatibility Degree, and Canonical Decomposition

The notion of denominator vectors can be extended to all generic basis elements of upper cluster algebras in a natural way. Under a weakened version of generic pairing assumption, we provide a representation-theoretic interpretation for this extended notion. We derive several consequences in this generality. We present a counterexample to the conjecture that distinct cluster monomials have distinct denominator vectors. Utilizing a new rank function called the Schur rank, we extend the notion of compatibility degree. As an application, we find a tropical method to compute the multiplicity of a real component in the canonical decomposition of $\delta$-vectors.

math.RT↗
arXiv2024

On AI's "semistable torsion classes and canonical decompositions"

In this short note, we give two-line proofs for main results in "Semistable torsion classes and canonical decompositions" by Asai-Iyama from a main result in "Tropical $F$-polynomials and general presentations", which appeared on the math arXiv 2 years earlier.

math.RT↗
arXiv2023

Crystal Structure of Upper Cluster Algebras

We describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety. We show that the crystal structure can be algebraically lifted to the generic basis of the upper cluster algebra. This can be viewed as an additive categorification of the crystal structure arising from cluster algebras. We introduce the biperfect bases in the cluster algebra setting and give a description of all biperfect bases, which are parametrized by lattice points in a product of polyhedral sets. We illustrate this theory from classical examples and new examples.

math.RT↗
arXiv2023

On the General Ranks of QP Representations

We propose a mutation formula for the general rank from a principal component ${\rm PC}(\delta)$ of representations to another one ${\rm PC}(\epsilon)$ for a quiver with potential. We give sufficient conditions for the formula to hold. In particular, the formula holds when any of $\delta$ and $\epsilon$ is reachable. We discover several related mutation invariants.

math.RT↗
arXiv2021

Combinatorics of $F$-polynomials

We use the stabilization functors to study the combinatorial aspects of the $F$-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the $F$-polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the $F$-polynomial is saturated for any rigid representation. We provide many examples and counterexamples, and pose several conjectures.

math.RT↗
arXiv2020

Mahler Measure of 3D Landau-Ginzburg Potentials

We express the Mahler measures of $23$ families of Laurent polynomials in terms of Eisenstein-Kronecker series. These Laurent polynomials arise as Landau-Ginzburg potentials on Fano $3$-folds, $16$ of which define $K3$ hypersurfaces of generic Picard rank $19$, and the rest are of generic Picard rank $< 19$. We relate the Mahler measure at each rational singular moduli to the value at $3$ of the $L$-function of some weight-$3$ newform. Moreover, we find $10$ exotic relations among the Mahler measures of these families.

math.NT↗
arXiv2019

Tropical $F$-polynomials and General Presentations

We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. We study its interplay with the general presentation for any finite-dimensional basic algebra. We give an interpretation of evaluating $f_M$ at a weight vector. As a consequence, we give a presentation of the Newton polytope ${\sf N}(M)$ of $M$. We study the dual fan and 1-skeleton of ${\sf N}(M)$. We propose an algorithm to determine the generic Newton polytopes, and show it works for path algebras. As an application, we give a representation-theoretic interpretation of Fock-Goncharov's duality pairing. We give an explicit construction of dual clusters, which consists of real Schur representations. We specialize the above general results to the cluster-finite algebras and the preprojective algebras of Dynkin type.

math.RT↗
arXiv2019

Tensor Product Multiplicities via Upper Cluster Algebras

For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver $Δ_Q^2$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of $Δ_Q^2$ is graded by the triple dominant weights $(μ,ν,λ)$ of $G$. We prove that when $G$ is simply-laced, the dimension of each graded component counts the tensor multiplicity $c_{μ,ν}^λ$. We conjecture that this is also true if $G$ is not simply-laced, and sketch a possible approach. Using this construction, we improve Berenstein-Zelevinsky's model, or in some sense generalize Knutson-Tao's hive model in type $A$.

math.RT↗
arXiv2018

Counting using Hall Algebras III. Quivers with Potentials

For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. The wall-crossing formula leads to a categorification of quantum cluster algebras under some assumption. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a counting formula relating the representation Grassmannians under sink-source reflections.

math.QA↗
arXiv2017

Cluster Algebras, Invariant Theory, and Kronecker Coefficients II

We prove that the semi-invariant ring of the standard representation space of the $l$-flagged $m$-arrow Kronecker quiver is an upper cluster algebra for any $l,m\in \mathbb{N}$. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient $g_{μ,ν}^λ$ with $λ$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the ${\rm g}$-vector cone, which is explicitly given.

math.RT↗
arXiv2017

Extending Upper Cluster Algebras

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm 1}]$. We try to understand under which conditions ${R}$ is an upper cluster algebra, and how the quiver of $R$ relates to that of $S$. Moreover, if the restriction of $(Δ,W)$ to some subquiver is a cluster model, we give a sufficient condition for $(Δ,W)$ itself being a cluster model. As an application, we show that the semi-invariant ring of any complete $m$-tuple flags is an upper cluster algebra whose quiver is explicitly given. Moreover, the quiver with its rigid potential is a polyhedral cluster model.

math.AC↗