arXiv subjects

Jiarui Fei

Publications and source records attributed to Jiarui Fei.

30 records · Page 2Linked to original sources

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↗

arXiv2016

Constructing Coherently G-invariant Modules

Let $G$ be a reductive group acting on a path algebra $kQ$ as automorphisms. We assume that $G$ admits a graded polynomial representation theory, and the action is polynomial. We describe the quiver $Q_G$ of the smash product algebra $kQ\# k[M_G]^*$, where $M_G$ is the associated algebraic monoid of $G$. We use $Q_G$-representations to construct $G$-invariant representations of $Q$. As an application, we construct algebraic semi-invariants on the quiver representation spaces from those $G$-invariant representations.

math.RT↗

arXiv2015

Cluster Algebras and Semi-invariant Rings II. Projections

Let ${\rm SI}_β(Q)$ be the semi-invariant ring of $β$-dimensional representations of a quiver $Q$. Suppose that $(Q,β)$ projects to another quiver with dimension vector $(Q',β')$ through an exceptional representation $E$. We show that if ${\rm SI}_β(Q)$ is the upper cluster algebra associated to an ice quiver $Δ$, then ${\rm SI}_{β'}(Q')$ is the upper cluster algebra associated to $Δ'$, where $Δ'$ is obtained from $Δ$ through simple operations depending on $E$. We also study the relation of their basis using the quiver with potential model.

math.AC↗

arXiv2015

Cluster Algebras, Invariant Theory, and Kronecker Coefficients I

We relate the $m$-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when $m=2$. Each {\sf g}-vector cone ${\sf G}_{\Diamond_l}$ of these cluster algebras controls the $2$-truncated Kronecker products for all symmetric functions of degree no greater than $l$. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all ${\sf G}_{\Diamond_l}$'s. As an application, we compute some invariant rings.

math.RT↗

arXiv2015

Counting using Hall Algebras II. Extensions from Quivers

We count the $\mathbb{F}_q$-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras -- one is one-point extended from a quiver $Q$, and the other is the Dynkin $A_2$ tensored with $Q$. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions.

math.RT↗

arXiv2015

General Presentations of Algebras

For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.

math.RA↗

arXiv2015

On Some Quiver Determinantal Varieties

We introduce certain quiver analogue of the determinantal variety. We study the Kempf-Lascoux-Weyman's complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen-Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equations of these varieties. When the variety has codimension one, the only irreducible polynomial function is a relative tensor invariant. As a by-product, we find some vanishing condition for the Kronecker coefficients. In the end, we make a generalization from the quiver setting to the tensor setting.

math.AC↗

arXiv2015

Cluster Algebras and Semi-invariant Rings I. Triple Flags

We prove that each semi-invariant ring of the complete triple flag of length $n$ is an upper cluster algebra associated to an ice hive quiver. We find a rational polyhedral cone ${\sf G}_n$ such that the generic cluster character maps its lattice points onto a basis of the cluster algebra. As an application, we use the cluster algebra structure to find a special minimal set of generators for these semi-invariant rings when $n$ is small.

math.AC↗

arXiv2013

Moduli of Representations I. Projections from Quivers

We use categorical method and birational geometry to study moduli spaces of quiver representations. From certain "representable" functor, we construct a birational transformation from the moduli space of representations of one quiver to another new quiver with one vertex less. The dimension vector and the stability for the new moduli are determined functorially. We introduce several relative notions of stability to study such a birational transformation. The essential case is proven to be the usual blow-ups. Moreover, we compare the induced ample divisors for the two moduli. We illustrate this theory by various examples.

math.AG↗

arXiv2012

Categorical Homotopy I. Quivers

We quiver-interpret the classical simplicial theory - including the cosimplex category $Δ$, Dold-Kan correspondence, and Hochschild homology - as a certain Q-homotopy theory of type $A$. For the cyclic and cubical theories, we proceed analogously. Subsequently, we present far-reaching generalizations, using different types of quivers. Moreover, we explain how to construct certain categories as analogs of $Δ$, and associate to each a Q-homotopy theory. We provide many examples, including such theories of type $D$.

math.AT↗

arXiv2012

Counting Using Hall Algebras I. Quivers

We survey some results on counting the rational points of moduli spaces of quiver representations. We then make generalizations to Grassmannians and flags of quiver representations. These results have nice applications to the cluster algebra. Along the way, we use the full Hopf structure of the Hall algebra of a quiver.

math.QA↗