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Wee Liang Gan

Publications and source records attributed to Wee Liang Gan.

At least 19 recordsLinked to original sources
arXiv2025

Bounding regularity of $\mathrm{VI}^m$-modules

Fix a finite field $\mathbb{F}$. Let $\mathrm{VI}$ be a skeleton of the category of finite dimensional $\mathbb{F}$-vector spaces and injective $\mathbb{F}$-linear maps. We study $\mathrm{VI}^m$-modules over a noetherian commutative ring in the nondescribing characteristic case. We prove that if a finitely generated $\mathrm{VI}^m$-module is generated in degree $\leqslant d$ and related in degree $\leqslant r$, then its regularity is bounded above by a function of $m$, $d$, and $r$. A key ingredient of the proof is a shift theorem for finitely generated $\mathrm{VI}^m$-modules.

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arXiv2025

Bounding regularity of $FI^m$-modules

Let $FI$ be a skeleton of the category of finite sets and injective maps, and $FI^m$ the product of $m$ copies of $FI$. We prove that if an $FI^m$-module is generated in degree $\leqslant d$ and related in degree $\leqslant r$, then its regularity is bounded above by a function of $m$, $d$, and $r$.

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arXiv2024

A noetherian criterion for sequences of modules

We prove a noetherian criterion for a sequence of modules with linear maps between them. This generalizes a noetherian criterion of Gan and Li for infinite EI categories. We apply our criterion to the linear categories associated to certain diagram algebras defined by Patzt.

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arXiv2021

Adjoint functors on the representation category of $\mathscr{OI}$

In this paper we study adjunction relations between some natural functors on the representation category of the category of finite linearly ordered sets and order-preserving injections. We also prove that the Nakayama functor induces an equivalence from the Serre quotient of the category of finitely generated modules by the category of finitely generated torsion modules to the category of finite dimensional modules.

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arXiv2020

An inductive method for $\mathrm{OI}$-modules

In this paper we introduce an inductive method to study $\mathrm{OI}$-modules presented in finite degrees, where $\mathrm{OI}$ is a skeleton of the category of finitely totally ordered sets and strictly increasing maps. As an application, we obtain an explicit upper bound for the Castelnuovo-Mumford regularity of $\mathrm{OI}$-modules.

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arXiv2020

Castelnuovo-Mumford regularity of representations of certain product categories

We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories $\mathrm{FI}^m$ and $\mathrm{FI}_G^m$.

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arXiv2019

Bounds on homological invariants of VI-modules

We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

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arXiv2019

Asymptotic behaviors of representations of graded categories with inductive functors

In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring $k$, via introducing inductive functors which generalize important properties of shift functors of $\mathrm{FI}$-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory are equipped with inductive functors, and hence the finiteness of Castelnuovo-Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when $k$ is a field of characteristic 0.

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arXiv2018

Koszulity of directed categories in representation stability theory

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero.

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arXiv2017

An application of Nakayama functor in representation stability theory

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI$_G$ and VI$_q$, and answer positively an open question of Nagpal on representation stability theory.

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arXiv2017

A representation stability theorem for VI-modules

A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable.

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arXiv2017

An inductive machinery for representations of categories with shift functors

We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories $\FI_G$ and $\OI_G$ which appear in representation stability theory.

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arXiv2017

On central stability

The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of $d$-step central stability, and prove that if the ideal of relations of a category is generated in degrees at most $d$, then every module presented in finite degrees is $d$-step centrally stable.

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arXiv2016

On the negative-one shift functor for FI-modules

We show that the negative-one shift functor on the category of FI-modules is a left adjoint of the shift functor and a right adjoint of the derivative functor. We also describe how the coinduction functor is related to the negative-one shift functor.

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