arXiv subjects

Huimin Chang

Publications and source records attributed to Huimin Chang.

10 recordsLinked to original sources

arXiv2025

A new generalization of the McKay conjecture for $p$-solvable groups

Let $P$ be a Sylow $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime. Using a normal $p$-series $\mathcal{N}$ of $G$, we introduce the notion of $(\mathcal{N},p)$-stable characters and prove that $G$ and ${\bf N}_G(P)$ have equal numbers of such characters, which gives a new generalization of the McKay conjecture for $p$-solvable groups. Also, we establish a canonical bijection between these characters in the case where $G$ has odd order. Our proofs depend heavily on the theory of self-stabilizing pairs founded by M. L. Lewis, as well as some results of $\pi$-special characters due to I. M. Isaacs.

math.GR↗

arXiv2025

The Grothendieck groups of n-cluster categories of type A_{\infty}

In this article, we investigate the Grothendieck groups $K_0(\C_{A_{\infty}}^n)$ of $n$-cluster categories $\C_{A_{\infty}}^n$ of type $A_{\infty}$ introduced by T.~Holm and P.~J{\o}rgensen. We prove that $K_0(\C_{A_{\infty}}^n)\cong\mathbb{Z}$ for an arbitrary $n\geq 1$. As an application, this generalizes a result of Murphy for $n=1$.

math.RT↗

arXiv2025

Mutation of $n$-cotorsion pairs in extriangulated categories

In this article, we introduce the notion of $n$-cotorsion pairs in extriangulated categories, which extends both the cotorsion pairs established by Nakaoka and Palu and the $n$-cotorsion pairs in triangulated categories developed by Chang and Zhou. We further prove that any mutation of an $n$-cotorsion pair remains an $n$-cotorsion pair. As applications, we provide a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A_{\infty}$, and we realize mutations of $n$-cotorsion pairs geometrically via rotations of certain configurations of $n$-admissible arcs.

math.RT↗

arXiv2025

Grothendieck groups of repetitive cluster categories

In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories $\mathcal D^b(\mathcal H)/\langle(\tau^{-1}\Sigma)^p\rangle$ for $1\leq p\in\mathbb{N}$, where $\mathcal H$ is a hereditary abelian category with tilting objects. In this paper, we compute partial but essential results on the Grothendieck groups of the repetitive cluster categories $\mathcal D^b({\rm mod}KA_n)/\langle(\tau^{-1}\Sigma)^p\rangle$ and $\mathcal D^b({\rm mod} KD_n)/\langle(\tau^{-1}\Sigma)^p\rangle$. Our results extend the known computations for classical cluster categories, reveal new structural patterns arising from the repetitive parameter $p$, and provide further evidence of the close interplay between Grothendieck groups, Auslander-Reiten theory, and Coxeter transformations.

math.RT↗

arXiv2024

Cotorsion pairs in $(d+2)$-angulated categories

Let $\mathcal C$ be a $(d+2)$-angulated category. In this paper, we define the notions of cotorsion pairs and weak cotorsion pairs in $\mathcal C$, which are generalizations of the classical cotorsion pairs in triangulated categories. As an application, we give a geometric characterization of weak cotorsion pairs in $(d+2)$-angulated cluster categories of type $A$. Moreover, we prove that any mutation of a (weak) cotorsion pair in $\mathcal C$ is again a (weak) cotorsion pair. When $d=1$, this result generalizes the work of Zhou and Zhu on classical cotorsion pairs in triangulated categories.

math.RT↗

arXiv2024

Mutation of $n$-cotorsion pairs in triangulated categories

In this article, we define the notion of $n$-cotorsion pairs in triangulated categories, which is a generalization of the classical cotorsion pairs. We prove that any mutation of an $n$-cotorsion pair is again an $n$-cotorsion pair. When $n=1$, this result generalizes the work of Zhou and Zhu for classical cotorsion pairs. As applications, we give a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A$ and give a geometric realization of mutation of $n$-cotorsion pairs via rotation of certain configurations of $n$-diagonals.

math.RT↗

arXiv2023

Ptolemy diagrams and torsion pairs in m-cluster categories of type D

In this paper, we give a complete classification of torsion pairs in m-cluster categories of type D when m is odd, denoted by CmDn, via a bijection to combinatorial objects called Ptolemy diagrams of type D. As applications, we classify m-rigid subcategories of CmDn, which gives Jacquet-Malo's classification of m-cluster tilting subcategories of CmDn. When m = 1, we generalizes the work of Holm, Jorgensen and Rubey for the classification of torsion pairs in cluster categories of type Dn.

math.RT↗

arXiv2023

Torsion pairs in repetitive cluster categories of type $A_n$

We give a complete classification of torsion pairs in repetitive cluster categories of type $A_n$, which were defined by Zhu as the orbit categories, via certain configurations of diagonals, called Ptolemy diagrams. As applications, we classify rigid subcategories, which gives Lamberti's classification of cluster tilting subcategories. When $p = 1$, this generalizes the work of Holm, Jorgensen and Rubey for the classification of torsion pairs in cluster categories of type $A_n$.

math.RT↗

arXiv2017

Cotorsion pairs in cluster categories of type $A_{\infty}^{\infty}$

In this paper, we give a complete classification of cotorsion pairs in a cluster category $\mathscr{C}$ of type $A^\infty_\infty$ via certain configurations of arcs, called $τ$-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify $t$-structures and functorially finite rigid subcategories in $\mathscr{C}$, respectively. We also deduce Liu-Paquette's classification of cluster tilting categories of $\mathscr{C}$ and Ng's classification of torsion pairs in the cluster category of type $A_\infty$.

math.RT↗

arXiv2017

Torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects

We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by $\mathcal{A}_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A, $ or type $D$, we give a geometric description of (co)torsion pairs in $\mathcal{A}_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to count the number of (co)torsion pairs in these categories. Finally, we determine the hearts of (co)torsion pairs in all finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations.

math.RT↗