Stability functions inducing stability data

Fang Li; Kangping Qu

Home/Archive/arXiv

arXiv · 2610.34657

Stability functions inducing stability data

Fang Li·Kangping Qu

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study when a stability data on an abelian length category is induced by a stability function. We give an intrinsic characterization of such stability data. For at most six phases, an inducing function exists if and only if every semistable subcategory is wide. An example with seven phases shows that this bound is sharp. For finite phase sets and finite rank, an inducing function exists if and only if an inducing red path exists. We also give a maximal green sequence of length eleven in type $D_4$ that no stability function induces. This disproves the red-path conjecture of Brüstle--Smith--Treffinger.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Fang Li, Kangping Qu. 2026-10-26. Stability functions inducing stability data. https://arxiv.org/abs/2610.34657

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Characters of symmetric groups: sharp bounds on virtual degrees and the Witten zeta function

We prove sharp bounds on the virtual degrees introduced by Larsen and Shalev. This leads to improved bounds on characters of symmetric groups. We then sharpen bounds of Liebeck and Shalev concerning the Witten zeta function. Our main application is a characterization of the fixed-point free conjugacy classes whose associated random walk mixes in 2 steps.

math.RT↗

arXiv2026

Endoscopy for representations of disconnected reductive groups over finite fields

Let G be the group of rational points of a connected reductive group over a finite field. Based on work of Lusztig and Yun, we make the Jordan decomposition for irreducible G-representations canonical. It comes in the form of an equivalence between the category of G-representations with a fixed semisimple parameter s and a category of essentially unipotent representations of an endoscopic group of G, enriched with an equivariant structure with respect to the component group of the centralizer of s. Next we generalize these results, replacing the connected reductive group by a smooth group scheme with reductive neutral component. Again we establish canonical equivalences for both the rational and the geometric series of G-representations, in terms of essentially unipotent representations of endoscopic groups.

math.RT↗

arXiv2026

Representations of generalized linear Reedy categories and abelian model structures

We study representations of generalized $k$-linear Reedy categories, extending both the linear Reedy categories of Dalezios and Št'ov\'ıček and $k$-linearizations of generalized Reedy categories of Berger and Moerdijk. We establish homological properties of standard modules and classify irreducible representations in terms of simple modules over the local endomorphism algebras. We give criteria for a decomposition of the representation category into a product of local module categories; under the hypotheses of our non-degeneracy criterion, this decomposition holds if and only if the category is Hom-finite over $k$. Applications include span categories and representations of combinatorial categories. Using Grothendieck bifibrations, we then lift families of complete cotorsion pairs on the local module categories to complete cotorsion pairs on the representation category. Under compatibility assumptions, this also lifts families of (hereditary) Hovey triples and yields abelian model structures, with applications to Gorenstein homological algebra.

math.RT↗