Resolution quivers and the QF-1 property of higher Auslander Nakayama algebras

Zhe Han; Dawei Shen; Jiawei Zhou

Home/Archive/arXiv

arXiv · 2610.34716

Resolution quivers and the QF-1 property of higher Auslander Nakayama algebras

Zhe Han·Dawei Shen·Jiawei Zhou

View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a resolution-quiver-theoretic proof of the QF-1 conjecture of Cruz and Marczinzik for higher Auslander Nakayama algebras of positive even global dimension. The proof combines syzygy calculations in Kupisch coordinates, which show that the projective dimensions of quotients of a fixed projective module with cyclic top strictly decrease as their lengths increase, with a torsion-pair argument and Morita's criterion.

Explore related subjects

math.RT math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zhe Han, Dawei Shen, Jiawei Zhou. 2026-10-26. Resolution quivers and the QF-1 property of higher Auslander Nakayama algebras. https://arxiv.org/abs/2610.34716

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↗