arXiv subjects

Calvin Pfeifer

Publications and source records attributed to Calvin Pfeifer.

4 recordsLinked to original sources

arXiv2025

Serre cyclotomic algebras

We introduce a class of proper differential graded algebras which we call Serre cyclotomic. They generalize fractionally Calabi-Yau algebras and categorify de la Pe\~na's algebras of cyclotomic type. Path algebras of affine type and (higher) canonical algebras are examples of Serre cyclotomic algebras. Our definition is related to Elias-Hogancamp's theory of categorical diagonalization. We compute the categorical entropy of their Serre functors, in the sense of Dimitrov-Haiden-Katzarkov-Kontsevich, and use this to determine which graded path algebras and which homologically smooth graded gentle algebras are Serre cyclotomic. Finally, we show that trivial extension algebras of Serre cyclotomic algebras have finite global complexity.

math.RT↗

arXiv2023

Remarks on $\tau$-tilted versions of the second Brauer-Thrall Conjecture

In this short note, we state a stable and a $\tau$-reduced version of the second Brauer-Thrall Conjecture. The former is a slight strengthening of a brick version of the second Brauer-Thrall Conjecture raised by Mousavand and Schroll-Treffinger-Valdivieso. The latter is stated in terms of Geiss-Leclerc-Schr\"oer's generically $\tau$-reduced components and provides a geometric interpretation of a question of Demonet. It follows that the stable second Brauer-Thrall Conjecture implies our $\tau$-reduced second Brauer-Thrall Conjecture. Finally, we prove the reversed implication for the class of E-tame algebras recently introduced by Asai-Iyama.

math.RT↗

arXiv2023

A generic classification of locally free representations of affine GLS algebras

Throughout, let $K$ be an algebraically closed field of characteristic $0$. We provide a generic classification of locally free representations of Geiss-Leclerc-Schr\"oer's algebras $H_K(C,D,\Omega)$ associated to affine Cartan matrices $C$ with minimal symmetrizer $D$ and acyclic orientation $\Omega$. Affine GLS algebras are "smooth" degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several "tame" phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type $\tilde{BC}_1$, we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a $1$-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular $\tau$-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free $H$-module is isomorphic to a direct sum of $\tau$-rigid modules and modules from our $1$-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types, and we obtain such a decomposition by "folding" the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are $E$-tame in the sense of Derksen-Fei and Asai-Iyama.

math.RT↗

arXiv2020

Wide subcategories and lattices of torsion classes

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $τ$-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal{U},\mathcal{T}]$ in the lattice $\operatorname{\mathsf{tors}} \mathcal{A}$ of torsion classes in $\mathcal{A}$ such that $\mathcal{W}:=\mathcal{U}^\perp \cap \mathcal{T}$ is a wide subcategory of $\mathcal{A}$; we call these intervals wide intervals. We prove that a wide interval $[\mathcal{U},\mathcal{T}]$ is isomorphic to the lattice $\operatorname{\mathsf{tors}} \mathcal{W}$ of torsion classes in the abelian category $\mathcal{W}$. We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet--Iyama--Reading--Reiten--Thomas; and second, in terms of the Ingalls--Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks--Šťovíček.

math.CT↗