Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Frobenius--Perron dimension via $τ$-tilting theory

Takahide Adachi, Ryoichi Kase

TL;DR

This work develops a combinatorial framework tying the Frobenius–Perron dimension $FPdim$ of finite-dimensional algebras to $ au$-tilting theory and semibricks, yielding finiteness results and precise bounds. It proves a fundamental inequality linking $FPdim(A)$ to the $FPdim$ of the finite lattice $ au$-tiltp$(A)$ and a BR-modulated self-extension parameter $d_b$, with equality when $d_b=0$. The authors establish sharp bounds for $ au$-tilting finite algebras of tame representation type and give exact $FPdim$ values for Nakayama algebras (0 or 1) and generalized preprojective algebras of Dynkin type (equal to the spectral radius $ ho(Q)$ of the Gabriel quiver), supported by Coxeter-theoretic and lattice-combinatorial methods. Collectively, the results illuminate how representation-type and lattice structure govern $FPdim$, provide explicit computations for central algebra families, and furnish a toolkit for further FPdim determinations in related settings.

Abstract

From the perspective of $τ$-tilting theory, we study Frobenius--Perron dimensions of finite-dimensional algebras. First, we evaluate the Frobenius--Perron dimensions of $τ$-tilting finite algebras by a combinatorial method in $τ$-tilting theory. Secondly, we give the upper bound for the Frobenius--Perron dimension for $τ$-tilting finite algebras of tame representation type. Thirdly, we determine the Frobenius--Perron dimensions of Nakayama algebras and generalized preprojective algebras of Dynkin type in the sense of Geiss--Leclerc--Schröer.

Frobenius--Perron dimension via $τ$-tilting theory

TL;DR

This work develops a combinatorial framework tying the Frobenius–Perron dimension of finite-dimensional algebras to -tilting theory and semibricks, yielding finiteness results and precise bounds. It proves a fundamental inequality linking to the of the finite lattice -tiltp and a BR-modulated self-extension parameter , with equality when . The authors establish sharp bounds for -tilting finite algebras of tame representation type and give exact values for Nakayama algebras (0 or 1) and generalized preprojective algebras of Dynkin type (equal to the spectral radius of the Gabriel quiver), supported by Coxeter-theoretic and lattice-combinatorial methods. Collectively, the results illuminate how representation-type and lattice structure govern , provide explicit computations for central algebra families, and furnish a toolkit for further FPdim determinations in related settings.

Abstract

From the perspective of -tilting theory, we study Frobenius--Perron dimensions of finite-dimensional algebras. First, we evaluate the Frobenius--Perron dimensions of -tilting finite algebras by a combinatorial method in -tilting theory. Secondly, we give the upper bound for the Frobenius--Perron dimension for -tilting finite algebras of tame representation type. Thirdly, we determine the Frobenius--Perron dimensions of Nakayama algebras and generalized preprojective algebras of Dynkin type in the sense of Geiss--Leclerc--Schröer.
Paper Structure (11 sections, 38 theorems, 95 equations)

This paper contains 11 sections, 38 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frobenius--Perron dimension of an algebra
  4. $\tau$-tilting theory
  5. Frobenius--Perron dimension of the $\tau$-tilting finite lattice
  6. Frobenius--Perron dimensions of $\tau$-tilting finite algebras of tame representation type
  7. Frobenius--Perron dimension of Nakayama algebras
  8. Frobenius--Perron dimension of generalized preprojective algebras of Dynkin type
  9. Cartan matrices and Coxeter groups
  10. Generalized preprojective algebras
  11. The Frobenius--Perron dimension

Key Result

Theorem 1.1

Let $Q$ be a finite connected quiver and let $A$ be its path algebra over an algebraically closed field. Then the following statements hold.

Theorems & Definitions (75)

  • Theorem 1.1: CGWZ19
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['thm:ul_bound']}
  • Theorem 1.4: Theorem \ref{['thm:ub_rf']}
  • Theorem 1.5: Theorems \ref{['thm:FPdim_of_Nakayama']} and \ref{['thm:FPdim-preproj']}
  • Definition 2.1
  • Proposition 2.2: R76
  • Lemma 2.3: CGWZ19
  • Definition 2.4
  • Remark 2.5
  • ...and 65 more