Frobenius--Perron dimension via $\tau$-tilting theory

TL;DR

This paper investigates Frobenius--Perron dimensions of finite-dimensional algebras using $ au$-tilting theory, providing combinatorial evaluations, bounds for tame types, and explicit calculations for specific algebra classes.

Contribution

It introduces a combinatorial method to evaluate Frobenius--Perron dimensions and establishes bounds for tame algebras, advancing the understanding of these dimensions via $ au$-tilting theory.

Findings

Evaluated Frobenius--Perron dimensions for $ au$-tilting finite algebras.

Provided an upper bound for dimensions in tame representation type.

Determined dimensions for Nakayama and generalized preprojective algebras of Dynkin type.

Abstract

From the perspective of $\tau$-tilting theory, we study Frobenius--Perron dimensions of finite-dimensional algebras. First, we evaluate the Frobenius--Perron dimensions of $\tau$-tilting finite algebras by a combinatorial method in $\tau$-tilting theory. Secondly, we give the upper bound for the Frobenius--Perron dimension for $\tau$-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--Schr\"oer.