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

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–Schröer.