On the Number of τ-Tilting Modules over Nakayama Algebras

Abstract

Let Λʳₙ be the path algebra of the linearly oriented quiver of type A with n vertices modulo the r-th power of the radical, and let Λ˜ʳₙ be the path algebra of the cyclically oriented quiver of type à with n vertices modulo the r-th power of the radical. Adachi gave a recurrence relation for the number of τ-tilting modules over Λʳₙ. In this paper, we show that the same recurrence relation also holds for the number of τ-tilting modules over Λ˜ʳₙ. As an application, we give a new proof for a result by Asai on recurrence formulae for the number of support τ-tilting modules over Λʳₙ and Λ˜ʳₙ.