On the Number of $\tau$-Tilting Modules over Nakayama Algebras

Abstract

Let ${\Lambda }_n^r$ be the path algebra of the linearly oriented quiver of type $\mathbb{A}$ with $n$ vertices modulo the $r$-th power of the radical, and let ${\tilde{\Lambda }}_n^r$ be the path algebra of the cyclically oriented quiver of type $\tilde{\mathbb{A}}$ with $n$ vertices modulo the $r$-th power of the radical. Adachi gave a recurrence relation for the number of $\tau$-tilting modules over ${\Lambda }_n^r$. In this paper, we show that the same recurrence relation also holds for the number of $\tau$-tilting modules over ${\tilde{\Lambda }}_n^r$. As an application, we give a new proof for a result by Asai on recurrence formulae for the number of support $\tau$-tilting modules over ${\Lambda }_n^r$ and ${\tilde{\Lambda }}_n^r$.

Cited by in corpus (2)

• Counting the number of $\tau$-exceptional sequences over Nakayama algebras
• $\tau$-Tilting modules over one-point extensions by a simple module at a source point