Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 058, 13 pages      arXiv:2002.02990
Contribution to the Special Issue on Cluster Algebras

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

Hanpeng Gao a and Ralf Schiffler b
a)  Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China
b)  Department of Mathematics, University of Connecticut, Storrs, CT 06269-1009, USA

Received March 06, 2020, in final form June 11, 2020; Published online June 18, 2020

Let $\Lambda^r_n$ 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 $\widetilde{\Lambda}^r_n$ be the path algebra of the cyclically oriented quiver of type $\widetilde{\mathbb{A}}$ with $n$ vertices modulo the $r$-th power of the radical. Adachi gave a recurrence relation for the number of $\tau$-tilting modules over $\Lambda^r_n$. In this paper, we show that the same recurrence relation also holds for the number of $\tau$-tilting modules over $\widetilde{\Lambda}^r_n$. 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 $\Lambda^r_n$ and $\widetilde{\Lambda}^r_n$.

Key words: $\tau$-tilting modules; support $\tau$-tilting modules; Nakayama algebras.

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