The classification of $\tau$-tilting modules over algebras of type $D_n$

Abstract

Let $\Lambda$ be an algebra whose quiver is

In this paper, we classify the $\tau$-tilting modules over $\Lambda$ when $l(P_1)\leqslant n-2$. Moreover, the following recurrence formula for the number of $\tau$-tilting $\Lambda$-modules holds:

$$\begin{aligned} |{\tau {-}\mathrm{tilt}\, }\Lambda |=\sum ^{l(P_1)}_{i=1}C_{i-1}\cdot |{\tau {-}\mathrm{tilt}\, }\Lambda /\langle e_{\leqslant i}\rangle |, \end{aligned}$$

where $e_{\leqslant i}:=e_1+e_2+\cdots +e_i$ and $C_i=\frac{1}{i+1}\left( {\begin{array}{c}2i\\ i\end{array}}\right)$ is the ith Catalan number.