Brauer tree algebras have $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ $2$-tilting complexes

Abstract

We show that any Brauer tree algebra has precisely $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ $2$-tilting complexes, where $n$ is the number of edges of the associated Brauer tree. More explicitly, for an external edge $e$ and an integer $j\ne 0$, we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is $\left(\genfrac{}{}{0ex}{}{2n-|j|-1}{n-1}\right)$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of $T$. To prove this, we use a geometric model of Brauer graph algebras on the closed oriented marked surfaces and a classification of $2$-tilting complexes due to Adachi-Aihara-Chan.

20 pages