Classifying torsion classes for algebras with radical square zero via sign decomposition

To study the set of torsion classes of a finite dimensional basic algebra, we use a decomposition, called sign-decomposition, parametrized by elements of $\{\pm1\}^n$ where $n$ is the number of simple modules. If $A$ is an algebra with radical square zero, then for each $\epsilon \in \{\pm1\}^n$ there is a hereditary algebra $A_{\epsilon}^!$ with radical square zero and a bijection between the set of torsion classes of $A$ associated to $\epsilon$ and the set of faithful torsion classes of $A_{\epsilon}^!$. Furthermore, this bijection preserves the property of being functorially finite. As an application in $\tau$-tilting theory, we prove that the number of support $\tau$-tilting modules over Brauer line algebras (resp. Brauer odd-cycle algebras) having $n$ edges is $\binom{2n}{n}$ (resp. $2^{2n-1}$).