Isotropic Schur roots

Abstract

In this paper, we study the isotropic Schur roots of an acyclic quiver $Q$ with $n$ vertices. We study the perpendicular category $\mathcal{A}(d)$ of a dimension vector $d$ and give a complete description of it when $d$ is an isotropic Schur $\delta$. This is done by using exceptional sequences and by defining a subcategory $\mathcal{R}(Q,\delta)$ attached to the pair $(Q,\delta)$. The latter category is always equivalent to the category of representations of a connected acyclic quiver $Q_{\mathcal{R}}$ of tame type, having a unique isotropic Schur root, say $\delta_{\mathcal{R}}$. The understanding of the simple objects in $\mathcal{A}(\delta)$ allows us to get a finite set of generators for the ring of semi-invariants SI$(Q,\delta)$ of $Q$ of dimension vector $\delta$. The relations among these generators come from the representation theory of the category $\mathcal{R}(Q,\delta)$ and from a beautiful description of the cone of dimension vectors of $\mathcal{A}(\delta)$. Indeed, we show that SI$(Q,\delta)$ is isomorphic to the ring of semi-invariants SI$(Q_{\mathcal{R}},\delta_{\mathcal{R}})$ to which we adjoin variables. In particular, using a result of Skowro\'nski and Weyman, the ring SI$(Q,\delta)$ is a polynomial ring or a hypersurface. Finally, we provide an algorithm for finding all isotropic Schur roots of $Q$. This is done by an action of the braid group $B_{n-1}$ on some exceptional sequences. This action admits finitely many orbits, each such orbit corresponding to an isotropic Schur root of a tame full subquiver of $Q$.