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 δ. This is done by using exceptional sequences and by defining a subcategory ℛ(Q, δ) attached to the pair (Q, δ). The latter category is always equivalent to the category of representations of a connected acyclic quiver Q ℛ of tame type, having a unique isotropic Schur root, say δ ℛ. The understanding of the simple objects in $$ \mathcal{A}\left(\delta \right) $$ allows us to get a finite set of generators for the ring of semiinvariants SI(Q, δ) of Q of dimension vector δ. The relations among these generators come from the representation theory of the category ℛ(Q, δ) and from a beautiful description of the cone of dimension vectors of $$ \mathcal{A}\left(\delta \right) $$ . Indeed, we show that SI(Q, δ) is isomorphic to the ring of semi-invariants SI(Q ℛ, δ ℛ) to which we adjoin variables. In particular, using a result of Skowronski and Weyman, the ring SI(Q, δ) 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.