Representation Type and Schurian Finiteness of Tits Algebras

Abstract

We determine the representation type and Schurian finiteness of the Tits algebra of a finite central real hyperplane arrangement over an algebraically closed field. The algebra is representation-finite when the arrangement has at most two hyperplanes, tame when it has three, and wild when it has at least four. Moreover, it is Schurian-finite if and only if the arrangement has at most two hyperplanes. Both classifications hold over algebraically closed fields of arbitrary characteristic. The proof combines Saliola's bound-quiver presentation with explicit module constructions, convex subquivers, essentialization, and deletion. In the $\mathbb{R}^2$ case, we describe all eleven indecomposable modules for two lines and construct a full exact embedding of modules over a free algebra for at least four lines.