Cyclic covers and non-orbit 3-representation-finite symmetric algebras

Abstract

Over an algebraically closed field of characteristic zero, we exhibit two symmetric algebras that are 3-representation-finite but are not orbit algebras of repetitive categories. The first is a 36-dimensional characteristic-zero lift $A$ of $Q(3A)^2_2$, the quaternion-type algebra that B\"ohmler and Marczinzik proved 3-representation-finite in characteristic 2; we construct an explicit 3-cluster-tilting module. We show that any orbit presentation of a connected symmetric algebra forces the algebra, or a connected cyclic cover of it, to admit a half-dimensional square-zero grading. Such gradings are detected by idempotent derivations, and a finite group of arrow characters constrains the possible covers. For $A$, only three double-cover candidates remain and the derivation obstruction excludes all of them. Thus $A$ is not an orbit algebra of any finite-dimensional algebra, regardless of global dimension, answering a question of Darp\"o and Iyama. The 84-dimensional 3-spherical weighted surface algebra is likewise 3-representation-finite and not an orbit algebra.