2024/05/06 by Jianmin Chen, Chen, Jianmin, Yiting Zheng +1 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2405.03395
By using the equivariant theory of group actions, we give a geometric model for the category of finite dimensional representations over a type \mathbbD quiver QD with n vertices and directional symmetry. Furthermore, we introduce the notion of maximal almost pre-rigid representations over QD, which form a family of objects counted by the generalized Catalan number. We present a geometric realization for maximal almost pre-rigid representations and prove that the endomorphism algebras of maximal almost pre-rigid representations are tilted algebras of type Q_D, where Q_D is a quiver obtained by adding n-2 new vertices and n-2 arrows to the quiver QD. Additionally, we define a partial order on the set of maximal almost pre-rigid representations, which therefore presents a representation-theoretic interpretation of the type-\mathbbD Cambrian lattice determined by QD. Meanwhile, we obtain a representation-theoretic interpretation of the type-\mathbbB Cambrian lattices.