2021/08/26 by Grzegorz Bobiński, Bobinski, Grzegorz, Grzegorz Zwara +1
Mathematics · #14L30 #16G20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2108.11722
openalex publication_date 2021/08/26 · openalex created_date 2021/08/30 · openalex updated_date 2026/07/28
Let \Bbbk be an algebraically closed field, Q a finite quiver, and denote by \mathoprepQd the affine \Bbbk-scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set OM of points in \mathoprepQd(\Bbbk) isomorphic as representations to M is an orbit under an action on \mathoprepQd(\Bbbk) of a product of general linear groups. The orbit OM and its Zariski closure OM, considered as reduced subschemes of \mathoprepQd, are contained in an affine scheme CM defined by rank conditions on suitable matrices associated to \mathoprepQd. For all Dynkin and extended Dynkin quivers, the sets of points of OM and CM coincide, or equivalently, OM is the reduced scheme associated to CM. Moreover, OM=CM provided Q is a Dynkin quiver of type \mathbbA, and this equality is a conjecture for the remaining Dynkin quivers (of type \mathbbD and 𝔼). Let Q be a Dynkin quiver of type \mathbbD and M a finite dimensional representation of Q. We show that the equality TNOM=TNCM of Zariski tangent spaces holds for any closed point N of OM. As a consequence, we describe the tangent spaces to OM in representation theoretic terms.