2013/07/31 by Ryan Kinser, Jenna Rajchgot
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Flag (linear algebra) #Isomorphism (crystallography) #Orbit (dynamics) #Quiver #Representation theory #Schubert variety #Space (punctuation) #Type (biology) #math.AG #math.CO #math.RT
paper · pdf · doi:10.1216/jca-2015-7-2-265
published as J. Commut. Algebra Volume 7, Number 2 (2015), 265-301 · 24 pages, comments welcome. v2: Section 3.2 new, Theorem 4.20 improved, references added
arxiv created 2013/08/31 · openalex publication_date 2015/06/01 · arxiv updated 2015/09/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This ``bipartite Zelevinsky map'' restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups. These identifications allow us to recover results of Bobi'nski and Zwara; namely, we see that orbit closures of type A quivers are normal, Cohen-Macaulay and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.