2006/08/14 by Allen Knutson, Mark Shimozono, Knutson, Allen +1
Mathematics · #16G20 #16G70 #19L47 #55N91 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.math/0608327
openalex publication_date 2006/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Kempf [1976] studied proper, G-equivariant maps from equivariant vector bundles over flag manifolds to G-representations V, which he called collapsings_. We give a simple formula for the G-equivariant cohomology class on V, ormultidegree_, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is 1, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, we can use the desingularization to give also a formula for the G-equivariant K-class of the image, whose leading term is the multidegree. Our application is to quiver loci and quiver polynomials. Let Q be a quiver of finite type (A, D, or E, in arbitrary orientation), and assign a vector space to each vertex. Let \Hom denote the (linear) space of representations of Q with these vector spaces. This carries an action of GL, the product of the general linear groups of the individual vector spaces. Aquiver locus_ Ωis the closure in \Hom of a GL-orbit, and its multidegree is the corresponding quiver polynomial_. Reineke [2004] proved that every ADE quiver locus is the image of a birational Kempf collapsing (giving a desingularization directly). Using Reineke's collapsings, we give formulae for ADE quiver polynomials, previously only computed in type A (though in this case, our formulae are new). In the A and D cases quiver loci are known to have rational singularities [Bobiński-Zwara 2002], so we also get formulae for their K-classes, which had previously only been computed in equioriented type A (and again our formulae are new).