Kempf collapsing and quiver loci
| dc.creator | Knutson, Allen | |
| dc.creator | Shimozono, Mark | |
| dc.date | 2006-08-14 | |
| dc.date.accessioned | 2026-07-07T07:21:44Z | |
| dc.date.available | 2026-07-07T07:21:44Z | |
| dc.description | 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, or_multidegree_, 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. A_quiver 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). | |
| dc.description | 21 pages | |
| dc.identifier | https://arxiv.org/abs/math/0608327 | |
| dc.identifier | http://arxiv.org/abs/math/0608327 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115404 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Combinatorics | |
| dc.subject | Rings and Algebras | |
| dc.subject | 16G20; 16G70; 55N91; 19L47 | |
| dc.title | Kempf collapsing and quiver loci | |
| dc.type | text |