[*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).
address:
- 'Department of Mathematics, University of California, San Diego'
- |
    Mathematics Department\
    Virginia Tech\
    Blacksburg, Virginia
author:
- Allen Knutson
- Mark Shimozono
title: Kempf collapsing and quiver loci
---

Introduction
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Kempf collapsings
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Let $G$ be a reductive algebraic group, and $P$ a parabolic subgroup. Let $Y$ be a linear representation of $G$, and $Z\leq Y$ a $P$-invariant subspace (or more generally, a closed subvariety with at worst rational singularities). In [@K], Kempf considers the map $$\begin{aligned}
  G \times^P Z &\overset{\kappa}{\to} Y & \qquad \hbox{here
    $G\times^P Z :=
    (G\times Z)\big/ \left \{ [g,\vec v] \sim [gp^{-1}, p\vec v] \right\}$}
  \\
  [g, \vec z] &\mapsto g\cdot \vec z &\end{aligned}$$ which he 