##### Department of Mathematics,

University of California San Diego

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### Lie Groups

## Allen Knutson

#### UCSD

## Kempf collapsing and quiver loci

##### Abstract:

Let $Q=(Q_0,Q_1)$ be a directed graph or ``quiver", and $i\mapsto$ $V_i$ a vector space assigned to each vertex in $Q_0$. The closures of the orbits of $\prod_{Q_0} GL(V_i)$ on $\prod_{Q_1}Hom(V_{ta},V_{ha})$ are called quiver loci. \vskip .1in \noindent In the case that $Q$ is an ADE Dynkin diagram, Reineke showed that each quiver loci is the proper image of a homogeneous vector bundle over a flag manifold. Kempf had earlier used this setup to prove geometric statements about such images. \vskip .1in \noindent I'll explain how we've extended Kempf's results to apply to this case, and a formula for the equivariant cohomology class (or ``multidegree'') of a quiver locus. These multidegrees were previously only known in type $A$. \vskip .1in \noindent This work is joint with Mark Shimozono.

Host:

### October 4, 2005

### 2:30 PM

### AP&M 7218

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