##### Department of Mathematics,

University of California San Diego

****************************

### Representation Theory

## Hanspeter Kraft

#### University of Basel, Switzerland

## Instability in many copies of a representation

##### Abstract:

The set of unstable vectors of a representation $V$ of a reductive group $G$, the so-called {\\it nullcone\\/} $N_V$, contains a lot of information about the geometry of the representation $V$. E.g. if $N_V$ contains finitely many orbits then this holds for every fiber of the quotient morphism $\\pi_V\\colon V \\to V/\\!\\!/ G$. The Hilbert-Mumford criterion allows to describe the nullcone as a union $\\bigcup G V_\\lambda$, using maximal unstable subspaces of $V_\\lambda \\subset V$ annihilated by a 1-parameter subgroup $\\lambda$ of $G$. They correspond to maximal unstable subsets of weights which allows some interesting combinatorics. We will give some methods how to determine the irreducible components $GV_\\lambda$ of the nullcone and will describe their behavior if one considers several copies of a given representation $V$. A rather complete picture is obtained for the so-called $\\theta$ representations studied by Kostant-Rallis and Vinberg. E.g. we were able to show that for the 4-qubits $Q_4:={\\bf C}^2\\otimes {\\bf C}^2\\otimes {\\bf C}^2 \\otimes {\\bf C}^2$ the nullcone has four irreducible components all of dimension 12 for one copy and 12 irreducible components for $k\\geq 2$ copies. These 12 components decompose into 3 orbits under the obvious action of $S_4$ on $Q_4$, each one consisting of 4 elements, of dimensions $8k+4$, $8k+3$ and $8k+1$. (This is joint work with Nolan Wallach.)

Host: Nolan Wallach

### October 7, 2003

### 2:30 PM

### AP&M 7321

****************************