##### Department of Mathematics,

University of California San Diego

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### Math 256 - Representation Theory

## Karin Baur

#### UCSD

## Representations of classical groups: tensor products and minimal orbits

##### Abstract:

We consider tensor products $V_{\\lambda}\\otimes V_{\\mu}$ of irreducible representations of a classical group $G$. In general, such a tensor product decomposes in irreducible components. It is a fundamental question how the components are embedded in the tensor product. Of special interest is the so-called Cartan component $V_{\\lambda+\\mu}$. It appears exactly once in the decomposition. On the other hand, one can look at decomposable tensors (tensors of the form $v\\otimes w$) in the tensor product. A natural question arising here is the following: are the decomposable tensors in the Cartan component given as the closure of the minimal orbit in $V_{\\lambda+\\mu}$? If this is the case we say that the Cartan component is small. We give a characterization and a combinatorial description of tensor products with small Cartan components. In particular, we show that for general $\\lambda$, $\\mu$, Cartan components are small.

Host: Wee Teck Gan

### November 4, 2003

### 1:30 PM

### AP&M 7321

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