##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Allen Knutson

#### UCSD

## Totally nonnegative matrices, juggling patterns, and the affine flag manifold

##### Abstract:

Consider $b\times n$ real matrices such that every $b\times b$ minor has nonnegative determinant. Since there are polynomial relations between these minors, not every pattern of zero vs. strictly positive is achievable. In a widely circulated prepreprint, Alex Postnikov gave many ways to index the patterns that are. I'll describe a new indexing, by ``bounded juggling patterns'', which will require a brief foray into the mathematics of juggling (with demonstrations). It turns out that many of the natural concepts from the matrix picture have been known to jugglers for 20 years. Unbounded juggling patterns form a group, the affine Weyl group, and thereby index the Schubert varieties on the (infinite-dimensional) affine flag manifold. I'll explain how the complicated finite-dimensional geometry of Postnikov's stratification is induced from what is actually much more familiar infinite-dimensional geometry.

### April 15, 2008

### 4:00 PM

### AP&M 6402

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