# Mathematics Colloquia and Seminars

### A scheme related to the Brauer loop model

Algebra & Discrete Mathematics

 Speaker: Brian Rothbach, UC Berkeley Location: 1147 MSB Start time: Thu, Jan 26 2006, 12:10PM

The Brauer loop model is a quantum integral stochastic process introduced by de Gier and Nienhuis. They observed that certain entries of the Perron-Frobenius eigenvector of this model matched degrees of the components of the upper-upper scheme of Knutson. In order to obtain all entries of the eigenvector, Knutson and Zinn-Justin introduced the Brauer loop scheme, which can be described as the set of all matrices with M · M=0, where · is a degeneration of normal matrix multiplication. They showed that the components correspond to states in the loop model, and the multidegrees of each component are shown inductively to agree with a polynomial generalization of the entries of the Perron-Frobenius eigenvector. In particular, this gives a formula for the multidegree of the commuting variety.

This talk will assume no prior knowledge of the Brauer loop model, the upper-upper scheme, or multidegrees, and hopefully should be accessible.