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# Q-systems and Generalizations in Representation Theory

### Mathematical Physics and Probability Seminar

Q-systems and Generalizations in Representation Theory
Series: Mathematical Physics and Probability Seminar
Location: ONLINE
Presenter: Darlayne Addabbo, University of Arizona

I will define hierarchies of difference equations whose solutions, called $\tau$-functions, are matrix elements for the action of loop groups, $\widehat{GL_n}$, on $n$-component fermionic Fock space. In the simplest case, $n=2$, these $\tau$-functions are determinants of Hankel matrices, and one can apply the famous Desnanot-Jacobi identity to see that they satisfy a $Q$-system. $Q$-systems are discrete dynamical systems that appear in many areas of mathematics, so it is interesting to study the more general, $n>2$ hierarchies. I will discuss these new hierarchies of difference equations and the progress I have made in investigating them. (The first part of this talk is based on joint work with Maarten Bergvelt.)

(https://arizona.zoom.us/j/99811812123)