# Calendar

Lie Groups Quantum Mathematics Seminar

## $$Q$$-systems and Generalizations in Representation Theory

Location:  Hill 705
Date & time: Friday, 07 October 2016 at 12:00PM -

### "$$Q$$-systems and Generalizations in Representation Theory"

Time: 12:00 PM
Location: Hill 705
Abstract: Date: Friday, October 7, 2016 Time: 12:00 Room: Hill 705 Tau-functions given as matrix elements for the action of the loop group, $$widehat{GL_2}$$, on two-component Fermionic Fock space are solutions to the $$A_{infty/2}$$ $$Q$$-system. Since $$Q$$-systems are of interest in representation theory and in combinatorics, it is interesting to ask what sort of relations are satisfied by analogous tau-functions, given as matrix elements for the action of $$widehat{GL_3}$$ on three-component Fermionic Fock space. In this talk, we will define these new $$widehat{GL_3}$$ tau-functions and derive the system of discrete equations that they satisfy, and will discuss the progress we have made in analyzing this new system of equations. We will present a conjecture for relations satisfied by $$widehat{GL_n}$$ tau-functions. Since loop groups can be embedded in infinite matrix groups, we can generalize the above by calculating tau-functions given as matrix elements for infinite matrix groups, which will satisfy more general systems of equations than those mentioned above. This is analogous to the way in which the KP and KdV hierarchies are related. We have begun to calculate some of these new systems of equations and will present preliminary results and conjectures in this direction.

This is joint work with Maarten Bergvelt.

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