# 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 - 12:11PM

Darlayne Addabbo, University of Illinois at Urbana-Champaign: Date: Friday, October 7, 2016Time: 12:00Room: Hill 705Tau-functions given as matrix elements for the action of the loopgroup, $$widehat{GL_2}$$, on two-component Fermionic Fock space aresolutions to the $$A_{infty~2}$$ $$Q$$-system. Since $$Q$$-systems are ofinterest in representation theory and in combinatorics, it isinteresting to ask what sort of relations are satisfied by analogoustau-functions, given as matrix elements for the action of$$widehat{GL_3}$$ on three-component Fermionic Fock space. In thistalk, we will define these new $$widehat{GL_3}$$ tau-functions andderive the system of discrete equations that they satisfy, and willdiscuss the progress we have made in analyzing this new system ofequations. 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 cangeneralize the above by calculating tau-functions given as matrixelements for infinite matrix groups, which will satisfy more generalsystems of equations than those mentioned above. This is analogous tothe way in which the KP and KdV hierarchies are related. We have begunto calculate some of these new systems of equations and will presentpreliminary results and conjectures in this direction.

This is joint work with Maarten Bergvelt.

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