Limit shapes of probability measures in representation theory of U_q(sl_2) at roots of unity

Abstract: Limit shape phenomenon emerges in systems with random behavior. It manifests a formation of the most probable state, where all other macroscopically different states are exponentially improbable. In this talk, we explore such phenomena in the Grothendieck ring of the category of tilting modules for the quantum group U_q(sl_2) with divided powers, where q is an even root of unity. Considering large tensor powers of the defining representation, we describe the most probable trajectory in the main Weyl chamber with respect to the character probability measure and analyze fluctuations around this limit shape.

This talk is based on arXiv:2404.03933, a joint work with A. Lachowska, O. Postnova and N. Reshetikhin.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

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