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Algebra Seminar

Tropical Fock-Goncharov coordinates for linear bases of SL3-skein algebras of surfaces

Daniel Douglas - Yale University

Location:  Hill Center Room 705
Date & time: Wednesday, 27 April 2022 at 2:00PM - 3:00PM

Abstract - For a finite-type surface S, we study a preferred basis for the commutative algebra C[R_SL3(S)] of regular functions on the SL3(C)-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface S. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety. This is joint work with Zhe Sun.

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