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Curves inside DAHA and EHA, skein relations, and symmetric functions

Algebra & Discrete Mathematics

Speaker: Pavel Galashin, UCLA
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Location: 1147 MSB
Start time: Mon, May 15 2023, 4:10PM

I will explain how a curve in a rectangle gives rise to a symmetric function depending on q and t, via a recent construction of Morton--Samuelson. The construction passes through the double affine Hecke algebra (DAHA) and the elliptic Hall algebra (EHA). An interesting special case arises when the curve is convex, in which case it corresponds to a positroid variety in the Grassmannian. Conjecturally, in that case, the symmetric function is Schur-positive, and one of the Schur coefficients computes Dyck paths above the curve. I will also briefly mention relations between this symmetric function and knot invariants. No background on the above objects will be assumed. Joint work with Thomas Lam.