Elliptic stable envelope for Hilbert scheme of points in the complex plane and 3D mirror symmetry

Abstract: In this talk I discuss the elliptic stable envelope classes of torus fixed points in the Hilbert scheme of points in the complex plane. I describe the 3D-mirror self-duality of the elliptic stable envelopes. The K-theoretic limits of these classes provide various special bases in the space of symmetric polynomials, including well known bases of Macdonald or Schur functions. The mirror symmetry then translates to new symmetries for these functions. In particular, I outline a proof of conjectures by E.Gorsky and A.Negut on "Infinitesimal change of stable basis'', which relate the wall R-matrices of the Hilbert scheme with the Leclerc-Thibon involution for $U_q(\mathfrak{gl}_b).$

algebraic geometry

Audience: researchers in the topic

UC Davis algebraic geometry seminar