Perverse sheaves on symmetric products of the plane, Schur algebras and K-theoryOberseminar Darstellungstheorie

by Tom Braden (UMass Amherst)

Friday Jun 20, 2025, 3:30 PM → 4:30 PM Europe/Berlin

1008 (Mathezentrum)

In analogy with the (generalized) Springer correspondence relating perverse sheaves on a nilpotent cone to representations of the Weyl group, we consider perverse sheaves on the symmetric product of n copies of the plane C^2, constructible with respect to the natural stratification by collision of points. This category is semisimple when the coefficients have characteristic zero, but with positive characteristic coefficients it can be very complicated. We show that this category is equivalent to modules over a convolution algebra given by K-theory of sheaves on the symmetric group, equivariant for the action of Young subgroups on the left and right. Up to Morita equivalence, this algebra has a Schur algebra as a quotient. I will also explain how this algebra arises using the K-theory of Hilbert schemes and a theorem due to Bezrukavnikov-Kaledin and Bridgeland-King-Reid. Joint work with Carl Mautner.