Jonathan Brundan

Highest weight categories arising from Khovanov's diagram algebra.

Abstract:

I will discuss the quasi-hereditary covers of Khovanov's diagram algebras and their
generalisations. These are standard Koszul algebras with an explicit cellular basis given in terms of a diagram calculus. They were discovered by Khovanov, as part of his work categorifying the Jones polynomial in knot theory. I will try to explain some of the connections between these algebras and Grassmannians, parabolic category O and the general linear supergroup. This is a report on joint work with Catharina Stroppel.

References:

A. Lascoux and M.-P. Schutzenberger, Polynomes de Kazhdan et Lusztig pour les grassmanniennes, Asterisque 87-88 (1981), 249-266.

M. Khovanov, A functor-valued invariant of tangles, Alg. Geom. Topology 2 (2002), 665-741.

T. Braden, Perverse sheaves on Grassmannians, Canad. J. Math. 54 (2002), 493-532.

J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2003), 185-231.

C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homology; arXiv:math/0608234v2.

J. Brundan and A. Kleshchev, Schur-Weyl duality for higher levels; arXiv:math/0605217v2.