A Categorification of the Vandermonde Determinant

Abstract

In the spirit of Bar Natan’s construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers, we construct a commutative diagram in the shape of the Bruhat order whose nodes are colored smoothings of a 2-strand torus link and whose arrows are colored cobordisms. An application of a TQFT to this diagram yields a chain complex whose Euler characteristic is the Vandermonde determinant evaluated at the sequence of positive integers we started with. A generalization to arbitrary link diagrams is given, producing categorifications of certain generalized Vandermonde determinants. We also address functoriality of this construction.