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A categorification of the Vandermonde determinant

Alex Chandler

Fakultät für Mathematik, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

https://doi.org/10.1142/S0218216521410054Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue — Dedicated to the 60th Birthday of J. H. Przvtycki
Guest Editors: M. K. Dabkowski, V. Harizanov, L. H. Kauffman, J. H. Przytycki, R. Sazdanovic and A. Sikora

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 $\vec{x} = (x_{1}, \dots, x_{n})$ $\vec{x} = (x_{1}, \dots, x_{n})$ , we construct a complex of colored smoothings of the two-strand torus link $T_{2, n}$ $T_{2, n}$ in the shape of the Bruhat order on $S_{n}$ $S_{n}$ , and apply a topological quantum field theory (TQFT) to obtain a chain complex whose Euler characteristic is equal to the Vandermonde determinant evaluated at $\vec{x}$ $\vec{x}$ . A generalization to arbitrary link diagrams is given, yielding categorifications of certain generalized Vandermonde determinants.

Keywords:

AMSC: 57M25, 57M27

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