Knizhnik-Zamolodchikov bundles are topologically trivial

TL;DR

This paper proves that certain vector bundles associated with braid group representations are topologically trivial, with partial extensions to generalized braid groups, and reveals a key algebraic property of the symmetric group's representation ring.

Contribution

It establishes the topological triviality of Knizhnik-Zamolodchikov bundles and generalizes this to broader braid group contexts, highlighting a fundamental algebraic structure.

Findings

Knizhnik-Zamolodchikov bundles are topologically trivial

Representation ring of symmetric group generated by alternating powers

Partial generalizations to generalized braid groups

Abstract

We prove that the vector bundles at the core of the Knizhnik-Zamolodchikov and quantum constructions of braid groups representations are topologically trivial bundles. We provide partial generalizations of this result to generalized braid groups. A crucial intermediate result is that the representation ring of the symmetric group on n letters is generated by the alternating powers of its natural n-dimensional representation.