Orthogonal basis for functions over a slice of the Boolean hypercube

TL;DR

This paper introduces an orthogonal basis for functions over a slice of the Boolean hypercube, which also serves as eigenvectors for Johnson and Kneser graphs, simplifying existing proofs of Friedgut's theorem.

Contribution

The paper develops a new orthogonal basis for functions on slices of the Boolean hypercube that connects to graph eigenvectors and simplifies theoretical proofs.

Findings

Basis is orthogonal for functions over the slice

Basis consists of eigenvectors for Johnson and Kneser graphs

Streamlines proof of Friedgut's theorem for slices

Abstract

We present an orthogonal basis for functions over a slice of the Boolean hypercube. Our basis is also an orthogonal basis of eigenvectors for the Johnson and Kneser graphs. As an application of our basis, we streamline Wimmer's proof of Friedgut's theorem for slices of the Boolean hypercube.