Boolean degree 1 functions on some classical association schemes

TL;DR

This paper classifies Boolean degree 1 functions across various classical association schemes, revealing their triviality in many cases and reducing complex classifications to simpler subproblems.

Contribution

It provides a comprehensive classification of Boolean degree 1 functions on multiple association schemes, including Johnson, Grassmann, polar spaces, and bilinear forms graphs, with new reduction techniques.

Findings

Boolean degree 1 functions are trivial on the multislice.

All Boolean degree 1 functions are trivial on Grassmann schemes for certain parameters.

Evidence suggests triviality of Boolean degree 1 functions on polar spaces and bilinear forms graphs.

Abstract

We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as \textit{completely regular strength 0 codes of covering radius 1}, \textit{Cameron--Liebler line classes}, and \textit{tight sets}. We classify all Boolean degree $1$ functions on the multislice. On the Grassmann scheme $J_q(n, k)$ we show that all Boolean degree $1$ functions are trivial for $n \geq 5$, $k, n-k \geq 2$ and $q \in \{ 2, 3, 4, 5 \}$, and that for general $q$, the problem can be reduced to classifying all Boolean degree $1$ functions on $J_q(n, 2)$. We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree $1$ functions are trivial for appropriate choices of the parameters.