Complexity Measures on the Symmetric Group and Beyond

TL;DR

This paper extends complexity measures to the symmetric group and related domains, establishing polynomial relations among them and applying these insights to characterize functions and intersecting families.

Contribution

It generalizes classical complexity measure relations to new domains and introduces methods to include sensitivity, also extending characterizations of specific functions and families.

Findings

Polynomial relations among complexity measures on the symmetric group.

Extension of Boolean degree 1 function characterization to the perfect matching scheme.

Simplified characterization of maximum-size t-intersecting families.

Abstract

We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang's sensitivity theorem using "pseudo-characters", which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size $t$-intersecting families in the symmetric group and the perfect matching scheme.