Hypercube subgroups of (outer) reduced Weyl groups of the Cuntz algebras

TL;DR

This paper introduces algebraic and combinatorial tools to analyze quadratic subgroups of the reduced Weyl group of Cuntz algebras, providing explicit descriptions and generators, especially for the case n=4.

Contribution

It offers a detailed theoretical framework for understanding specific subgroups of the Weyl group, including a classification and generator sets, complementing previous computational results.

Findings

Explicit description of quadratic subgroups of the Weyl group

Generators for these subgroups are provided

A new interpretation of previously tabulated groups for n=4

Abstract

We develop some tools, of an algebraic and combinatorial nature, which enable us to obtain a detailed description of certain quadratic subgroups of the (outer) reduced Weyl group of the Cuntz algebra ${\mathcal O}_n$. In particular, for $n=4$ our findings give a self-contained theoretical interpretation of the groups tabulated in [AJS18], which were obtained with the help of a computer. For each of these groups we provide a set of generators. A prominent role in our analysis is played by a certain family of subgroups of the symmetric group of a discrete square which we call bicompatible.