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

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.

💡 Research Summary

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This paper develops a suite of algebraic and combinatorial tools to give a complete, computer‑free description of certain finite subgroups of the outer reduced Weyl group of the Cuntz algebras (\mathcal O_n). The authors focus on the case (n=4) and level (t=2), where previous work (