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

Francesco Brenti, Roberto Conti, Gleb Nenashev

TL;DR

This work advances the combinatorial understanding of automorphisms of Cuntz algebras by analyzing finite-order subgroups of the outer reduced Weyl group via level-2 permutations on the hypercube $[n]^t$, with a detailed, explicit description for ${\cal O}_4$. It introduces and exploits bicompatible subgroups of $S([n]^2)$ to construct and classify 46 maximal subgroups of ${\rm Out}({\cal O}_4)$ arising from ${\cal P}_4^2$, providing complete generators and proving a conjecture from prior work. The approach unifies several families of subgroups (lines 1–7) and embeds them in a broader framework that explains when such subgroups are distinct in outer automorphism groups. The results pave the way for analogous classifications at other levels and for larger $n$, with potential applications to the broader study of automorphisms of Cuntz algebras and their symmetry structures.

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.

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

TL;DR

This work advances the combinatorial understanding of automorphisms of Cuntz algebras by analyzing finite-order subgroups of the outer reduced Weyl group via level-2 permutations on the hypercube , with a detailed, explicit description for . It introduces and exploits bicompatible subgroups of to construct and classify 46 maximal subgroups of arising from , providing complete generators and proving a conjecture from prior work. The approach unifies several families of subgroups (lines 1–7) and embeds them in a broader framework that explains when such subgroups are distinct in outer automorphism groups. The results pave the way for analogous classifications at other levels and for larger , with potential applications to the broader study of automorphisms of Cuntz algebras and their symmetry structures.

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 . In particular, for 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.
Paper Structure (16 sections, 38 theorems, 138 equations, 1 table)

This paper contains 16 sections, 38 theorems, 138 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Horizontal and vertical cycles
  4. Subgroups and compatibility
  5. Quadratic subgroups
  6. The first line
  7. The second line
  8. The third line
  9. The fourth line
  10. The fifth line
  11. The sixth line
  12. The seventh line
  13. Bicompatible subgroups and embeddings
  14. Bicompatible subgroups
  15. Bicompatible embeddings of $S_4$ in $S([4]^2)$
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $u \in {\cal P}_n^t$ be such that $\lambda_u \in {\rm Aut}({\cal O}_n)$, and let $v \in {\cal P}_n^t$ and $w \in {\cal U}({\cal O}_n)$ be such that ${\rm Ad}(w) \lambda_u = \lambda_v$ (i.e., $\pi(\lambda_u) = \pi(\lambda_v)$). Then $w \in {\cal P}_n^{t-1}$.

Theorems & Definitions (71)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 61 more