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A note on the Cuntz algebra automorphisms

Junyao Pan

TL;DR

The paper addresses the combinatorial characterization of stable rank-1 involutions on $[n]^2$ that induce permutative automorphisms of the Cuntz algebra $ O_n$, proving Conjecture 12.2 of Brenti and Conti. It leverages the equivalence that rank-1 stability is captured by the commutation relation $(v\\otimes 1)(1\\otimes v)=(1\\otimes v)(v\\otimes 1)$ in $S([n]^3)$ and applies a contradiction-based analysis to delineate necessary conditions. The main result shows that the involution $u=((a_1,b_1),(a_1,b_2))((a_2,b_3),(a_2,b_4))$ is stable of rank $1$ precisely when either (i) the $a$-indices are disjoint from the $b$-indices or (ii) the $a$-indices form the same set as the $b$-indices, yielding a new six-parameter family of automorphisms for any $n>1$. This advances the explicit construction of reduced Weyl group elements for $Aut( O_n)$ and contributes to the combinatorial understanding of permutative automorphisms.

Abstract

Permutative automorphisms of the Cuntz algebras $\mathcal{O}_n$ are in bijection with the stable permutations of $[n]^k$. They are also the elements of the restricted Weyl group of $Aut(\mathcal{O}_n)$. In this note, we characterize a class of stable involutions of $[n]^2$. More precisely, we prove Conjecture 12.2 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], and thus providing a new family (with $6$ degrees of freedom) of automorphisms of the Cuntz algebras $\mathcal{O}_n$ for any $n>1$.

A note on the Cuntz algebra automorphisms

TL;DR

The paper addresses the combinatorial characterization of stable rank-1 involutions on that induce permutative automorphisms of the Cuntz algebra , proving Conjecture 12.2 of Brenti and Conti. It leverages the equivalence that rank-1 stability is captured by the commutation relation in and applies a contradiction-based analysis to delineate necessary conditions. The main result shows that the involution is stable of rank precisely when either (i) the -indices are disjoint from the -indices or (ii) the -indices form the same set as the -indices, yielding a new six-parameter family of automorphisms for any . This advances the explicit construction of reduced Weyl group elements for and contributes to the combinatorial understanding of permutative automorphisms.

Abstract

Permutative automorphisms of the Cuntz algebras are in bijection with the stable permutations of . They are also the elements of the restricted Weyl group of . In this note, we characterize a class of stable involutions of . More precisely, we prove Conjecture 12.2 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], and thus providing a new family (with degrees of freedom) of automorphisms of the Cuntz algebras for any .
Paper Structure (3 sections, 4 theorems, 20 equations)

This paper contains 3 sections, 4 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.2
  3. Acknowledgement

Key Result

Theorem 1.2

Let $(a_1, b_1),(a_1, b_2),(a_2, b_3),(a_2, b_4)$ be four distinct elements in $[n]^2$ with $a_1 \neq a_2$. Then the involution $((a_1, b_1),(a_1, b_2)) ((a_2, b_3),(a_2, b_4))\in S([n]^2)$ is stable of rank $1$ if and only if one of the following conditions is satisfied: $(i)$$\{a_1, a_2\} \cap \{b

Theorems & Definitions (5)

  • Conjecture 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3