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Maximality Levels of the classical permutation group in the quantum permutation group

J. P. McCarthy

Abstract

Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka--Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category $\mathcal{NC}$ of non-crossing partitions and the category $\mathcal{P}$ of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by $\mathcal{NC}$ together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, at $N=6$, there is no exotic category containing a linear combination of 31 crossing-partition vectors that is distinguished from $\mathcal{NC}$ or $\mathcal{P}$ at moments of order six.

Maximality Levels of the classical permutation group in the quantum permutation group

Abstract

Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka--Krein duality, any counterexample to this Maximality Conjecture must arise from a category strictly intermediate between the category of non-crossing partitions and the category of all partitions. Any such exotic category must therefore contain a linear combination of crossing-partition vectors. The categories generated by together with some such vectors are studied, with a number of generation results. It is shown that no exotic category can contain a linear combination of three crossing-partition vectors, and, at , there is no exotic category containing a linear combination of 31 crossing-partition vectors that is distinguished from or at moments of order six.
Paper Structure (4 sections, 14 theorems, 50 equations)

This paper contains 4 sections, 14 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Compact matrix quantum groups
  3. Partitions via Index Permutation Symmetry
  4. Partitions

Key Result

Theorem 1.1

The classical permutation group $S_4$ is a maximal quantum subgroup of the quantum permutation group $S_4^+$.

Theorems & Definitions (29)

  • Theorem 1.1: Banica--Bichon
  • Conjecture 1.2: The Maximality Conjecture
  • Theorem 1.3: Banica--Curran--Speicher bcs
  • Theorem 1.4: Banica (ban; Th. 7.10, Prop. 8.3)
  • Theorem 1.5: Freslon--Speicher
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1: Woronowicz wo1
  • Definition 2.2
  • ...and 19 more