Quantum permutation groups

2020/12/20 by Teodor Banica, Banica, Teo · 2 citations
Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2012.10975

openalex publication_date 2020/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The permutation group S_N has a quantum analogue S_N⁺, which is infinite at N≥4. We review the known facts regarding S_N⁺, and notably its easiness property, Weingarten calculus, and the isomorphism S₄⁺=SO₃^-1 and its consequences. We discuss then the structure of the closed subgroups G⊂ S_N⁺, and notably of the quantum symmetry groups of finite graphs G⁺(X)⊂ S_N⁺, with particular attention to the quantum reflection groups H_N^s+. We also discuss, more generally, the quantum symmetry groups S_Z⁺ of the finite quantum spaces Z, and their closed subgroups G⊂ S_Z⁺, with particular attention to the quantum graph case, and to quantum reflection groups.

Cited by

• Duals of finite quantum permutation groups
• Quantum partial automorphisms of finite graphs

Related

• Infinite quantum permutations
• Models of quantum permutations
• Duals of finite quantum permutation groups
• Topological generation and matrix models for quantum reflection groups
• Quantum automorphism groups of finite graphs