The Frucht property in the quantum group setting

“…4.3. In fact, it seems that certain automorphism groups are "quantum excluding" [26]. Generalizing the above question, we may ask: Problem 3.10 Can you find a graph whose automorphism group is either the trivial one {e}, a cyclic group Z k or the symmetric group S 3 (also for the alternating groups A n , in particular for A 5 , it is open) and a quantum automorphism matrix of such that one of its entries u i j with i = j is nonzero?…”

Section: Problem 38 Can You Find a Quantum Permutation Matrix With No...mentioning

confidence: 99%

Quantum Permutation Matrices

Weber

2023

Multivariable Operator Theory

View full text Add to dashboard Cite

Quantum permutations arise in many aspects of modern "quantum mathematics". However, the aim of this article is to detach these objects from their context and to give a friendly introduction purely within operator theory. We define quantum permutation matrices as matrices whose entries are operators on Hilbert spaces; they obey certain assumptions generalizing classical permutation matrices. We give a number of examples and we list many open problems. We then put them back in their original context and give an overview of their use in several branches of mathematics, such as quantum groups, quantum information theory, graph theory and free probability theory.

show abstract

“…An interesting question to ask is which quantum permutation groups can be realized as the quantum automorphism group of some graph. This question has, for example, been considered in [4].…”

Section: Introductionmentioning

confidence: 99%

Solution group representations as quantum symmetries of graphs

Roberson

Schmidt

2022

Journal of London Math Soc

View full text Add to dashboard Cite

In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex-and edge-colored graph with certain properties, we construct an uncolored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the first-known example of a graph that has quantum symmetry and finite quantum automorphism group. Furthermore, we construct a pair of quantum isomorphic, non-isomorphic graphs that both have no quantum symmetry. M S C 2 0 2 0 46LXX (primary), 20B25, 05CXX (secondary)

show abstract

The Frucht property in the quantum group setting

Cited by 6 publications

References 33 publications

Quantum symmetries of Cayley graphs of abelian groups

Quantum symmetries of Cayley graphs of abelian groups

Quantum Permutation Matrices

Solution group representations as quantum symmetries of graphs

Contact Info

Product

Resources

About