arXiv subjects

J. P. McCarthy

Publications and source records attributed to J. P. McCarthy.

9 recordsLinked to original sources

arXiv2026

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

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.

math.QA↗

arXiv2025

Analysis for idempotent states on quantum permutation groups

Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele's proof yields an idempotent state in any non-empty weak*-compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

math.OA↗

arXiv2024

Tracing the orbitals of the quantum permutation group

Using a suitably noncommutative flat matrix model, it is shown that the quantum permutation group has free orbitals: that is, a monomial in the generators of the algebra of functions can be zero for trivial reasons only. It is shown that any strictly intermediate quantum subgroup between the classical and quantum permutation groups must have free three-orbitals. This is used to give explicit formulae for the Haar state on degree four monomials that hold for such intermediate quantum subgroups as well as the quantum permutation group itself.

math.QA↗

arXiv2021

A state-space approach to quantum permutations

An exposition of quantum permutation groups where an alternative to the 'Gelfand picture' of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and posits that states on the algebra of continuous functions on a quantum permutation group can be interpreted as quantum permutations. This interpretation allows talk of an element of a compact quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated with the Kac-Paljutkin quantum group, the duals of finite groups, as well as by other finite quantum group phenomena.

math.QA↗

arXiv2021

The Frucht property in the quantum group setting

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

math.OA↗

arXiv2021

The Ergodic Theorem for Random Walks on Finite Quantum Groups

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

math.QA↗

arXiv2018

Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups

A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group $\widehat{S_n}$ as well as on the truly quantum groups of Sekine.

math.QA↗

arXiv2018

Random Walks on Finite Quantum Groups: Diaconis-Shahshahani Theory for Quantum Groups

The concept of a random walk on a finite group converging to random - and a way of measuring the distance to random after $k$ transitions - is generalised from the classical case to the case of random walks on finite quantum groups. A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis--Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups $\mathbb{Z}_n$, $\mathbb{Z}_2^n$, the dual group $\widehat{S_n}$ as well as the `truly' quantum groups of Kac & Paljutkin, and Sekine.

math.QA↗

arXiv2015

The Cut-Off Phenomenon in Random Walks on Finite Groups

How many shuffles are needed to mix up a deck of cards? This question may be answered in the language of a random walk on the symmetric group, $S_{52}$. This generalises neatly to the study of random walks on finite groups, themselves a special class of Markov chains. Ergodic random walks exhibit nice limiting behaviour, and both the quantitative and qualitative aspects of the convergence to this limiting behaviour is examined. A particular qualitative behaviour, the cut-off phenomenon, occurs in many examples. For random walks exhibiting this behaviour, after a period of time, convergence to the limiting behaviour is abrupt.

math.PR↗