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I gave the same talk to the Non local games seminar and C*-Days in Prague:
Abstract: The first part of the talk will give a post hoc motivation for Banica’s 2005 definition of the quantum automorphism group of a finite graph, and in doing so attempt to build a good intuition for quantum automorphism. Frucht in 1939 showed that every finite group is the automorphism group of a finite graph, and a natural pursuit in the theory of quantum automorphism groups is to establish quantum analogues of this result. Based on a joint work with Banica, the second part of the talk will address this question.
The slides are subtly different: the one for C*-Days is the latest version:
Abstract: Using a suitably non-commutative 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 strict intermediate quantum subgroup between the classical and quantum permutation groups must have free three orbitals, and this is used to derive some elementary bounds for the Haar state on degree four monomials in such quantum permutation groups.
Abstract: 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.
BCRI Mini-Symposium: Noncommutative Probability & Quantum Information
Monday, 10th October 2022 from 12:00 to 15:00
Organizers: Claus Koestler (UCC), Stephen Wills (UCC)
SPEAKER: J.P. McCarthy (Munster Technological University)
TITLE: The Kawada-Itô theorem for finite quantum groups.
ABSTRACT: 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 compact quantum groups, where a state on the algebra of functions plays the role of the driving probability. A random walk on a compact quantum group can fail to be irreducible without being concentrated on a proper quantum subgroup. In this talk we will explore this phenomenon. Time allowing, we will talk about periodicity, and as a conclusion, I give necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group in terms of the support projection of the driving state.
In the end the talk (below) didn’t quite match the abstract.
Quantum Group Seminar, Monday 24 January, 2022.
Abstract: A classical theorem of Frucht states that every finite group is the automorphism group of a finite graph. Is every quantum permutation group the quantum automorphism group of a finite graph? In this talk we will answer this question with the help of orbits and orbitals.
This talk is based on joint work with Teo Banica.
This article has been accepted to Expositiones Mathematicae.
Abstract
In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.
Anyone clicking here before March 04, 2022 will be taken directly to the latest version of the article on ScienceDirect, which they are welcome to read or download. No sign up, registration or fees are required.
Abstract
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.
with Teo Banica, Glasgow Math J., to appear. Arxiv link here.
Abstract
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.
Arxiv link here.
Abstract
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.
Link to journal here.
J.P. McCarthy: Math Page 
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