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In May 2017 I wrote down some problems that I hoped to look at in my study of random walks on quantum groups. Following work of Amaury Freslon, a number of these questions have been answered. In exchange for solving these problems, Amaury has very kindly suggested some other problems that I can work on. The below hopes to categorise some of these problems and their status.

## Solved!

- Show that the total variation distance is equal to the projection distance. Amaury has an a third proof. Amaury suggests that this should be true in more generality than the case of being absolutely continuous (of the form for all and a unique ). If the Haar state is no longer tracial Amaury’s proof breaks down (and I imagine so do the two others in the link above). Amaury believes this is true in more generality and says perhaps the Jordan decomposition of states will be useful here.
- Prove the Upper Bound Lemma for compact quantum groups of Kac type. Achieved by Amaury.
- Attack random walks with conjugate invariant driving probabilitys: achieved by Amaury.
- Look at quantum generalisations of ‘natural’ random walks and shuffles. Solved is probably too strong a word, but Amaury has started this study by looking at a generalisation of the random transposition shuffle. As I suggested in Seoul, Amaury says: “One important problem in my opinion is to say something about analogues of classical random walks on (for instance the random transpositions or riffle shuffle)”. Amaury notes that “we are blocked by the counit problem. We must therefore seek bounds for other distances. As I suggest in my paper, we may look at the norm of the difference of the transition operators. The -estimate that I give is somehow the simplest thing one can do and should be thought of as a “spectral gap” estimate. Better norms would be the norms as operators on or even better, the completely bounded norm. However, I have not the least idea of how to estimate this.”

## Results to be Improved

- I have recently received an email from Isabelle Baraquin, a student of Uwe Franz, pointing out a small error in the thesis (a basis-error with the Kac-Paljutkin quantum groups).
- Recent calculations suggest that the lower bound for the random walk on the dual of is effective at while the upper bound shows the walk is random at time order . This is still a very large gap but at least the lower bound shows that this walk does converge very slowly.
- Get a much sharper lower bound for the random walk on the Sekine family of quantum groups studied in Section 5.7. Projection onto the ‘middle’ of the factor may provide something of use. On mature reflection, recognising that the application of the upper bound lemma is dominated by one set of terms in particular, it should be possible to use cruder but more elegant estimates to get the same upper bound except with lighter calculations (and also a smaller — see Section 5.7).

## More Questions on Random Walks

- Irreducibility is harder than the classical case (where ‘not concentrated’ on a subgroup is enough). Can anything be said about aperiodicity in the quantum case? (U. Franz).
- Prove an Ergodic Theorem (Theorem 1.3.2) for Finite Quantum Groups. Extend to Compact Quantum Groups. It is expected that the conditions may be more difficult than the classical case. However, it may be possible to use Diaconis-Van Daele theory to get some results in this direction. It should be possible to completely analyse some examples (such as the Kac-Paljutkin quantum group of order 8).This will involve a study of subgroups of quantum groups as well as
*normal*quantum subgroups and cosets. - Look at a random walk on the Sekine quantum groups with an -dependent driving probability and see if the cut-off phenomenon (Chapter 4) can be detected. This will need good lower bounds for , some cut-off time.
- Convolutions of the Random Distribution: such a study may prove fruitful in trying to find the Ergodic Theorem. See Section 6.5.
**Amaury**mentions the problem of considering random walks associated to non-central states (in the compact case). “The difficulty is first to build non-central states (I do not have explicit examples at hand but Uwe Franz said he had some) and second to be able to compute their Fourier transform. Then, the computations will certainly be hard but may still be doable.”- A study of the Cesaro means: see Section 6.6.
- Spectral Analysis: it should be possible to derive crude bounds using the spectrum of the stochastic operator. More in Section 6.2.

## Future Work (for which I do not yet have the tools to attack)

**Amaury/Franz**Something perhaps more accessible is to investigate quantum homogeneous spaces. The free sphere is a noncommutative analogue of the usual sphere and a quantum homogeneous space for the free orthogonal quantum group. We can therefore define random walks on it and the whole machinery of Gelfand pairs might be available. In particular, Caspers gave a Plancherel theorem for Gelfand pairs of locally compact quantum groups which should apply here yielding an Upper Bound Lemma and then the problem boils down to something which should be close to my computations. There are probably works around this involving Adam Skalski and coauthors.**Amaury:**If one can prove a more general total variation distance equal to half one norm result, then Amaury suggests one can consider random walks on compact quantum groups which are not of Kac type. The Upper Bound Lemma will then involve matrices measuring the modular theory of the Haar state and some (but not all) dimensions in the formulas must be replaced by quantum dimensions. The main problem here is to define explicit central states since there is no Haar-state preserving conditional expectation onto the central algebra. However, there are tools from monoidal equivalence to do this.

Amaury Freslon has put a pre-print on the arXiv, *Cut-off phenomenon for random walks on free orthogonal quantum groups*, that answers so many of these questions, some of which appeared as natural further problems in my PhD thesis.

It really is a fantastic paper and I am delighted to see my PhD work cited: it appears that while I may have taken some of the low hanging fruit, Amaury has really extended these ideas and has developed some fantastic examples: all beyond my current tools.

This pre-print gives me great impetus to draft a pre-print of my PhD work, hopefully for publication. I am committed to improving my results and presentation, and Amaury’s paper certainly provides some inspiration is this direction.

As things stand I do not have to tools to develop results as good as Amaury’s. Therefore I am trying to develop my understanding of compact quantum groups and their representation theory. Afterwards I can hopefully study some of the remaining further problems mentioned in the thesis.

As suggested by Uwe Franz, representation theoretic methods (such as presented by Diaconis (1988) for the classical case), might be useful for analysing random walks on quantum homogeneous spaces.

## Distances between Probability Measures

Let be a finite quantum group and be the set of states on the -algebra .

The algebra has an invariant state , the dual space of .

Define a (bijective) map , by

,

for .

Then, where and , define the total variation distance between states by

.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative machinary shows that:

.

(supremum presentation)

In the classical case, using the test function , where , we have the probabilists’ preferred definition of total variation distance:

.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in , and therefore the classical total variation distance is equal to:

.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

## Quantum Projection Distance Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a factor of might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

Just back from a great workshop at Seoul National University, I am just going to use this piece to outline in a relaxed manner my key goals for my work on random walks on quantum groups for the near future.

In the very short term I want to try and get a much sharper lower bound for my random walk on the Sekine family of quantum groups. I believe the projection onto the ‘middle’ of the might provide something of use. On mature reflection, recognising that the application of the upper bound lemma is dominated by one set of terms in particular, it should be possible to use cruder but more elegant estimates to get the same upper bound except with lighter calculations (and also a smaller — see Section 5.7).

I also want to understand how sharp (or otherwise) the order convergence for the random walk on the dual of is — sounds awfully high. Furthermore it should be possible to get a better lower bound that what I have.

It should also be possible to redefine the quantum total variation distance as a supremum over projections subsets via . If I can show that for a positive linear functional that then using these ideas I can. More on this soon hopefully. No, this approach won’t work. (I have since completed this objective with some help: see here).

The next thing I might like to do is look at a random walk on the Sekine quantum groups with an -dependent driving probability and see if I can detect the cut-off phenomenon (Chapter 4). This will need good lower bounds for , some cut-off time.

Going back to the start, the classical problem began around 1904 with the question of Markov:

Which card shuffles mix up a deck of cards and cause it to ‘go random’?

For example, the perfect riffle shuffle does not mix up the cards at all while a riffle shuffle done by an amateur will.

In the context of random walks on classical groups this question is answered by the Ergodic Theorem 1.3.2: when the driving probability is not concentrated on a subgroup (irreducibility) nor the coset of a normal subgroup (aperiodicity).

Necessary and sufficient conditions on the driving probability for the random walk on a *quantum *group to converge to random are required. It is expected that the conditions may be more difficult than the classical case. However, it may be possible to use Diaconis-Van Daele theory to get some results in this direction. It should be possible to completely analyse some examples (such as the Kac-Paljutkin quantum group of order 8).

This will involve a study of subgroups of quantum groups as well as *normal *quantum subgroups.

It should be straightforward to extend the Upper Bound Lemma (Lemma 5.3.8) to the case of compact Kac algebras. Once that is done I will want to look at quantum generalisations of ‘natural’ random walks and shuffles.

I intend also to put the PhD thesis on the Arxiv. After this I have a number of options as regard to publishing what I have or maybe waiting a little while until I solve the above problems — this will all depend on how my further study progresses.

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

**Abstract** *A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & 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. These ideas are generalised to the case of finite quantum groups.*

After a long time I have finally completed my PhD studies when I handed in my hardbound thesis (a copy of which you can see here).

It was a very long road but thankfully now the pressure is lifted and I can enjoy my study of quantum groups and random walks thereon for many years to come.

**The following is taken (almost) directly from the first draft of my PhD thesis.**

## The Quantisation Functor

This functor can be used to motivate the correct notion of (the algebra of functions on) a *quantum group.* Note that the ‘quantised’ objects that are arrived at via this ‘categorical quantisation’ are nothing but the established definitions so this section should be considered as little more than a motivation. The author feels that introductory texts on quantum groups could include these ideas and that is why they are included here. This quantisation is the translation of statements about a finite group, into statements about the algebra of functions on , .

This notion of quantisation sits naturally in category theory where two functors — the functor and the dual functor — lead towards a satisfactory quantisation.

I have finally finished the first draft of my PhD thesis. My advisor Dr Stephen Wills is presently reading through it and will get back to me with his comments in the next few weeks. The project was successful in that I managed to prove the Diaconis-Shahshahani Upper Bound Lemma for finite quantum groups… how successful my application of the Lemma to concrete examples is probably open to debate. First draft of abstract and introduction — without references — below the fold.

Let be a finite quantum group described by with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) . Under the assumption of finiteness, there is a unique Haar state, on .

# Representation Theory

A *representation* of is a linear map that satisfies

The dimension of is given by . If has basis then we can define the *matrix elements* of by

One property of these that we will use it that .

Two representations and are said to be *equivalent*, , if there is an invertible *intertwiner* between them. An intertwiner between and is a map such that

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if is a maximal family of pairwise inequivalent irreducible representation that is a basis of . When we refer to “the matrix elements” we always refer to such a family. We define the span of as , the *space of matrix elements of* .

Given a representation , we define its *conjugate*, , where is the conjugate vector space of , by

so that the matrix elements of are .

Timmermann shows that the matrix elements have the following orthogonality relations:

- If and are inequivalent then for all and .
- If is such that the conjugate, , is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

This second relation is more complicated without the assumption and refers to the entries and trace of an intertwiner from to the coreprepresention with matrix elements . If , then this intertwiner is simply the identity on and so the the entries and the trace is .

Denote by the set of unitary equivalence classes of irreducible unitary representations of . For each , let be a representative of the class where is the finite dimensional vector space on which acts.

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