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Slides of a talk given to the Functional Analysis seminar in Besancon.

Some of these problems have since been solved.

### “e in support” implies convergence

Consider a $\nu\in M_p(G)$ on a finite quantum group such that where

$M_p(G)\subset \mathbb{C}\varepsilon \oplus (\ker \varepsilon)^*$,

$\nu=\nu(e)\varepsilon+\psi$ with $\nu(e)>0$. This has a positive density of trace one (with respect to the Haar state $\int_G\in M_p(G)$), say

$\displaystyle a_\nu=\nu(e)\eta+b_\psi\in \mathbb{C}\eta\oplus \ker \varepsilon$,

where $\eta$ is the Haar element.

An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so $b_\psi\geq 0$. Assume that $b_\psi\neq 0$ (if $b_\psi=0$, then $\psi=0\Rightarrow \nu=\varepsilon\Rightarrow \nu^{\star k}=\varepsilon$ for all $k$ and we have trivial convergence)

Therefore let

$\displaystyle a_{\tilde{\psi}}:=\frac{b_\psi}{\int_G b_\psi}$

be the density of $\tilde{\psi}\in M_p(G)$.

Now we can explicitly write

$\displaystyle \nu=\nu(e)\varepsilon+(1-\nu(e))\tilde{\psi}$.

This has stochastic operator

$P_\nu=\nu(e)I_{F(G)}+(1-\nu(e))P_{\tilde{\psi}}$.

Let $\lambda$ be an eigenvalue of $P_\nu$ of eigenvector $a$. This yields

$\nu(e)a+(1-\nu(e))P_{\tilde{\psi}}(a)=\lambda a$

and thus

$\displaystyle P_{\tilde{\psi}}a=\frac{\lambda-\nu(e)}{1-\nu(e)}a$.

Therefore, as $a$ is also an eigenvector for $P_{\tilde{\psi}}$, and $P_{\tilde{\psi}}$ is a stochastic operator (if $a$ is an eigenvector of eigenvalue $|\lambda|>1$, then $\|P_\nu a\|_1=|\lambda|\|a\|_1\leq \|a\|_1$, contradiction), we have

$\displaystyle \left|\frac{\lambda-\nu(e)}{1-\nu(e)}\right|\leq 1$

$\Rightarrow |\lambda-\nu(e)|\leq 1-\nu(e)$.

This means that the eigenvalues of $P_\nu$ lie in the ball $B_{1-\nu(e)}(\nu(e))$ and thus the only eigenvalue of magnitude one is $\lambda=1$, which has (left)-eigenvector the stationary distribution of $P_\nu$, say $\nu_\infty$.

If $\nu$ is symmetric/reversible in the sense that $\nu=\nu\circ S$, then $P_\nu$ is self-adjoint and has a basis of (left)-eigenvectors $\{\nu_\infty=:u_1,u_2,\dots,u_{|G|}\}\subset \mathbb{C}G$ and we have, if we write $\nu=\sum_{t=1}^{|G|}a_tu_t$,

$\displaystyle \nu^{\star k}=\sum_{t=1}^{|G|}a_t\lambda_t^ku_t$,

which converges to $a_1\nu_\infty$ (so that $a_1=1$).

If $\nu$ is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers $P_{\nu}^k$ converge and thus so do the $\nu^{\star k}$ $\bullet$

### Total Variation Decrasing

Uses Simeng Wang’s $\|a\star_Ab\|_1\leq \|a\|_1\|b\|_1$. Result holds for compact Kac if the state has a density.

### Periodic $e^2$ is concentrated on a coset of a proper normal subgroup of $\mathfrak{G}_0$

$e_2+e_4$ is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by $\langle e_1,e_3\rangle$

### Supported on Subgroup implies Reducible

I believe I have a full proof that reducible is equivalent to supported on a pre-subgroup.

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum GroupsJournal of Fourier Analysis and Applications, doi: 10.1007/s00041-019-09670-4 (earlier preprint available here)

Abstract

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 Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that 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.

In a recent preprint, Haonan Zhang shows that if $\nu\in M_p(Y_n)$ (where $Y_n$ is a Sekine Finite Quantum Group), then the convolution powers, $\nu^{\star k}$, converges if

$\nu(e_{(0,0)})>0$.

The algebra of functions $F(Y_n)$ is a multimatrix algebra:

$F(Y_n)=\left(\bigoplus_{i,j\in\mathbb{Z}_n}\mathbb{C}e_{(i,j)}\right)\oplus M_n(\mathbb{C})$.

As it happens, where $a=\sum_{i,j\in\mathbb{Z}_n}x_{(i,j)}e_{(i,j)}\oplus A$, the counit on $F(Y_n)$ is given by $\varepsilon(a)=x_{(0,0)}$, that is $\varepsilon=e^{(0,0)}$, dual to $e_{(0,0)}$.

To help with intuition, making the incorrect assumption that $Y_n$ is a classical group (so that $F(Y_n)$ is commutative — it’s not), because $\varepsilon=e^{(0,0)}$, the statement $\nu(e_{(0,0)})>0$, implies that for a real coefficient $x^{(0,0)}>0$,

$\nu=x^{(0,0)}\varepsilon+\cdots= x^{(0,0)}\delta^e+\cdots$,

as for classical groups $\varepsilon=\delta^e$.

That is the condition $\nu(e_{(0,0)})>0$ is a quantum analogue of $e\in\text{supp}(\nu)$.

Consider a random walk on a classical (the algebra of functions on $G$ is commutative) finite group $G$ driven by a $\nu\in M_p(G)$.

The following is a very non-algebra-of-functions-y proof that $e\in \text{supp}(\nu)$ implies that the convolution powers of $\nu$ converge.

Proof: Let $H$ be the smallest subgroup of $G$ on which $\nu$ is supported:

$\displaystyle H=\bigcap_{\underset{\nu(S_i)=1}{S_i\subset G}}S_i$.

We claim that the random walk on $H$ driven by $\nu$ is ergordic (see Theorem 1.3.2).

The driving probability $\nu\in M_p(G)$ is not supported on any proper subgroup of $H$, by the definition of $H$.

If $\nu$ is supported on a coset of proper normal subgroup $N$, say $Nx$, then because $e\in \text{supp}(\nu)$, this coset must be $Ne\cong N$, but this also contradicts the definition of $H$.

Therefore, $\nu^{\star k}$ converges to the uniform distribution on $H$ $\bullet$

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist $\nu\in M_p(G)$ that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of $e\in\text{supp}(\nu)$.

So we have some questions:

• Is there a proof of the classical result (above) in the language of the algebra of functions on $G$, that necessarily bypasses talk of points and of subgroups?
• And can this proof be adapted to the quantum case?
• Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

## Quantum Subgroups

Let $C(G)$ be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication $\Delta$) and $\nu\in M_p(G)$ a state on $C(G)$. We say that a quantum group $H$ with algebra of function $C(H)$ (with comultiplication $\Delta_H$) is a quantum subgroup of $G$ if there exists a surjective unital *-homomorphism $\pi:C(G)\rightarrow C(H)$ such that:

$\displaystyle \Delta_H\circ \pi=(\pi\otimes \pi)\circ \Delta$.

## The Classical Case

In the classical case, where the algebras of functions on $G$ and $H$ are commutative,

$\displaystyle \pi(\delta_g)=\left\{\begin{array}{cc}\delta_g & \text{ if }g\in H \\ 0 & \text{ otherwise}\end{array}\right..$

There is a natural embedding, in the classical case, if $H$ is open (always true for $G$ finite) (thanks UwF) of $\imath: C(H) \xrightarrow\, C(G)$,

$\displaystyle \sum_{h\in H}a_h \delta_h \mapsto \sum_{g\in G} a_g \delta_g$,

with $a_g=a_h$ for $h\in G$, and $a_g=0$ otherwise.

Furthermore, $\pi$ is has the property that

$\pi\circ\imath\circ \pi=\pi$,

which resembles $\pi^2=\pi$.

In the case where $\nu$ is a probability on a classical group $G$, supported on a subgroup $H$, it is very easy to see that convolutions $\nu^{\star k}$ remain supported on $H$. Indeed, $\nu^{\star k}$ is the distribution of the random variable

$\xi_k=\zeta_k\cdots \zeta_2\cdot \zeta_1$,

where the i.i.d. $\zeta_i\sim \nu$. Clearly $\xi_k\in H$ and so $\nu^{\star k}$ is supported on $H$.

We can also prove this using the language of the commutative algebra of functions on $G$, $C(G)$. The state $\nu\in M_p(G)$ being supported on $H$ implies that

$\nu\circ\imath\circ \pi=\nu\imath\pi=\nu$.

Consider now two probabilities on $G$ but supported on $H$, say $\mu,\,\nu$. As they are supported on $H$ we have

$\mu=\mu\imath\pi$ and $\nu=\nu\imath\pi$.

Consider

$(\mu\star \nu)\imath\pi=(\mu\otimes \nu)\circ \Delta\circ \imath\pi$

$=((\mu\imath\pi)\otimes(\nu\imath\pi))\circ \Delta\circ\imath\pi =(\mu\imath\otimes \nu\imath)(\pi\circ \pi)\Delta\circ\imath\pi$

$=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi\circ \imath\circ \pi)=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi)$

$=(\mu\imath\otimes \nu\imath)\circ (\pi\circ \pi)\circ\Delta=(\mu\imath\pi\otimes \nu\imath\pi)\circ\Delta$

$=(\mu\otimes\nu)\circ\Delta=\mu\star \nu$,

that is $\mu\star \nu$ is also supported on $H$ and inductively $\nu^{\star k}$.

## Some Questions

Back to quantum groups with non-commutative algebras of functions.

• Can we embed $C(H)$ in $C(G)$ with a map $\imath$ and do we have $\pi\circ \imath\circ \pi=\pi$, giving the projection-like quality to $\pi$?
• Is $\nu\circ\imath\circ \pi=\nu$ a suitable definition for $\nu$ being supported on the subgroup $H$.

If this is the case, the above proof carries through to the quantum case.

• If there is no such embedding, what is the appropriate definition of a $\nu\in M_p(G)$ being supported on a quantum subgroup $H$?
• If $\pi$ does not have the property of $\pi\circ \imath\circ \pi=\pi$, in this or another definition, is it still true that $\nu$ being supported on $H$ implies that $\nu^{\star k}$ is too?

## Edit

UwF has recommended that I look at this paper to improve my understanding of the concepts involved.

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

Abstract Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.

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 $\nu$ being absolutely continuous (of the form $\nu(x)=\int_G xa_{\nu}$ for all $x\in C(G)$ and a unique $a_{\nu}\in C(G)$). 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 $S_n$ (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 $\mathcal{L}^2$-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 $\mathcal{L}^\infty$ 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 $S_n$ is effective at $k\sim (n-1)!$ while the upper bound shows the walk is random at time order $n!$.  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 $M_n(\mathbb{C})$ 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 $\alpha$ — 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 $n$-dependent driving probability and see if the cut-off phenomenon (Chapter 4) can be detected. This will need good lower bounds for $k\ll t_n$, some cut-off time.
• Convolutions Factorisations 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 $Q$ 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 $G$ be a finite quantum group and $M_p(G)$ be the set of states on the $\mathrm{C}^\ast$-algebra $F(G)$.

The algebra $F(G)$ has an invariant state $\int_G\in\mathbb{C}G=F(G)^\ast$, the dual space of $F(G)$.

Define a (bijective) map $\mathcal{F}:F(G)\rightarrow \mathbb{C}G$, by

$\displaystyle \mathcal{F}(a)b=\int_G ba$,

for $a,b\in F(G)$.

Then, where $\|\cdot\|_1^{F(G)}=\int_G|\cdot|$ and $\|\cdot\|_\infty^{F(G)}=\|\cdot\|_{\text{op}}$, define the total variation distance between states $\nu,\mu\in M_p(G)$ by

$\displaystyle \|\nu-\mu\|=\frac12 \|\mathcal{F}^{-1}(\nu-\mu)\|_1^{F(G)}$.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative $\mathcal{L}^p$ machinary shows that:

$\displaystyle \|\nu-\mu\|=\sup_{\phi\in F(G):\|\phi\|_\infty^{F(G)}\leq 1}\frac12|\nu(\phi)-\mu(\phi)|$.

(supremum presentation)

In the classical case, using the test function $\phi=2\mathbf{1}_S-\mathbf{1}_G$, where $S=\{\nu\geq \mu\}$, we have the probabilists’ preferred definition of total variation distance:

$\displaystyle \|\nu-\mu\|_{\text{TV}}=\sup_{S\subset G}|\nu(\mathbf{1}_S)-\mu(\mathbf{1}_S)|=\sup_{S\subset G}|\nu(S)-\mu(S)|$.

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

$\displaystyle \|\nu-\mu\|_P=\sup_{p\text{ a projection}}|\nu(p)-\mu(p)|$.

(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 $\neq$ 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 $M_n(\mathbb{C})$ factor of $F(G)$ 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 $M_n(\mathbb{C})$ 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 $\alpha$ — see Section 5.7).

I also want to understand how sharp (or otherwise) the order $n^n$ convergence for the random walk on the dual of $S_n$ is — $n^n$ 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 $\sim$ subsets via $G \supset S\leftrightarrow \mathbf{1}_S$. If I can show that for a positive linear functional $\rho$ that $|\rho(a)|\leq \rho(|a|)$ 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 $n$-dependent driving probability and see if I can detect the cut-off phenomenon (Chapter 4). This will need good lower bounds for $k\ll t_n$, 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 $\nu\in M_p(\mathbb{G})$ 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.