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### 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.

In the hope of gleaning information for the study of aperiodic random walks on (finite) quantum groups, which I am struggling with here, by couching Freslon’s Proposition 3.2, in the language of Fagnola and Pellicer, and in the language of my (failed) attempts (see here, here, and here) to find the necessary and sufficient conditions for a random walk to be aperiodic. It will be necessary to extract the irreducibility and aperiodicity from Freslon’s rather ‘unilateral’ result.

Well… I have an inkling that because dual groups satisfy what I would call the condition of abelianness (under the ‘quantisation’ functor), all (quantum) subgroups are normal… this is probably an obvious thing to write down (although I must search the literature) to ensure it is indeed known (or is untrue?). **Edit: **Wang had it already, see the last proposition here.

Let be a the algebra of functions on a finite classical (as opposed to quantum) group . This has the structure of both an algebra and a coalgebra, with an appropriate relationship between these two structures. By taking the dual, we get the *group algebra**,* . The dual of the pointwise-multiplication in is a coproduct for the algebra of functions on the dual group … this is all well known stuff.

Recall that the set of probabilities on a finite quantum group is the set of states , and this lives in the dual, and the dual of is , and so probabilities on are functions on . To be positive is to be positive definite, and to be normalised to one is to have .

The ‘simplicity’ of the coproduct,

,

means that for ,

,

so that, inductively, is equal to the (pointwise-multiplication power) .

The Haar state on is equal to:

,

and therefore necessary and sufficient conditions for the convergence of is that is *strict. *It can be shown that for any that . Strictness is that this is a strict inequality for , in which case it is obvious that .

Here is a finite version of Freslon’s result which holds for discrete groups.

### Freslon’s Ergodic Theorem for (Finite) Group Algebras

*Let be a probability on the dual of finite group. The random walk generated by is ergodic if and only if is not-concentrated on a character on a non-trivial subgroup .*

Freslon’s proof passes through the following equivalent condition:

*The random walk on driven by is *not *ergodic if is *bimodular*with respect to a non-trivial subgroup , in the sense that*

.

Before looking at the proof proper, we might note what happens when is abelian, in which case is a classical group, the set of characters on .

To every positive definite function , we can associate a probability such that:

.

This is Bochner’s Theorem for finite abelian groups. This implies that positive definite functions on finite abelian groups are exactly convex combinations of characters.

Freslon’s condition says that to be not ergodic, must be a character on a non-trivial subgroup . Such characters can be extended in ways.

Therefore, if is not ergodic, .

For , we have

,

dividing both sides by yields:

.

As , and , this implies that is supported on characters such that, for all :

,

such that . The set of such is the annihilator of in , and it is a subgroup. Therefore is concentrated on the coset of a normal subgroup (as all subgroups of an abelian group are normal).

This, via Pontragin duality, is not looking at the ‘support’ of , but rather of . Although we denote , and when is abelian, is a group (unnaturally, of characters) isomorphic to . Is it the case though that,

gives the same object in as

?

Well… of course this is true because .

We could proceed to look at Fagnola & Pellicer’s work but first let us prove Freslon’s result, hopefully in the finite case the analysis disappears…

*Proof: *Assume that is not strict and let

.

There exists a unitary representation and a unit vector such that

Cauchy-Schwarz implies that

.

If is not strict there is an such that this is an inequality and so is colinear to , it follows that .

This implies for and :

,

and so is closed under multiplication. Also and so and so is a subgroup. It follows that is a character on , which is not trivial because is not strict.

I don’t really need to go through the third equivalent condition. If coincides with a character on a subgroup , for

,

and so is not strict

Now let us look at the language of Fagnola and Pellicer. What is a projection in ? First note the involution in is . The second multiplication is the convolution. This means projections in the algebra are symmetric with respect to the group inverses and they are also idempotents. They are actually equal to Haar states on finite subgroups.

I think periodicity is also wrapped up in Freslon’s result as I think all subgroups of dual groups are normal. Perhaps, oddly, not being concentrated on a subgroup means that the positive function (probability on the dual) is one on that subgroup…

Well… let us start with irreducible. Suppose fails to be ergodic because it is irreducible. This means there is a projection such that that (and support less than ?)

Let us look at the first condition:

.

What now is the support of ? Well… some work I have done offline shows that the special projections, the group-like projections, correspond to to for a subgroup of . If is reducible, it is concentrated on such a quasi-subgroup, and this means that coincides with a *trivial *character on . In terms of Fagnola Pellicer, .

Now let us tackle aperiodicity. It is going to correspond, I think, with being concentrated on a ‘coset’ of a non-trivial character on …

Well, we can show that if is periodic, there is a subset such that for all . We can use Freslon’s proof to show that is in a subgroup on which .

Now what I want to do is put this in the language of ‘inclusion’ matrices… but the inclusions for cocommutative quantum groups are trivial so no go…

We can reduce Freslon’s conditions down to irreducible and aperiodic: not coinciding with a trivial character, and not coinciding with a character.

In the case of a finite classical group , we can show that if we have i.i.d. random variables , that if , for a coset of a proper normal subgroup , that the random walk on driven by , the random variables:

,

exhibits a periodicity because

.

This shows that a necessary condition for ergodicity of a random walk on a finite classical group driven by is that the support of not be concentrated on the coset of a proper normal subgroup.

I had hoped that something similar might hold for the case of random walks on finite *quantum *groups but alas I think I have found a barrier.

Slides of a talk given at Munster Groups 2019, WIT.

**Abstract: ***It is a folklore theorem that necessary and sufficient conditions for a random walk on a finite group to converge in distribution to the uniform distribution – “to random” – are that the driving probability is not concentrated on a proper subgroup nor the coset of a proper normal subgroup. This is the Ergodic Theorem for Random Walks on Finite Groups. In this talk we will outline the rarely written down proof, and explain why, for example, adjacent transpositions can never mix up a deck of cards. From here we will, in a very leisurely and natural fashion, introduce (and motivate the definition of) finite quantum groups, and random walks on them. We will see how the group algebra of a finite group is the algebra of functions on a finite quantum group. Freslon has very recently proved the Ergodic Theorem in this setting, and we present ongoing work towards an Ergodic Theorem in the more general finite quantum group setting; a result that would generalise both the folklore and group algebra Ergodic Theorems.*

### Introduction

Every finite quantum group has finite dimensional algebra of functions:

.

At least one of the factors must be one-dimensional to account for the counit , and if this factor is denoted , the counit is given by the dual element . There may be more and so reorder the index so that for , and for :

,

Denote by the states of . The *pure *states of arise as pure states on single factors.

In the case of the Kac-Paljutkin and Sekine quantum groups, the convolution powers of pure states exhibit a periodicity of sorts. Recall for these quantum groups that consists of a single matrix factor.

In these cases, for pure states of the form , that is supported on (and we can say a little more than is necessary), the convolution remains supported on because

.

If we have a pure state supported on , then because

,

then must be supported on, because of , .

Inductively all of the are supported on and the are supported on . This means that the convolutions powers of a pure state, in these cases, cannot converge to the Haar state.

The question is, do the results above about the image of and under the coproduct hold more generally? I believe that the paper of Kac and Paljutkin shows that this is the case whenever consists of a single factor… but does it hold more generally?

To find out we go back and do some sandboxing with the paper of Kac and Paljutkin. Which is a pleasure because that paper is beautiful. The blue stuff is my own scribbling.

## Finite Ring Groups

Let be a finite quantum group with notation on the algebra of functions as above. Note that is commutative. Let

,

which is a central idempotent.

### Lemma 8.1

.

*Proof: *If , then for some , and , the mapping is a non-zero homomorphism from into commutative which is impossible.

If , then one of the , with ‘something’ in . Using the centrality and projectionality of , we can show that the given map is indeed a homomorphism.

It follows that , and so

### Lemma 8.2

*Proof: *Suppose that for some non-commutative . This means that there exists an index such that . Then for that factor,

is a non-null homomorphism from the non-commutative into the commutative.

We see that for all . Putting we get the result

The following says that is a group-like projection. We know from previous work that if a state is supported on a group-like projection that it will remain supported on it. In particular, any state supported on will remain there.

### Lemma 8.3

.

*Proof: *Since is a homomorphism, is an idempotent in . I do not understand nor require the rest of the proof.

### Lemma 8.4

* is the algebra of functions on finite group with elements , and we write . The coproduct is given by .*

We have:

,

,

,

as .

The element is a sum of four terms, lying in the subalgebras:

.

We already know what is going on with the first summand. Denote the second by . From the group-like-projection property, the last two summands are zero, so that

.

Since the are symmetric () mutually orthogonal idempotents, has similar properties:

for .

At this point Kac and Paljutkin restrict to , that is there is only one summand. Here we *try *to keep arbitrarily (finitely) many summands in .

Let the summand have matrix units , where . Kac and Paljutkin now do something which I think is a little dodgy, but basically that the integral over is equal on each of the , equal on each of the , and then zero off the diagonal.

It does follow from above that each is a projection.

Now I am stuck!

## Background

I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by is *ergodic* if the convolution powers converge to the Haar state .

The classical theorem for finite groups:

#### Ergodic Theorem for Random Walks on Finite Groups

A random walk on a finite group driven by a probability is ergodic if and only if is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.

Not concentrated on a proper subgroup gives *irreducibility*. A random walk is *irreducible *if for all , there exists such that .

Not concentrated on the coset of a proper normal subgroup gives *aperiodicity. *Something which should be equivalent to aperiodicity is if

is equal to one (perhaps via invariance ).

If is concentrated on the coset a proper normal subgroup , specifically on , then we have periodicity (), and , the order of .

In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.

The theorem in the quantum case should look like:

#### Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group driven by a state is ergodic if and only if “X”.

## Irreducibility

At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to *group-like-projections,* which I will say correspond to *quasi-subgroups *following Kasprzak & Sołtan.

I have shown that if is concentrated on a proper quasi-subgroup , in the sense that for a group-like-projection , that so are the . The analogue of irreducible is that for all projections in , there exists such that . If is concentrated on a quasi-subgroup , then for all , , where .

I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means

,

converge to an idempotent state . If for all then the also, so that (as the Haar state is faithful). I was able to prove that is supported on the quasi-subgroup given by the idempotent .

I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.

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 on a *finite* quantum group such that where

,

with . This has a positive density of trace one (with respect to the Haar state ), say

,

where 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 . Assume that (if , then for all and we have trivial convergence)

Therefore let

be the density of .

Now we can explicitly write

.

This has stochastic operator

.

Let be an eigenvalue of of eigenvector . This yields

and thus

.

Therefore, as is also an eigenvector for , and is a stochastic operator (if is an eigenvector of eigenvalue , then , contradiction), we have

.

This means that the eigenvalues of lie in the ball and thus the only eigenvalue of magnitude one is , which has (left)-eigenvector the stationary distribution of , say .

If is symmetric/reversible in the sense that , then is self-adjoint and has a basis of (left)-eigenvectors and we have, if we write ,

,

which converges to (so that ).

If is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers converge and thus so do the

### Total Variation Decrasing

Uses Simeng Wang’s . Result holds for compact Kac if the state has a density.

### Periodic is concentrated on a coset of a proper normal subgroup of

is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by

### Supported on Subgroup implies Reducible

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

*Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups, *Journal 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 as well as on the truly quantum groups of Sekine.*

In a recent preprint, Haonan Zhang shows that if (where is a Sekine Finite Quantum Group), then the convolution powers, , converges if

.

The algebra of functions is a multimatrix algebra:

.

As it happens, where , the counit on is given by , that is , dual to .

To help with intuition, making the incorrect assumption that is a classical group (so that is commutative — it’s not), because , the statement , implies that for a real coefficient ,

,

as for classical groups .

That is the condition is a quantum analogue of .

Consider a random walk on a classical (the algebra of functions on is commutative) *finite* group driven by a .

The following is a very non-algebra-of-functions-y proof that implies that the convolution powers of converge.

*Proof: *Let be the smallest subgroup of on which is supported:

.

We claim that the random walk on driven by is *ergordic* (see Theorem 1.3.2).

The driving probability is not supported on any proper subgroup of , by the definition of .

If is supported on a coset of proper normal subgroup , say , then because , this coset must be , but this also contradicts the definition of .

Therefore, converges to the uniform distribution on

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 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 .

So we have some questions:

- Is there a proof of the classical result (above) in the language of the algebra of functions on , 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 be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication ) and a state on . We say that a quantum group with algebra of function (with comultiplication ) is a quantum subgroup of if there exists a surjective unital *-homomorphism such that:

.

## The Classical Case

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

There is a natural embedding, in the classical case, if is open (always true for finite) (thanks UwF) of ,

,

with for , and otherwise.

Furthermore, is has the property that

,

which resembles .

In the case where is a probability on a classical group , supported on a subgroup , it is very easy to see that convolutions remain supported on . Indeed, is the distribution of the random variable

,

where the i.i.d. . Clearly and so is supported on .

We can also prove this using the language of the commutative algebra of functions on , . The state being supported on implies that

.

Consider now two probabilities on but supported on , say . As they are supported on we have

and .

Consider

,

that is is also supported on and inductively .

## Some Questions

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

- Can we embed in with a map and do we have , giving the projection-like quality to ?
- Is a suitable definition for being supported on the subgroup .

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 being supported on a quantum subgroup ?
- If does not have the property of , in this or another definition, is it still true that being supported on implies that is too?

## Edit

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

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