Slides of a talk given at CIT School of Science Seminar.

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 $F(G)$ be a the algebra of functions on a finite classical (as opposed to quantum) group $G$. 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, $\mathbb{C}G=:F(\widehat{G})$. The dual of the pointwise-multiplication in $F(G)$ is a coproduct for the algebra of functions on the dual group $\widehat{G}$… this is all well known stuff.

Recall that the set of probabilities on a finite quantum group is the set of states $M_p(G):=\mathcal{S}(F(G))$, and this lives in the dual, and the dual of $F(\widehat{G})$ is $F(G)$, and so probabilities on $\widehat{G}$ are functions on $G$. To be positive is to be positive definite, and to be normalised to one is to have $u(\delta^e)=1$.

The ‘simplicity’ of the coproduct,

$\Delta(\delta^g)=\delta^g\otimes\delta^g$,

means that for $u\in M_p(\widehat{G})$,

$(u\star u)(\delta^g)=(u\otimes u)\Delta(\delta^g)=u(\delta^g)^2$,

so that, inductively, $u^{\star k}$ is equal to the (pointwise-multiplication power) $u^k$.

The Haar state on $\widehat{G}$ is equal to:

$\displaystyle \pi:=\int_{\hat{G}}:=\delta_e$,

and therefore necessary and sufficient conditions for the convergence of $u^{\star k}\rightarrow \pi$ is that $u$ is strict. It can be shown that for any $u\in M_p(G)$ that $|u(\delta^g)|\leq u(\delta^e)=1$. Strictness is that this is a strict inequality for $g\neq e$, in which case it is obvious that $u^{\star k}\rightarrow \delta_e$.

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

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

Let $u\in M_p(\widehat{G})$ be a probability on the dual of finite group. The random walk generated by $u$ is ergodic if and only if $u$ is not-concentrated on a character on a non-trivial subgroup $H\subset G$.

Freslon’s proof passes through the following equivalent condition:

The random walk on $\widehat{G}$ driven by $u\in M_p(\widehat{G})$ is not ergodic if $u$ is bimodularwith respect to a non-trivial subgroup $H\subset G$, in the sense that

$\displaystyle u(\underbrace{\delta^g\delta^h}_{=\delta^{gh}})=u(\delta^g)u(\delta^h)=u\left(\underbrace{\delta^h\delta^g}_{=\delta^{hg}}\right)$.

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

To every positive definite function $u\in M_p(\widehat{G})$, we can associate a probability $\nu_u\in M_p(\widehat{G})$ such that:

$\displaystyle u(\delta^g)=\sum_{\chi\in\hat{G}} \chi(\delta^g) \nu_u(\chi)$.

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, $u$ must be a character on a non-trivial subgroup $H\subset G$. Such characters can be extended in $[G\,:\,H]$ ways.

Therefore, if $u$ is not ergodic, $u_{\left|H\right.}=\eta\in \widehat{H}$.

For $h\in H$, we have

$\displaystyle u(h)=\sum_{\chi\in\widehat{G}}\chi(h)\nu_u(\chi)$,

dividing both sides by $u(h)=\eta(h)\neq 0$ yields:

$\displaystyle\sum_{\chi \in \widehat{G}} (\eta^{-1}\chi)(h)\nu_u (\chi)=1$.

As $\nu_u\in M_p(\widehat{G})$, and $(\eta^{-1}\chi)(h)\in \mathbb{T}$, this implies that $\nu_u$ is supported on characters such that, for all $h\in H$:

$\eta^{-1}(h)\chi(h)=1\Rightarrow \chi=\eta\tilde{\chi}$,

such that $\tilde{\chi}(H)=\{1\}$. The set of such $\tilde{\chi}$ is the annihilator of $H$ in $\widehat{G}$, and it is a subgroup. Therefore $\nu_u$ 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 $u$, but rather of $\nu_u$. Although we denote $\mathbb{C}G=:F(\widehat{G})$, and when $G$ is abelian, $\widehat{G}$ is a group (unnaturally, of characters) isomorphic to $G$. Is it the case though that,

$\Delta(\chi)=\chi\otimes\chi$

gives the same object in as

$\displaystyle\Delta(\chi)=\sum_{g\in G}\chi(\delta^g)\Delta(\delta_g)$

$\displaystyle =\sum_{g\in G}\chi(\delta^g)\sum_{t\in G}\delta_{gt^{-1}}\otimes \delta_t$?

Well… of course this is true because $\chi(gh)=\chi(g)\chi(h)$.

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 $u$ is not strict and let

$\Lambda:=|u|^{-1}(\{1\})$.

There exists a unitary representation $\Phi:G\rightarrow B(H)$ and a unit vector $\xi$ such that

$u(g)=\langle \Phi(g)\xi,\xi\rangle$

Cauchy-Schwarz implies that

$|u(g)|\leq \|\Phi(g)\xi\|\|\xi\|=\|\xi\|^2$.

If $h$ is not strict there is an $h$ such that this is an inequality and so $\Phi(h)\xi$ is colinear to $\xi$, it follows that $\Phi(h)\xi=u(h)\xi$.

This implies for $h\in \Lambda$ and $g\in G$:

$|u(gh)|=|\langle \Phi(gh)\xi,\xi\rangle|=|u(h)||\langle \Phi(g)\xi,\xi\rangle|=|u(g)|$,

and so $\Lambda$ is closed under multiplication. Also $u(g^{-1})=\overline{u(g)}$ and so $\Lambda$ and so $\Lambda$ is a subgroup. It follows that $u$ is a character on $\Lambda$, which is not trivial because $u$ is not strict.

I don’t really need to go through the third equivalent condition. If $u$ coincides with a character on a subgroup $\Lambda$, for $h\in \Lambda$

$|u(h)|^2=u(h)\overline{u(h)}=u(h)u(h^{-1})=u(e)=1$,

and so $u$ is not strict $\bullet$

Now let us look at the language of Fagnola and Pellicer. What is a projection in $\mathbb{C}G$? First note the involution in $\mathbb{C}G$ is $(\delta^g)^*=\delta^{g^{-1}}$. 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 $u$ fails to be ergodic because it is irreducible. This means there is a projection $p_H=\int_H$ such that that $P_u(p_H)=p_H$ (and support $u$ less than $p_H$?)

Let us look at the first condition:

$P_u(p_H)=(u\otimes I)\Delta(p_H)=\cdots=\frac{1}{|H|}\sum_{h\in H}u(h)\delta^h=p_H\Rightarrow u_{\left|H\right.}=1$.

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

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

Well, we can show that if $u$ is periodic, there is a subset $S\subset G$ such that $u(s)=e^{2\pi i a_s/d}$ for all $s\in S$. We can use Freslon’s proof to show that $S$ is in a subgroup on which $|u|=1$.

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.

• The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
• Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
• For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

2. does my answer make sense? If no, say so, and then try and fix your solution.
3. check your answer (e.g. if you are looking at a general true, look at a special case; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Week 12

We will finish our study of Graph Theory by looking at Eulerian graphsHamiltonian graphs, and Dirac’s Theorem.

We had the test on Tuesday.

On Friday we had a look at this graph, a Chapter 4 question:

• The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
• Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
• For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

2. does my answer make sense? If no, say so, and then try and fix your solution.
3. check your answer (e.g. for a fitted curve or beam function, input values and see do they make sense; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Tutorial Split

I am not sure what is going on with Leonard’s classes but I will be present this Friday, tomorrow 6 December for tutorials in A243L (11:00) and A213B (12:00).

## Week 12

On Monday we finished off Chapter 4 by looking at Error Analysis. Better exercises than the book here (including corrections to the sheet handed out in class). We then had over two and a half hours of tutorial time for lectures, and another tutorial on Friday.

• The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
• Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
• For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

2. does my answer make sense?
3. check your answer (e.g. differentiate/antidifferentiate an antiderivative/derivative, substitute your solution into equations, check your answer against a rough estimate, or what a picture is telling you, etc)

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Test 2 – Results

Have been emailed to you along with final CA results.

## Week 12

We looked at centroids of laminas and centres of gravity of solids of revolution.

We had one and a half classes of tutorial time.

## Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

• Monday 16:00
• Tuesday 09:00
• Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

• Monday at 09:00 in B180
• Monday at 17:00 in B189
• Wednesday at 10:00 in F1. 3

The exam is on Friday 13 December. Past exam papers (MATH6040 runs in Semester 1 and Semester 2) may be found here.

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

If you go into Canvas, and go into MATH6055 and the ‘Algebra’ unit, you might see online practise questions for Test 2.

## Week 11

On Monday we half-finished the Examples of Functions (we will finish this off on Friday) mini-chapter before starting the final chapter, the easy chapter, on Network (Graph) Theory.

## Week 12

We will finish our study of Graph Theory by looking at Eulerian graphsHamiltonian graphs, and Dirac’s Theorem.

The Test is on Tuesday.

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Tutorial Split

I am not sure what is going on with Leonard’s classes but I will be present this Friday and next Friday for tutorials in A243L (11:00) and A213B (12:00).

## Assessment 2 — Results

These have been sent to you. Comments to follow.

## Week 11

We had a tutorial dedicated to differentiation on Monday.

We tried to look at a Probability and Statistics Word Cloud on Wednesday. It was a disaster as my computer froze… anyway:

My answer: when $A$ and $B$ are independent.

A difficult one. My answer: Every probability/area under a bell curve can be calculated by transforming the area to a $z$ curve, and calculating the area ‘there’.

My answer: To infer things about the population, e.g. the population mean.

My answer: An interval that we believe the population mean is in (with a certain confidence).

On Wednesday and Thursday we had a look at more general Taylor Series: not just near $a=0$, before doing a revision of partial differentiation, and linking at error analysis.

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Test 2 – Results

Possibly early next week. I am a little behind in my corrections after being sick last week though. Definitely Thursday.

## Week 11

We had our test on Monday, then will looked at completing the square, and work on Tuesday and Thursday.

Here is some video of revision antidifferentiation.

## Week 12

We will look at centroids of laminas and centres of gravity of solids of revolution. Any spare lecture time will be given over to tutorial time.

## Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

• Monday 16:00
• Tuesday 09:00
• Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

• Monday at 09:00 in B180
• Monday at 17:00 in B189
• Wednesday at 10:00 in F1. 3

The exam is on Friday 13 December.

## Transposition Project – Survey

With regard to TRANSPOSITION (pages 86-97 of the manual), please fill out this survey.

## Transposition Project – Part II

October 14 you took a quiz as part of the Transposition Project that the Mathematics Department is undertaking in an effort to improve our teaching.

You will have another 15 minute quiz on Monday.

If you do not have an internet ready device, or did not do the first quiz, you may leave class early.

Thank you again for your participation.

## Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

## Exercises on Canvas

If you go into Canvas, and go into MATH6055 and the ‘Algebra’ unit, you might see online practise questions for Test 2.

## Week 10

We worked with logarithms and started a quick look at Examples of Functions (which we needed Algebra to talk about).

## Week 11

On Monday we will finish the Examples of Functions mini-chapter before starting the final chapter, the easy chapter, on Network (Graph) Theory.

## Tutorial Split

See my email of 12 November regarding the Friday tutorial split for the rest of the semester.

Tomorrow we go back to the normal room of A213B for the 12:00 tutorial.

## Assessment 2 — Corrections

Hopefully I can get these to ye within the week. I have some ‘man flu’ at the moment but hopefully that goes away.

## Week 10

We had an extra tutorial on Monday. Most people concentrated on the Sample Question 3s on Probability and Statistics.

We finished off Chapter 3 — and spoke about the Bad and Good and Bad News — before we began Chapter 4 with a Revision of Differentiation and had a look at Maclaurin Series.

Oh: one thing — I never told ye what a p-value was (Sample Paper I Q. 3 (c) B ii.). It is the same as the level of significance, the probability of making a Type I Error… the answer is 5%.

## Week 11

We will have a tutorial dedicated to differentiation on Monday.

We might look at a Probability and Statistics Concept MCQ on Wednesday.

On Wednesday and Thursday we will have a quick look at more general Taylor Series: not just near $a=0$, before doing a revision of partial differentiation.