After completing a piece of work, I like to record some things that I would like to work on next. The previous time was a little over 11 months ago, and was heavily geared towards random walks on quantum groups. The need at that time was to start properly learning some compact quantum groups. I made a start on this: my plan was to write up some notes on compact quantum groups… however these notes [these are real rough], were abandoned like the Marie Celeste in May 2020. What happened was that I had a technical question about the construction of the reduced algebra $C_r(\mathbb{G})$ of continuous functions on a compact quantum group $\mathbb{G}$, and the expert who helped me suggested some intuition that I could use for quantum permutations… this kind of set off a quest to find a better interpretation for quantum permutations that started with this talk, then led to this monstrosity, and finally to this paper that I am proud of (but not sure if journals will concur).

So anyway some problems and brief thoughts. I tried and failed to resist the urge to use the non-standard notation and interpretation used in this new paper… I guess this post is for me… if you want to understand the weird “$\varsigma$ is a quantum permutation” and “$\varsigma\in\mathbb{G}$“, etc you will have to read the paper.

### No Quantum Alternating Group

So the big long crazy draft of has some stuff about why there is no quantum cyclic or alternating groups but these are arguments rather than proofs. A no-go theorem here looks as follows:

A finite group $G has no quantum version if whenever $\mathbb{G} is a quantum permutation group with group of characters of $C_u(\mathbb{G})$ equal to $G$, then $\mathbb{G}=G$.

I know the question for $A_N$ is open… is it formally settled that there is no quantum $\mathbb{Z}_N$? Proving that would be a start. A possible strategy would be to construct from $G<\mathbb{G}$ and a quantum permutation like $\varsigma_{e_5}\in\mathfrak{G}_0$ a character on $C_u(\mathbb{G})$ that is in the complement of $G$ in $S_N$. The conclusion being that there can be no quantum permutation in $C_u(\mathbb{G})$ that is $C_u(\mathbb{G})$ is commutative. There might be stuff here coming from alternating (pun not intended) projections theory that could help.

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

From April 2020:

My proof of the Ergodic Theorem leans heavily on the finiteness assumption but a lot of the stuff in that paper (and there are many partial results in that paper also) should be true in the compact case too. How much of the proof/results carry into the compact case? A full Ergodic Theorem for Random Walks on Compact Quantum Groups is probably quite far away at this point, but perhaps partial results under assumptions such as (co?)amenability might be possible. OR try and prove ergodic theorems for specific compact quantum groups.

What I would be interested in doing here is seeing can I maybe use the language from the Ergodic Theorem to prove some partial results in this direction. The analysis is possibly a little harder than I am used to. What I might want to show is that if $\phi\in C(\mathbb{G})$ is an idempotent state (I am not sure are there group-like projections $\mathbf{1}_{\phi}$ lying around, I think there are maybe here), and $p_{\phi}$ its support projection in $\ell^{\infty}(\mathbb{G})$, that the convolution of quantum permutations $\varsigma_1,\varsigma_2$ such that $p_\phi(\varsigma_i)=1$, that $p_\phi(\varsigma_2\star \varsigma_1)=1$. This would almost certainly require the use of a group-like projection. Possibly restricting to to quantum permutation groups we would then have a good understanding of non-Haar idempotent “quasi-subgroups”:

A quasi-subgroup of a compact (permutation?) quantum group is a subset of $\mathbb{G}$ that (for $C_u(\mathbb{G})$) contains the identity, is closed under reversal, and closed under the quantum group law. A quasi-subgroup is a quantum subgroup precisely when it is closed under wave function collapse.

The other thing that could be done here would be to refine two aspects of the finite theory. One, from the direction of cyclic shifts, in order to make the definition of a cyclic coset more intrinsic (it is currently defined with respect to a state), and two to further study the idea of an idempotent commuting with something like a “finite order” deterministic state (see p.27).

### Maximality Conjecture

For $N\leq 5$, there is no intermediate quantum subgroup $S_N<\mathbb{G}. That is there is no comultiplication intertwining surjective *-homomorphism $\pi:C_u(S_N^+)\twoheadrightarrow C_u(\mathbb{G})$ to noncommutative $C_u(\mathbb{G})$ such that there is in addition there is another such map $\pi_C:C_u(\mathbb{G})\twoheadrightarrow F(S_N)$.

This well-known conjecture is that there is no intermediate quantum subgroup $S_N<\mathbb{G} at any $N$.

Let $\mathfrak{G}_0$ be the Kac–Paljutkin quantum group of order eight and consider the quantum permutation $\varsigma_0:= (\varepsilon+ \varsigma_{e_5})/2\in \mathfrak{G}_0$, the state space of the algebra of functions on $\mathfrak{G}_0$, $F(\mathfrak{G}_0)$. Where $u^{\mathfrak{G}_0 and $\pi_{\mathfrak{G}_0}:C(S_N^+)\twoheadrightarrow F(\mathfrak{G}_0)$, $\varsigma\circ \pi_{\mathfrak{G}_0}\in S_N^+$. Where $\pi_C:C(S_N^+)\rightarrow F(S_N)$ and (by abuse of notation) $h_{S_N}:=h_{S_N}\circ\pi_C\in S_N^+$ the Haar state of $S_N$ in $S_N^{+}$. Where $\varsigma_1:=h_{S_N}\star \varsigma_0$, consider the idempotent state on $C(S_N^+)$:

$\displaystyle \varsigma:=w^{*}-\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\varsigma_1^{\star k}$.

There are three possibilities and all three are interesting:

$\varsigma=h_{S_N^+}$ — this is what I expect to be true. If this could be proven, the approach would be to hope that it might be possible to construct a state like $\varsigma_{e_5}$ on any compact quantum group. The state $\varsigma_{e_5}$ has a nice “constraint” property: I can only map 1 or 2 to 3 or 4 and vice versa. A starting idea in the construction might be to take the unital $\mathrm{C}^*$ algebra generated by a non-commuting pair $u^{\mathbb{G}}_{i_1j_1},u^{\mathbb{G}}_{i_2j_2}$. Using Theorem 4.6, there is a *-representation $\pi_2:\mathrm{C}^*(u^{\mathbb{G}}_{i_1j_1},u^{\mathbb{G}}_{i_2j_2})\rightarrow B(\mathbb{C}^2)$ such that for some $t\in (0,1)$ we have a representation and from this representation we have a nice vector state that is something like $\varsigma_{e_5}$. Well it doesn’t have the “constraint” property… one idea which I haven’t thought through is to condition this lovely quantum permutation in a clever enough way… we could condition it to only map $j_1$ to $i_1$ or some $i_3$… but that seems to only be the start of it.

$\varphi=h_{\mathbb{G}}$ for $S_N<\mathbb{G} — this would obviously be a counterexample to the maximality conjecture.

$\varphi$ is a non-Haar idempotent — this is a possibility that I don’t think many have thought of. It wouldn’t disprove the conjecture but would be an interesting example. This might be something non-zero on e.g. $|u_{31}u_{22}u_{11}|^2$ but zero on some strictly positive $|f|^2\in C(S_N^+)$.

### Doing something with abelian quantum permutation groups

I can’t really describe this so instead I quote from the paper:

It could be speculated that the dual of a discrete group $\Gamma=\langle\gamma_1,\dots,\gamma_k\rangle$ could model a $k$ particle “entangled” quantum system, where the $p$-th particle, corresponding to the block $B_p$, has $|\gamma_p|$ states, labelled $1,\dots,|\gamma_p|$. Full information about the state of all particles is in general impossible, but measurement with $x(B_p)$ will see collapse of the $p$th particle to a definite state. Only the deterministic permutations in $\widehat{\Gamma}$ would correspond to classical states.

### Quantum Automorphism Groups of Graphs

So I want to read Schmidt’s PhD thesis and maybe answer the question of whether or not there is a graph whose quantum automorphism group is the Kac–Paljutkin quantum group. Also see can anything be done in the intersection of random walks and quantum automorphism groups.

### Write locally compact to compact dictionary

I want to able to extract results on locally compact quantum groups to compact quantum groups.

### Use of Stopping Times and Classical Probabilistic Methods for Random Walks on Quantum Permutation Groups

This is a bit mad… bottom of p.35 to p.36.

Maybe take $x_0:=1\in\{1,\dots,N\}$ and define a Markov chain on $\{1,\dots,N\}$ using a quantum permutation. So for example you measure $\varsigma$ with $x(1)$ to get $x_1$. Then measure $x(x_1)$ and iterate. The time taken to reach $N$ or some other $k\in\{1,\dots,N\}$ or maybe hit all of $\{1,\dots,N\}$… this is a random time $T$. Is there any relationship between the expectation of $T$ and the distance of the convolution powers of $\varsigma$ to the Haar state. Lots of things to think about here.

### Alternating Measurement

More mad stuff. See Section 8.5.

### Infinite Discrete Duals with no Finite Order Generators

If $\Gamma$ has a finite set of finite-order generators it is a quantum permutation group. What about if there are only infinite order generators? I guess this isn’t a quantum permutation group (subgroup of some $S_N^+$) but maybe using Goswami and Skalski such a dual can be given the structure of a quantum permutation group on infinite many symbols. At the other end of the scale… are the Sekine quantum groups quantum permutation groups?

### Other random walk questions

From April 2020:

1. Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac? — No interest in this by me at the moment
2. Following Urban, study convolution factorisations of the Haar state. — ditto.
3. Examples of non-central random walks on compact quantum groups — Freslon and coauthors have cornered the market on interesting examples of random walks on compact quantum groups… I don’t think I will be spending time on this.

Note that Simeng Wang has sorted: extending the Upper Bound Lemma to the non-Kac case. There are a handful of other problems here, here, and here that I am no longer interested in.