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An expository piece, the watered down version of the madness here… should be on the Arxiv Thursday.

Click here for pdf.

In this post here, I outlined some things that I might want to prove. Very bottom of that list was:

Extending the Upper Bound Lemma to the non-Kac case. As I speak, this is beyond what I am capable of. This also requires work on the projection and quantum total variation distances (i.e. show they are equal in this larger category).

After that post, Simeng Wang wrote with the, for me, exciting news that he had proven the Upper Bound Lemma in the non-Kac case, and had an upcoming paper with Amaury Freslon and Lucas Teyssier. At first glance the paper is an intersection of the work of Amaury in random walks on compact quantum groups, and Lucas’ work on limit profiles, a refinement in the understanding of how the random transposition random walk converges to uniform. The paper also works with continuous-time random walks but I am going to restrict attention to what it does with random walks on S_N^+.

Introduction

For the case of a family of Markov chains (X_N)_{N\geq 1} exhibiting the cut-off phenomenon, it will do so in a window of width w_N about a cut-off time t_N, in such a way that w_N/t_N\rightarrow 0, and, where d_N(t) is the ‘distance to random’ at time t, d_N(t_N+cw_N) will be close to one for c<0 and close to zero for c>0. The cutoff profile of the family of random walks is a continuous function f:\mathbb{R}\rightarrow[0,1] such that as N\rightarrow \infty,

\displaystyle d_N(t_N+cw_N)\rightarrow f(c).

I had not previously heard about such a concept, but the paper gives a number of examples in which the analysis had been carried out. Lucas however improved the Diaconis–Shahshahani Upper Bound Lemma and this allowed him to show that the limit profile for the random transpositions random walk is given by:

\displaystyle d_{\text{TV}}(\text{Poi}[1+e^{-c}],\text{Poi}[1])

Without looking back on Lucas’ paper, I am not sure exactly how this d_{\text{TV}} works… I will guess it is, where \xi_1\sim \text{Poi}[1+e^{-c}] and \xi_2\sim \text{Poi}[2], and so:

\displaystyle f_{\text{RT}}(c)=\frac{1}{2}\sum_{k=0}^{\infty}\left|\mathbb{P}[\xi_1=k]-\mathbb{P}[\xi_2=k]\right|,

and I get f(0)\approx 0.330, f(2)\approx 0.0497 and f(-2)\approx 0.949 on CAS. Looking at Lucas’ paper thankfully this is correct.

The article confirms that Lucas’ work is the inspiration, but the study will take place with infinite compact quantum groups. The representation theory carries over so well from the classical to quantum case, and it is representation theory that is used to prove so many random-walk results, that it might have been and was possible to study limit profiles for random walks on quantum groups.

More importantly, technical issues which arise as soon as N>4 disappear if the pure quantum transposition random walk is considered. This is a purely quantum phenomenon because the random walk driven only by transpositions in the classical case is periodic and does not converge to uniform. I hope to show in an upcoming work how something which might be considered a quantum transposition behaves very differently to a classical/deterministic transposition. My understanding at this point, in a certain sense (see here)) is that a quantum transposition has N-2 fixed points in the sense that it is an eigenstate (with eigenvalue N-2) of the character \text{fix}=\sum_i u_{ii}. I am hoping to find a dual with a quantum transposition that for example does not square to the identity (but this is a whole other story). This would imply in a sense that there is no quantum alternating group.

The paper will show that the quantum version of the ordinary random transposition random walk and of this pure random transposition walk asymptotically coincide. They will detect the cutoff at time \frac12 N\ln N, and find an explicit limit profile (which I might not be too interested in).

I will skip the stuff on O_N^+ but as there are some similarities between the representation theory of quantum orthogonal and quantum permutation groups I may have to come back to these bits.

Quantum Permutations

Character Theory

At this point I will move away and look at this Banica tome on quantum permutations for some character theory.

Read the rest of this entry »

What started about ten months ago as a technical question to an expert, led to a talk, and led to me producing this weird production here.

Now that it is complete, although I really like all its contents (well except for the note to reader and introduction I spilled out very hastily), I can see on reflection it represents rather than a cogent piece of mathematics, almost a log of all the things I have learnt in the process of writing it. It also includes far too much speculation and conjecture. So I am going to post it here and get to work on editing it down to something a little more useful and cogent.

EDIT: Edited down version here.

I have been working for a number of months on a paper/essay based on this talk (edit: big mad long draft here). After the talk the seminar host Teo Banica suggested a number of things that the approach could be used to look at, and one of these was orbits and orbitals. These are nice, intuitive ideas introduced by Lupini, Mancinska (missing an accent), and Roberson introduced orbitals.

I went to Teo’s quantum permutations tome, and Chapter 13 (p.297) orbits and orbitals are introduced, and it is remarked that, where we are studying quantum permutation groups \mathbb{G}< S_N^+, a certain relation on N\times N\times N is believed not to be transitive. This belief is expressed also in the fantastic nonlocal games and quantum permutations paper, as well as by Teo here.

One of the things that the paper has had me doing is using CAS to write up the magic unitaries for a number of group duals, and I said, hey, why don’t I try and see is there any counterexamples there. My study of the \widehat{S_3}<S_4^+ led me to believe there would be no counterexamples there. The next two to check would be \widehat{S_4}<S_5^+ and the dual of the quaternion group \widehat{Q}<S_8^+. I didn’t get called JPQ by Professor Des MacHale for nothing… I had to look there. OK, time to explain what the hell I am talking about.

I guess ye will have to wait for the never-ending paper to see exactly how I think about quantum permutation groups… so for the moment I am going to assume that you know what compact matrix quantum groups… but maybe I can put in some of the new approach, which can be gleaned from the above talk, in bold italics. A quantum permutation group \mathbb{G}\leq S_N^+ is a compact matrix quantum group whose fundamental representation u^{\mathbb{G}}\in M_N(C(\mathbb{G})) is a magic unitary. The relation that was believed not to be transitive is:

(j_3,j_2,j_1)\sim_3 (i_3,i_2,i_1)\Leftrightarrow u_{j_3i_3}u_{j_2i_2}u_{j_1i_1}\neq 0,

that is the indices are related when there is a quantum permutation \varsigma that has a non-zero probability of mapping:

(\varsigma(j_3)=i_3)\succ(\varsigma(j_2)=i_2)\succ (\varsigma(j_1)=i_1). (*)

This relation is reflexive and symmetric. If we work with the universal (or algebraic) level, then e\in\mathbb{G} will fix all indices giving reflexivity, if a quantum permutation \varsigma\in\mathbb{G} can map as per (*), it’s reverse \varsigma^{-1}:=\varsigma\circ S will map, with equal probability of \varsigma doing (*):

(\varsigma^{-1}(i_3)=j_3)\succ(\varsigma^{-1}(i_2)=j_2)\succ (\varsigma^{-1}(i_1)=j_1),

so that \sim_3 is symmetric.

Now, to transitivity. We’re going to work with the algebra of functions on the dual of the quaternions, F(\widehat{Q}):=\mathbb{C}Q. Working here is absolutely fraught what with coefficients i and -1 and elements of Q of the same symbol. Therefore we will use the \delta^g notation. Consider the following vector in F(\widehat{Q}):

\displaystyle (u^{\widehat{\langle j\rangle}})_{,1}:=\frac{1}{4}\left[\begin{array}{c}\delta^1+\delta^j+\delta^{-1}+\delta^{ij}\\ \delta^1+i\delta^{j}-\delta^{-1}-i\delta^{-j} \\ \delta^1-\delta^{j}+\delta^{-1}-\delta^{-j} \\ \delta^{1}-i\delta^{j}-\delta^{-1}+i\delta^{-j}\end{array}\right].

This vector is the first column of a magic unitary u^{\widehat{\langle j\rangle}} for \widehat{\langle j\rangle}\cong \widehat{\mathbb{Z}_4}\cong \mathbb{Z}_4, and the rest of the magic unitary is made by making a circulant matrix from this. Do the same with k\in\widehat{Q}, another magic unitary u^{\widehat{\langle k\rangle}}, and so we have \widehat{Q}<S_8^+ via:

u^{\widehat{Q}}=\left(\begin{array}{cc}u^{\widehat{\langle j\rangle}} & 0 \\0 & u^{\widehat{\langle k\rangle}} \end{array}\right).

Now for the counterexample: u^{\widehat{Q}}_{67}u^{\widehat{Q}}_{41}u^{\widehat{Q}}_{87}\neq 0 so  (6,4,8)\sim_3 (7,1,7) and u_{78}^{\widehat{Q}}u_{14}^{\widehat{Q}}u_{78}^{\widehat{Q}}\neq0 so (7,1,7)\sim_3(8,4,8), but u^{\widehat{Q}}_{68}u^{\widehat{Q}}_{44}u^{\widehat{Q}}_{88}=0 so (6,4,8) is not related to (8,4,8) and so \sim_3 is not transitive.

That u^{\widehat{Q}}_{68}u^{\widehat{Q}}_{44}u^{\widehat{Q}}_{88}=0 is a bit of algebra, and I guess the others are too… but instead we can exhibit states \varsigma_2,\,\varsigma_1\in S(F(\widehat{Q})) such that \varsigma_2(|u^{\widehat{Q}}_{67}u^{\widehat{Q}}_{41}u^{\widehat{Q}}_{87}|^2)>0 and \varsigma_1(|u_{78}^{\widehat{Q}}u_{14}^{\widehat{Q}}u_{78}^{\widehat{Q}}|^2)>0 instead. The algebra structure of F(\widehat{Q}) is:

\displaystyle F(\widehat{Q})=\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus M_2(\mathbb{C})\subset B(\mathbb{C}^6).

Define \varsigma_2 to be the vector state associated with \xi_2:=(0,0,0,0,1/\sqrt{2},1/\sqrt{2}). Then:

\|u^{\widehat{Q}}_{67}u^{\widehat{Q}}_{41}u^{\widehat{Q}}_{87}(\xi_2)\|^2=\frac14.

\varsigma_2\in\widehat{Q} is a quantum permutation such that:

\mathbb{P}[(\varsigma_2(7)=6)\succ (\varsigma_2(1)=4)\succ (\varsigma_2(7)=8)]=\frac{1}{4}.

Similarly the vector state \varsigma_1 given by \xi_1:=(0,0,0,0,0,1) has

\|u^{\widehat{Q}}_{78}u^{\widehat{Q}}_{14}u^{\widehat{Q}}_{78}(\xi_1)\|^2=:\mathbb{P}[(\varsigma_1(8)=7)\succ (\varsigma_1(4)=1)\succ (\varsigma_1(8)=7)]>0.

Now, classically we might expect that \varsigma_2\star \varsigma_1 (convolution) might have the property that:

(\varsigma_2\star\varsigma_1)(|u^{\widehat{Q}}_{68}u^{\widehat{Q}}_{44}u^{\widehat{Q}}_{88}|^2)>0,

but as we have seen the product in question is zero.

Edit: The reason this phenomenon happens is that u_{11}^{\widehat{Q_8}} and u_{55}^{\widehat{Q_8}} are projections to random/classical permutations! I have also found a counterexample in the Kac-Paljutkin quantum group for similar reasons.

In the paper under preparation I think I should be able to produce nice, constructive, proofs of the transitivity of \sim_1 and \sim_2, constructive in the sense that in both cases I think I can exhibit states on C(\mathbb{G}) that are non-zero on suitable products of u_{ij}, using I think the conditioning of states:

\displaystyle\varsigma\mapsto \frac{\varsigma(u_{ij}\cdot u_{ij})}{\varsigma(u_{ij})}.

There is also something here to say about the maximality of S_N<S_N^+. All must wait for this paper though (no I don’t have a proof of this)!

I am currently (slowly) working on an essay/paper where I expand upon the ideas in this talk. In this post I will try and explain in this framework why there is no quantum cyclic group, no quantum S_3, and ask why there is no quantum alternating group.

Quantum Permutations Basics

Let A be a unital \mathrm{C}^*-algebra. We say that a matrix u\in M_N(A) is a magic unitary if each entry is a projection u_{ij}=u_{ij}^2=u_{ij}^*, and each row and column of u is a partition of unity, that is:

\displaystyle \sum_ku_{ik}=\sum_k u_{kj}=1_A.

It is necessarily the case (but not for *-algebras) that elements along the same row or column are orthogonal:

u_{ij}u_{ik}=\delta_{j,k}u_{ij} and u_{ij}u_{k j}=\delta_{i,k}u_{ij}.

Shuzou Wang defined the algebra of continuous functions on the quantum permutation group on N symbols to be the universal \mathrm{C}^*-algebra C(S_N^+) generated by an N\times N magic unitary u. Together with (leaning heavily on the universal property) the *-homomorphism:

\displaystyle \Delta:C(S_N^+)\rightarrow C(S_N^+)\underset{\min}{\otimes}C(S_N^+), u_{ij}\mapsto \sum_{k=1}^N u_{ik}\otimes u_{kj},

and the fact that u and (u)^t are invertible (u^{-1}=u^t)), the quantum permutation group S_N^+ is a compact matrix quantum group.

Any compact matrix quantum group generated by a magic unitary is a quantum permutation group in that it is a quantum subgroup of the quantum permutation group. There are finite quantum groups (finite dimensional algebra of functions) which are not quantum permutation groups and so Cayley’s Theorem does not hold for quantum groups. I think this is because we can have quantum groups which act on algebras such as M_N(\mathbb{C}) rather than \mathbb{C}^N — the algebra of functions equivalent of the finite set \{1,2,\dots,N\}.

This is all basic for quantum group theorists and probably unmotivated for everyone else. There are traditional motivations as to why such objects should be considered algebras of functions on quantum groups:

  • find a presentation of an algebra of continuous functions on a group, C(G), as a commutative universal \mathrm{C}^*-algebra. Study the the same object liberated by dropping commutativity. Call this the quantum or free version of G, G^+.
  • quotient C(S_N^+) by the commutator ideal, that is we look at the commutative \mathrm{C}^*-algebra generated by an N\times N magic unitary. It is isomorphic to F(S_N), the algebra of functions on (classical) S_N.
  • every commutative algebra of continuous functions on a compact matrix quantum group is the algebra of functions on a (classical) compact matrix group, etc.

Here I want to take a very different direction which while motivationally rich might be mathematically poor.

Weaver Philosophy

Take a quantum permutation group \mathbb{G} and represent the algebra of functions as bounded operators on a Hilbert space \mathsf{H}. Consider a norm-one element \varsigma\in P(\mathsf{H}) as a quantum permutation. We study the properties of the quantum permutation by making a series of measurements using self-adjoint elements of C(\mathbb{G}).

Suppose we have a finite-spectrum, self-adjoint measurement f\in C(\mathbb{G})\subset B(\mathsf{H}). It’s spectral decomposition gives a partition of unity (p^{f_i})_{i=1}^{|\sigma(f)|}. The measurement of \varsigma with f gives the value f_i with probability:

\displaystyle \mathbb{P}[f=f_i\,|\,\varsigma]=\langle\varsigma,p^{f_i}\varsigma\rangle=\|p^{f_i}\varsigma\|^2,

and we have the expectation:

\displaystyle \mathbb{E}[f\,|\,\varsigma]=\langle\varsigma,f\varsigma\rangle.

What happens if the measurement of \varsigma with f yields f=f_i (which can only happen if p^{f_i}\varsigma\neq 0)? Then we have some wavefunction collapse of

\displaystyle \varsigma\mapsto p^{f_i}\varsigma\equiv \frac{p^{f_i}\varsigma}{\|p^{f_i}\varsigma\|}\in P(\mathsf{H}).

Now we can keep playing the game by taking further measurements. Notationally it is easier to describe what is happening if we work with projections (but straightforward to see what happens with finite-spectrum measurements). At this point let me quote from the essay/paper under preparation:

Suppose that the “event” p=\theta_1 has been observed so that the state is now p^{\theta_1}(\psi)\in P(\mathsf{H}). Note this is only possible if p=\theta_1 is non-null in the sense that

\displaystyle \mathbb{P}[p=\theta_1\,|\,\psi]=\langle\psi,p^\theta(\psi)\rangle\neq 0.

The probability that measurement produces q=\theta_2, and p^{\theta_1}(\psi)\mapsto q^{\theta_2}p^{\theta_1}(\psi)\in P(\mathsf{H}), is:

\displaystyle \mathbb{P}\left[q=\theta_2\,|\,p^{\theta_1}(\psi)\right]:=\left\langle \frac{p^{\theta_1}(\psi)}{\|p^{\theta_1}(\psi)\|},q^{\theta_2}\left(\frac{p^{\theta_1}(\psi)}{\|p^{\theta_1}(\psi)\|}\right)\right\rangle=\frac{\langle p^{\theta_1}(\psi),q^{\theta_2}(p^{\theta_1}(\psi))\rangle}{\|p^{\theta^1}(\psi)\|^2}.

Define now the event \left((q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right), said “given the state \psi, q is measured to be \theta_2 after p is measured to be \theta_1“. Assuming that p=\theta_1 is non-null, using the expression above a probability can be ascribed to this event:

\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]:=\mathbb{P}[p=\theta_1\,|,\psi]\cdot \mathbb{P}[q=\theta_2\,|\,p^{\theta_1}(\psi)]
\displaystyle =\langle\psi,p^{\theta_1}(\psi)\rangle\frac{\langle p^{\theta_1}(\psi),q^{\theta_2}(p^{\theta_1}(\psi))\rangle}{\|p^{\theta^1}(\psi)\|^2}
=\|q^{\theta_2}p^{\theta_1}\psi\|^2.

Inductively, for a finite number of projections \{p_i\}_{i=1}^n, and \theta_i\in{0,1}:

\displaystyle \mathbb{P}\left[(p_n=\theta_n)\succ\cdots \succ(p_1=\theta_1)\,|\,\psi\right]=\|p_n^{\theta_n}\cdots p_1^{\theta_1}\psi\|^2.

In general, pq\neq qp and so

\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]\neq \mathbb{P}\left[(p=\theta_1)\succ (q=\theta_1)\,|\,\psi\right],

and this helps interpret that q and p are not simultaneously observable. However the sequential projection measurement q\succ p is “observable” in the sense that it resembles random variables with values in \{0,1\}^2. Inductively the sequential projection measurement p_n\succ \cdots\succ p_1 resembles a \{0,1\}^n-valued random variable, and

\displaystyle \mathbb{P}[p_n\succ \cdots\succ p_1=(\theta_n,\dots,\theta_1)\,|\,\psi]=\|p_n\cdots p_1(\psi)\|^2.

If p and q do commute, they share an orthonormal eigenbasis, and it can be interpreted that they can “agree” on what they “see” when they “look” at \mathsf{H}, and can thus be determined simultaneously. Alternatively, if they commute then the distributions of q\succ p and p\succ q are equal in the sense that

\displaystyle \mathbb{P}\left[(q=\theta_2)\succ (p=\theta_1)\,|\,\psi\right]= \mathbb{P}\left[(p=\theta_1)\succ (q=\theta_1)\,|\,\psi\right],

it doesn’t matter what order they are measured in, the outputs of the measurements can be multiplied together, and this observable can be called pq=qp.

Consider the (classical) permutation group S_N or moreover its algebra of functions F(S_N). The elements of F(S_N) can be represented as bounded operators on \ell^2(S_N), and the algebra is generated by a magic unitary u^{S_N}\in M_N(B(\ell^2(S_N))) where:

u_{ij}^{S_N}(e_\sigma)=\mathbf{1}_{j\rightarrow i}(e_\sigma)e_{\sigma}.

Here \mathbf{1}_{j\rightarrow i}\in F(S_N) (‘unrepresented’) that asks of \sigma… do you send j\rightarrow i? One for yes, zero for no.

Recall that the product of commuting projections is a projection, and so as F(S_N) is commutative, products such as:

\displaystyle p_\sigma:=\prod_{j=1}^Nu_{\sigma(j)j}^{S_N},

There are, of, course, N! such projections, they form a partition of unity themselves, and thus we can build a measurement that will identify a random permutation \varsigma\in P(\ell^2(S_N)) and leave it equal to some e_\sigma after measurement. This is the essence of classical… all we have to do is enumerate n:S_N\rightarrow \{1,\dots,N!\} and measure using:

\displaystyle f=\sum_{\sigma\in S_N}n(\sigma)p_{\sigma}.

A quantum permutation meanwhile is impossible to pin down in such a way. As an example, consider the Kac-Paljutkin quantum group of order eight which can be represented as F(\mathfrak{G}_0)\subset B(\mathbb{C}^6). Take \varsigma=e_5\in \mathbb{C}^6. Then

\displaystyle\mathbb{P}[(\varsigma(1)=4)\succ(\varsigma(3)=1)\succ(\varsigma(1)=3)]=\frac{1}{8}.

If you think for a moment this cannot happen classically, and the issue is that we cannot know simultaneously if \varsigma(1)=3 and \varsigma(3)=1… and if we cannot know this simultaneously we cannot pin down \varsigma to a single element of S_N.

No Quantum Cyclic Group

Suppose that \varsigma\in \mathsf{H} is a quantum permutation (in S_N^+). We can measure where the quantum permutation sends, say, one to. We simply form the self-adjoint element:

\displaystyle x(1)=\sum_{k=1}^Nku_{k1}.

The measurement will produce some k\in \{1,\dots,N\}… but if \varsigma is supposed to represent some “quantum cyclic permutation” then we already know the values of \varsigma(2),\dots,\varsigma(N) from \varsigma(1)=k, and so, after measurement,

u_{k1}\varsigma \in \bigcap_{m=1}^N \text{ran}(u_{m+k-1,m}), u_{k1}\varsigma\equiv k-1\in\mathbb{Z}_N.

The significance of the intersection is that whatever representation of C(S_N^+) we have, we find these subspaces to be C(S_N^+)-invariant, and can be taken to be one-dimensional.

I believe this explains why there is no quantum cyclic group.

Question 1

Can we use a similar argument to show that there is no quantum version of any abelian group? Perhaps using F(G\times H)=F(G)\otimes F(H) together with the structure theorem for finite abelian groups?

No Quantum S_3

Let C(S_3^+) be represented as bounded operators on a Hilbert space \mathsf{H}. Let \varsigma\in P(\mathsf{H}). Consider the random variable

x(1)=u_{11}+2u_{21}+3u_{31}.

Assume without loss of generality that u_{31}\varsigma\neq0 then measuring \varsigma with x(1) gives x(1)\varsigma=3 with probability \langle\varsigma,u_{31}\varsigma\rangle, and the quantum permutation projects to:

\displaystyle \frac{u_{31}\varsigma}{\|u_{31}\varsigma\|}\in P(\mathsf{H}).

Now consider (for any \varsigma\in P(\mathsf{H}), using the fact that u_{21}u_{31}=0=u_{32}u_{31} and the rows and columns of u are partitions of unity:

u_{31}\varsigma=(u_{12}+u_{22}+u_{32})u_{31}\varsigma=(u_{21}+u_{22}+u_{23})u_{31}\varsigma

\Rightarrow u_{12}u_{31}\varsigma=u_{23}u_{31}\varsigma (*)

Now suppose, again without loss of generality, that measurement of u_{31}\varsigma\in P(\mathsf{H}) with x(2)=u_{12}+2u_{22}+3u_{33} produces x(2)u_{31}\varsigma=1, then we have projection to u_{12}u_{31}\varsigma\in P(\mathsf{H}). Now let us find the Birkhoff slice of this. First of all, as x(2)=1 has just been observed it looks like:

\Phi(u_{12}u_{31}\varsigma)=\left[\begin{array}{ccc}0 & 1 & 0 \\ \ast & 0 & \ast \\ \ast & 0 & \ast \end{array}\right]

In light of (*), let us find \Phi(u_{12}u_{31}\varsigma)_{23}. First let us normalise correctly to

\displaystyle \frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|}

So

\displaystyle\Phi(u_{12}u_{31}\varsigma)_{23}=\left\langle\frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|},u_{23}\frac{u_{12}u_{31}\varsigma}{\|u_{12}u_{31}\varsigma\|}\right\rangle

Now use (*):

\displaystyle\Phi(u_{12}u_{31}\varsigma)_{23}=\left\langle\frac{u_{23}u_{31}\varsigma}{\|u_{23}u_{31}\varsigma\|},u_{23}\frac{u_{23}u_{31}\varsigma}{\|u_{23}u_{31}\varsigma\|}\right\rangle=1

\displaystyle \Rightarrow \Phi(u_{12}u_{31}\varsigma)=\left[\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 1 \\ \Phi(u_{12}u_{31}\varsigma)_{31} & 0 & 0 \end{array}\right],

and as \Phi maps to doubly stochastic matrices we find that \Phi(u_{12}u_{31}\varsigma) is equal to the permutation matrix (132).

Not convincing? Fair enough, here is proper proof inspired by the above:

Let us show u_{11}u_{22}=u_{22}u_{11}. Fix a Hilbert space representation C(S_3^+)\subset B(\mathsf{H}) and let \varsigma\in\mathsf{H}.

The basic idea of the proof is, as above, to realise that once a quantum permutation \varsigma is observed sending, say, 3\rightarrow 2, the fates of 2 and 1 are entangled: if you see 2\rightarrow 3 you know that 1\rightarrow 1.

This is the conceptional side of the proof.

Consider u_{23}\varsigma which is equal to both:

(u_{11}+u_{21}+u_{31})u_{23}\varsigma=(u_{31}+u_{32}+u_{33})u_{23}\varsigma\Rightarrow u_{11}u_{23}\varsigma=u_{32}u_{23}\varsigma.

This is the manifestation of, if you know 3\rightarrow 2, then two and one are entangled. Similarly we can show that u_{22}u_{13}\varsigma=u_{31}u_{13}\varsigma and u_{22}u_{33}=u_{11}u_{33}.

Now write

\varsigma=u_{13}\varsigma+u_{23}\varsigma+u_{33}\varsigma

\Rightarrow u_{11}\varsigma=u_{11}u_{23}\varsigma+u_{11}u_{33}\varsigma=u_{32}u_{23}\varsigma+u_{22}u_{33}\varsigma

\Rightarrow u_{22}u_{11}\varsigma=u_{22}u_{33}\varsigma.

Similarly,

u_{22}\varsigma=u_{22}u_{13}\varsigma+u_{22}u_{33}\varsigma=u_{31}u_{13}\varsigma+u_{22}u_{33}\varsigma

\Rightarrow u_{11}u_{22}\varsigma=u_{11}u_{22}u_{33}\varsigma=u_{11}u_{11}u_{33}\varsigma=u_{11}u_{33}\varsigma=u_{22}u_{33}\varsigma

Which is equal to u_{22}u_{11}x, that is u_{11} and u_{22} commute.

Question 2

Is it true that if every quantum permutation in a \mathsf{H} can be fully described using some combination of u_{ij}-measurements, then the quantum permutation group is classical? I believe this to be true.

Quantum Alternating Group

Freslon, Teyssier, and Wang state that there is no quantum alternating group. Can we use the ideas from above to explain why this is so? Perhaps for A_4.

A possible plan of attack is to use the number of fixed points, \text{tr}(u), and perhaps show that \text{tr}(u) commutes with x(1). If you know these two simultaneously you nearly know the permutation. Just for completeness let us do this with (\text{tr}(u),x(1)):

tr(u)\x(1)1234
0(12)(34)(13)(24)(14)(23)
1(234),(243)(134),(143)(124),(142)(123),(132)
4e

The problem is that we cannot assume that that the spectrum of \text{tr}(u) is \{0,1,4\}, and, euh, the obvious fact that it doesn’t actually work.

What is more promising is

x(1)\x(2)1234
1e(234)(243)
2(12)(34)(123)(124)
3(132)(134)(13)(24)
4(142)(143)(14)(23)

However while the spectrums of x(1) and x(2) are cool (both in \{1,2,3,4\}), they do not commute.

Question 3

Are there some measurements that can identify an element of A_4 and via a positive answer to Question 3 explain why there is no quantum A_4? Can this be generalised to A_n.

Giving a talk 17:00, September 1 2020:

See here for more.

Slides.

This post follows on from this one. The purpose of posts in this category is for me to learn more about the research being done in quantum groups. This post looks at this paper of Schmidt.

Preliminaries

Compact Matrix Quantum Groups

The author gives the definition and gives the definition of a (left, quantum) group action.

Definition 1.2

Let G be a compact matrix quantum group and let C(X) be a \mathrm{C}^*-algebra. An (left) action of G on X is a unital *-homomorphism \alpha: C(X)\rightarrow C(X)\otimes C(G) that satisfies the analogue of g_2(g_1x)=(g_2g_1)x, and the Podlés density condition:

\displaystyle \overline{\text{span}(\alpha(C(X)))(\mathbf{1}_X\otimes C(G))}=C(X)\otimes C(G).

Quantum Automorphism Groups of Finite Graphs

Schmidt in this earlier paper gives a slightly different presentation of \text{QAut }\Gamma. The definition given here I understand:

Definition 1.3

The quantum automorphism group of a finite graph \Gamma=(V,E) with adjacency matrix A is given by the universal \mathrm{C}^*-algebra C(\text{QAut }\Gamma) generated by u\in M_n(C(\text{QAut }\Gamma)) such that the rows and columns of u are partitions of unity and:

uA=Au.

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The difference between this definition and the one given in the subsequent paper is that in the subsequent paper the quantum automorphism group is given as a quotient of C(S_n^+) by the ideal given by \mathcal{I}=\langle Au=uA\rangle… ah but this is more or less the definition of universal \mathrm{C}^*-algebras given by generators E and relations R:

\displaystyle \mathrm{C}^*(E,R)=\mathrm{C}^*( E)/\langle \mathcal{R}\rangle

\displaystyle \Rightarrow \mathrm{C}^*(E,R)/\langle I\rangle=\left(\mathrm{C}^*(E)/R\right)/\langle I\rangle=\mathrm{C}*(E)/(\langle R\rangle\cap\langle I\rangle)=\mathrm{C}^*(E,R\cap I)

where presumably \langle R\rangle \cap \langle I \rangle=\langle R\cap I\rangle all works out OK, and it can be shown that I is a suitable ideal, a Hopf ideal. I don’t know how it took me so long to figure that out… Presumably the point of quotienting by (a presumably Hopf) ideal is so that the quotient gives a subgroup, in this case \text{QAut }\Gamma\leq S_{|V|}^+ via the surjective *-homomorphism:

C(S_n^+)\rightarrow C(S_n^+)/\langle uA=Au\rangle=C(\text{QAut }\Gamma).

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Compact Matrix Quantum Groups acting on Graphs

Definition 1.6

Let \Gamma be a finite graph and G a compact matrix quantum group. An action of G on \Gamma is an action of G on V (coaction of C(G) on C(V)) such that the associated magic unitary v=(v_{ij})_{i,j=1,\dots,|V|}, given by:

\displaystyle \alpha(\delta_j)=\sum_{i=1}^{|V|} \delta_i\otimes v_{ij},

commutes with the adjacency matrix, uA=Au.

By the universal property, we have G\leq \text{QAut }\Gamma via the surjective *-homomorphism:

C(\text{QAut }\Gamma)\rightarrow C(G), u\mapsto v.

Theorem 1.8 (Banica)

Let X_n=\{1,\dots,n\}, and \alpha:F(X_n)\rightarrow F(X_n)\otimes C(G), \alpha(\delta_j)=\sum_i\delta_i\otimes v_{ij} be an action, and let F(K) be a linear subspace given by a subset K\subset X_n. The matrix v commutes with the projection onto F(K) if and only if \alpha(F(K))\subseteq F(K)\otimes C(G)

Corollary 1.9

The action \alpha: F(V)\rightarrow F(V)\otimes C(\text{QAut }\Gamma) preserves the eigenspaces of A:

\alpha(E_\lambda)\subseteq E_\lambda\otimes C(\text{QAut }\Gamma)

Proof: Spectral decomposition yields that each E_\lambda, or rather the projection P_\lambda onto it, satisfies a polynomial in A:

\displaystyle P_\lambda=\sum_{i}c_iA^i

\displaystyle \Rightarrow P_\lambda A=\left(\sum_i c_i A^i\right)A=A P_\lambda,

as A commutes with powers of A \qquad \bullet

A Criterion for a Graph to have Quantum Symmetry

Definition 2.1

Let V=\{1,\dots,|V|\}. Permutations \sigma,\,\tau: V\rightarrow V are disjoint if \sigma(i)\neq i\Rightarrow \tau(i)=i, and vice versa, for all i\in V.

In other words, we don’t have \sigma and \tau permuting any vertex.

Theorem 2.2

Let \Gamma be a finite graph. If there exists two non-trivial, disjoint automorphisms \sigma,\tau\in\text{Aut }\Gamma, such that o(\sigma)=n and o(\tau)=m, then we get a surjective *-homomorphism C(\text{QAut }\Gamma)\rightarrow C^*(\mathbb{Z}_n\ast \mathbb{Z}_m). In this case, we have the quantum group \widehat{\mathbb{Z}_n\ast \mathbb{Z}_m}\leq \text{QAut }\Gamma, and so \Gamma has quantum symmetry.

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Warning: This is written by a non-expert (I know only about finite quantum groups and am beginning to learn my compact quantum groups), and there is no attempt at rigour, or even consistency. Actually the post shows a wanton disregard for reason, and attempts to understand the incomprehensible and intuit the non-intuitive. Speculation would be too weak an adjective.

Groups

A group is a well-established object in the study of mathematics, and for the purposes of this post we can think of a group G as the set of symmetries on some kind of space, given by a set X together with some additional structure D(X). The elements of G  act on X as bijections:

G \ni g:X\rightarrow X,

such that D(X)=D(g(X)), that is the structure of the space is invariant under g.

For example, consider the space (X_n,|X_n|), where the set is X_n=\{1,2,\dots,n\}, and the structure is the cardinality. Then the set of all of the bijections X_n\rightarrow X_n is a group called S_n.

A set of symmetries G, a group, comes with some structure of its own. The identity map e:X\rightarrow X, x\mapsto x is a symmetry. By transitivity, symmetries g,h\in G can be composed to form a new symmetry gh:=g\circ h\in G. Finally, as bijections, symmetries have inverses g^{-1}, g(x)\mapsto x.

Note that:

gg^{-1}=g^{-1}g=e\Rightarrow (g^{-1})^{-1}=g.

A group can carry additional structure, for example, compact groups carry a topology in which the composition G\times G\rightarrow G and inverse {}^{-1}:G\rightarrow G are continuous.

Algebra of Functions

Given a group G together with its structure, one can define an algebra A(G) of complex valued functions on G, such that the multiplication A(G)\times A(G)\rightarrow A(G) is given by a commutative pointwise multiplication, for s\in G:

(f_1f_2)(s)=f_1(s)f_2(s)=(f_2f_1)(s).

Depending on the class of group (e.g. finite, matrix, compact, locally compact, etc.), there may be various choices and considerations for what algebra of functions to consider, but on the whole it is nice if given an algebra of functions A(G) we can reconstruct G.

Usually the following transpose maps will be considered in the structure of A(G), for some tensor product \otimes_\alpha such that A(G\times G)\cong A(G)\otimes_\alpha A(G), and m:G\times G\rightarrow G, (g,h)\mapsto gh is the group multiplication:

\begin{aligned}  \Delta: A(G)\rightarrow A(G)\otimes_{\alpha}A(G)&,\,f\mapsto f\circ m,\,\text{the comultiplication}  \\ S: A(G)\rightarrow A(G)&,\, f\mapsto f\circ {}^{-1},\,\text{ the antipode}  \\ \varepsilon: A(G)\rightarrow \mathbb{C}&,\, f\mapsto f\circ e,\,\text{ the counit}  \end{aligned}

See Section 2.2 to learn more about these maps and the relations between them for the case of the complex valued functions on finite groups.

Quantum Groups

Quantum groups, famously, do not have a single definition in the same way that groups do. All definitions I know about include a coassociative (see Section 2.2) comultiplication \Delta: A(G)\rightarrow A(G)\otimes_\alpha A(G) for some tensor product \otimes_\alpha (or perhaps only into a multiplier algebra M(A(G)\otimes_\alpha A(G))), but in general that structure alone can only give a quantum semigroup.

Here is a non-working (quickly broken?), meta-definition, inspired in the usual way by the famous Gelfand Theorem:

A quantum group G is given by an algebra of functions A(G) satisfying a set of axioms \Theta such that:

  • whenever A(G) is noncommutative, G is a virtual object,
  • every commutative algebra of functions satisfying \Theta is an algebra of functions on a set-of-points group, and
  • whenever commutative algebras of functions A(G_1)\cong_{\Theta} A(G_2), G_1\cong G_2 as set-of-points groups.

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In May 2017, shortly after completing my PhD and giving a talk on it at a conference in Seoul, I wrote a post describing the outlook for my research.

I can go through that post paragraph-by-paragraph and thankfully most of the issues have been ironed out. In May 2018 I visited Adam Skalski at IMPAN and on that visit I developed a new example (4.2) of a random walk (with trivial n-dependence) on the Sekine quantum groups Y_n with upper and lower bounds sharp enough to prove the non-existence of the cutoff phenomenon. The question of developing a walk on Y_n showing cutoff… I now think this is unlikely considering the study of Isabelle Baraquin and my intuitions about the ‘growth’ of Y_n (perhaps if cutoff doesn’t arise in somewhat ‘natural’ examples best not try and force the issue?). With the help of Amaury Freslon, I was able to improve to presentation of the walk (Ex 4.1) on the dual quantum group \widehat{S_n}. With the help of others, it was seen that the quantum total variation distance is equal to the projection distance (Prop. 2.1). Thankfully I have recently proved the Ergodic Theorem for Random Walks on Finite Quantum Groups. This did involve a study of subgroups (and quasi-subgroups) of quantum groups but normal subgroups of quantum groups did not play so much of a role as I expected. Amaury Freslon extended the upper bound lemma to compact Kac algebras. Finally I put the PhD on the arXiv and also wrote a paper based on it.

Many of these questions, other questions in the PhD, as well as other questions that arose around the time I visited Seoul (e.g. what about random transpositions in S_n^+?) were answered by Amaury Freslon in this paper. Following an email conversation with Amaury, and some communication with Uwe Franz, I was able to write another post outlining the state of play.

This put some of the problems I had been considering into the categories of Solved, to be Improved, More Questions, and Further Work. Most of these have now been addressed. That February 2018 post gave some direction, led me to visit Adam, and I got my first paper published.

After that paper, my interest turned to the problem of the Ergodic Theorem, and in May I visited Uwe in Besancon, where I gave a talk outlining some problems that I wanted to solve. The main focus was on proving this Ergodic Theorem for Finite Quantum Groups, and thankfully that has been achieved.

What I am currently doing is learning my compact quantum groups. This work is progressing (albeit slowly), and the focus is on delivering a series of classes on the topic to the functional analysts in the UCC School of Mathematical Sciences. The best way to learn, of course, is to teach. This of course isn’t new, so here I list some problems I might look at in short to medium term. Some of the following require me to know my compact quantum groups, and even non-Kac quantum groups, so this study is not at all futile in terms of furthering my own study.

I don’t really know where to start. Perhaps I should focus on learning my compact quantum groups for a number of months before tackling these in this order?

  1. 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.
  2. Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac?
  3. Following Urban, study convolution factorisations of the Haar state.
  4. Examples of non-central random walks on compact groups.
  5. Extending the Upper Bound Lemma to the non-Kac case. As I speak, this is beyond what I am capable of. This also requires work on the projection and quantum total variation distances (i.e. show they are equal in this larger category)

Finally cracked this egg.

Preprint here.

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