Guest post by Amaury Freslon

At a recent conference celebrating Roland Speicher’s birthday, I gave a talk on a problem from the theory of compact quantum groups that has fascinated me for years and on which Roland and I have been working, albeit unsuccessfully. I wrote detailed notes introducing the problem, which are available here and Roland suggested I write a blog post to publicize them. So here I am !

Let’s begin by unpacking the terminology in the title. The quantum permutation group SN+, introduced by Sh. Wang, is defined via a universal ∗-algebra 𝒪(SN+), which is generated by entries of a matrix (pij). These entries are subject to relations that resemble those of classical permutation matrices, but with the key difference that they do not commute. More precisely, the defining properties are as follows:

• Each generator is idempotent and self-adjoint;
• They are pairwise orthogonal on rows and columns;
• They add up to 1 on rows and columns.

In contrast, let 𝒪(SN) be the algebra of all complex-valued function on the classical permutation group SN. The connection between SN+ and SN is given by a canonical ∗-homomorphism πab: 𝒪(SN+) → 𝒪(SN), which is defined by making all generators commute. In that sense, SN+ can be thought of as the free or quantum version of SN. We can now state our main problem:

Does there exist a compact quantum group 𝔾 such that SN<𝔾<SN+? In other words, is there a Hopf ∗-algebra 𝒪(𝔾) through which the map πab factors?

In algebraic terms, the problem asks whether there exists a non-trivial Hopf ideal inside the commutator ideal of 𝒪(SN+). However, understanding Hopf ideals in general is notoriously difficult, so we must look for alternative approaches. One promising direction involves exploiting the fact that the quantum permutation group is an easy quantum groups in the sense that all the relations in the algebra 𝒪(SN+) are encoded by partitions of finite sets. While we will not delve into the details here (see the notes and references therein), we can sketch a potential strategy.

1. The relations in 𝒪(SN+) are exactly those given by linear combinations of non-crossing partitions;
2. The relations in 𝒪(SN) are exactly those given by linear combinations of arbitrary partitions;
3. The problem therefore boils down to this: if we are given a relation corresponding to a linear combination of arbitrary partitions, can we combine it with relations coming from non-crossing partitions to obtain all possible relations from partitions?

While this approach seems reasonable, it has not proven efficient thus far. The only known results on the problem, which rely on different techniques, are as follows:

• For N<4, there is no intermediate quantum permutation group, because SN=SN+ in that case;
• For N=4, there is again no intermediate quantum permutation group, as can be observed on the complete list of quantum subgroups of S4+;
• For N=5, the absence of intermediate quantum permutation group was obtained by T. Banica relying on deep results from the classification of subfactors.

The notes outline an alternative approach developed by Roland Speicher and myself which is inspired by noncommutative probability theory. More precisely, any quantum subgroup 𝔾 of SN+ comes with a distinguished linear form, known as its Haar state. Pulling it back to 𝒪(SN+), we obtain a map with lots of symmetry properties coming from the fact that 𝔾 is assumed to contain SN. One might therefore try to prove that such a highly symmetric state must coincide with the Haar state of SN+ or with that of SN, thereby answering negatively the question.

To tackle this, we focus on the moments of the Haar state, that is, its values on elements of the form x = pi1j1pi2j2...pikjk. These moments satisfy algebraic relations that arise from the ones satisfied by the generators, along with the symmetries induced by SN. Using these, we were able to show that the moments of order up to 5 of such a state must either match the corresponding moments of SN+, or those of SN. Although this is encouraging, our method breaks down at order 6, were we encounter complicated systems of quadratic equations. Nonetheless, there is a glimmer of hope because it may be possible to incorporate an additional ingredient: the fact that the Haar state sends elements of the form x*x to positive real numbers.

I hope that this motivates the reader to take a closer look at these notes and perhaps try their luck at the problem.