
On the classification of partition quantum groups, to appear in Exp. Math. 39 (2021), n^{o} 2, 238270 : link and pdf (old version).
Abstract
This is a survey on some results obtained recently in the classification of compact quantum groups associated to partitions, with a focus on the noncrossing case.
We take a global look at the main results in the subject and highlight some key features of the methods used.
We conclude by several suggestions for pushing further the classification.

Positive definite functions and cutoff for discrete groups, to appear in Canad. Math. Bull. 64 (2021), n^{o} 2, 306322 : link and pdf.
Abstract
We consider the sequence of powers a fixed positive definite function on a discrete group.
Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cutoff phenomenon occurs for this sequence,
including free groups and infinite Coxeter groups.
We also give examples of absence of cutoff using surface groups and free groups again.

Topological generation and matrix models for quantum reflection groups (with M. Brannan and A. Chirvasitu), Adv. Math. 363 (2020) : link and pdf.
Abstract
We establish several new topological generation results for the quantum permutation groups \(S_N^+\) $S$_{N}^{+} and the quantum reflection groups $H$_{N}^{s+}.
We use these results to show that these quantum groups admit sufficiently many "matrix models".
In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear),
and certain "flat" matrix models for $S$_{N}^{+} are inner faithful.

On twocoloured noncrossing partition quantum groups, Trans. Amer. Math. Soc. 372 (2019), n^{o} 6, 44714508 : link and pdf (old version).
Abstract
We classify compact quantum groups associated to noncrossing partitions coloured with two elements
$x$ and $y$ which are their own inverses.
Together with the work of P. Tarrago and M. Weber,
this completes the classification of all noncrossing quantum groups on two colours.
We also give some general structure results concerning noncrossing quantum groups
and suggest some more general classification statements.

On the representation theory of some noncrossing partition quantum groups, Algebr. Represent. Theory. 23 (2019), n^{o} 3, 483492 : link and pdf (old version).
Abstract
We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products.
This illustrates the efficiency of the methods developed in our previous joint work with M. Weber.

Quantum reflections, random walks and cutoff, Internat. J. Math. 29 (2018), n^{o} 14, 1850101 : link and pdf (old version).
Abstract
We study the cutoff phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections.
We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of such reflections.
We also study random walks on quantum reflection groups and more generally free wreath products of finite group by quantum permutation groups.

Cutoff phenomenon for random walks on free orthogonal quantum groups, Probab. Theory Related Fields 174 (2019), n^{o}34, 731760 : link and pdf (old version).
Abstract
We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups.
As a consequence, we prove that the analogue of the random rotation walk on this quantum group has a cutoff at $Nln(N)/2(1cos(\theta ))$.
This is the first result of this type for genuine compact quantum groups.

Torsion and Ktheory for some free wreath products (with R. Martos), Int. Math. Res. Not. 2020 (2020), n^{o} 6, 16391670 : link and pdf (old version).
Abstract
We classify torsion actions of free wreath products of arbitrary compact quantum groups and use this to prove that if $G$ is a torsionfree compact
quantum group satisfying the strong BaumConnes property, then $G\; \wr S$_{N}^{+} also satisfies the strong BaumConnes property.
We then compute the Ktheory of free wreath products of classical and quantum free groups by $SO$_{q}(3).

Modelling questions for quantum permutations (with T. Banica), Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), n^{o} 2, 126 : link and pdf (old version).
Abstract
Given a quantum permutation group $G\; \subset \; S$_{N}^{+}, with orbits having the same size $K$,
we construct a universal matrix model $\pi \; :\; C(G)\; \to \; M$_{K}(C(X)),
having the property that the images of the standard coordinates $u$_{ij} ∊ C(G)
are projections of rank $\le \; 1$.
Our conjecture is that this model is inner faithful under suitable algebraic assumptions,
and is in addition stationary under suitable analytic assumptions.
We fully discuss this conjecture for classical groups and prove it for several families of duals of discrete groups.

The radial MASA in free orthogonal quantum groups (with R. Vergnioux), J. Funct. Anal. 271 (2016), n^{o} 10, 27762807: link and pdf (old version).
Abstract
We prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing,
and we compute the associated bimodule.
The proof relies on new properties of the JonesWenzl projections and on an estimate of certain scalar products of
coefficients of irreducible representations.

Wreath products of finite groups by quantum groups (with A. Skalski), J. Noncommut. Geom. 12 (2018), n^{o} 1, 2968 : link and pdf (old version).
Abstract
We introduce a notion of partition wreath product of a finite group by a partition quantum group,
a construction motivated on the one hand by classical wreath products
and on the other hand by the free wreath product of J. Bichon.
We identify the resulting quantum group in several cases,
establish some of its properties and show that when the finite group in question is abelian,
the partition wreath product is itself a partition quantum group.
This allows us to compute its representation theory, using earlier results of the first named author.

On the partition approach to SchurWeyl duality and free quantum groups (with an appendix by A. Chirvasitu), Transform. Groups 22 (2017), n^{o} 3, 707751 : link and pdf (old version).
Abstract
We give a general definition of classical and quantum groups whose representation theory is "determined by partitions"
and study their structure.
This encompasses many examples of classical groups for which SchurWeyl duality is described with diagram algebras
as well as generalizations of P. Deligne's interpolated categories of representations.
Our setting is inspired by many previous works on easy quantum groups
and appears to be wellsuited to the study of free fusion semirings.
We classify free fusion semirings and prove that they can always be realized through our construction,
thus solving several open questions.
This suggests a general decomposition result for free quantum groups which in turn gives information on the compact groups whose
SchurWeyl duality is implemented by partitions.
The paper also contains an appendix by A. Chirvasitu proving simplicity results for the reduced C*algebras of some free quantum groups.

On bifree De Finetti theorems (with M. Weber), Ann. Math. Blaise Pascal 23 (2016), n^{o} 1, 2151 : link and pdf (old version).
Abstract
We investigate possible generalizations of the de Finetti theorem to bifree probability.
We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bifreeness.
We then study properties of families of pairs of variables which are invariant under this action,
both in the binoncommutative setting and in the usual noncommutative setting.
We do not have a completely satisfying analogue of the de Finetti theorem,
but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of $n$freeness.

Permanence of approximation properties for discrete quantum groups, Ann. Inst. Fourier 65 (2015), n^{o} 4, 14371467 : link and pdf (old version).
Abstract
We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups.
In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup.
We also define a notion of relative amenability for discrete quantum groups
and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.

Fusion (semi)rings arising from quantum groups, J. Algebra 417 (2014), 161197 : link and pdf (revised version).
Abstract
We study the fusion rings arising from easy quantum groups.
We classify all the possible free ones, answering a question of T. Banica and R. Vergnioux.
We then classify the possible groups of onedimensional representations for free easy quantum groups.
As an application, we give a unified proof of the Haagerup property for a broad class of easy quantum groups,
recovering as special cases previous results by M. Brannan and F. Lemeux.

On the representation theory of partition (easy) quantum groups (with M. Weber), J. Reine Angew. Math. 720 (2016), 155197 : link and pdf (old version).
Abstract
Compact matrix quantum groups are strongly determined by their intertwiner spaces,
due to a result by S.L. Woronowicz. In the case of easy quantum groups,
the intertwiner spaces are given by the combinatorics of partitions,
see the inital work of T. Banica and R. Speicher.
The philosophy is that all quantum algebraic properties of these objects should be visible in
their combinatorial data. We show that this is the case for their fusion rules
(i.e. for their representation theory). As a byproduct, we obtain a unified approach to the
fusion rules of the quantum permutation group $S$_{N}^{+},
the free orthogonal quantum group $O$_{N}^{+} as well as the
hyperoctahedral quantum group $H$_{N}^{+}.

Graphs of quantum groups and Kamenability (with P. Fima), Adv. Math. 260 (2014), 233280 : link and pdf (old version).
Abstract
Building on a construction of JP. Serre, we associate to any graph of C*algebras a maximal
and a reduced fundamental C*algebra and use this theory to construct
the fundamental quantum group of a graph of discrete quantum groups.
To illustrate the properties of this construction, we then prove that if all the vertex quantum groups are amenable,
the fundamental quantum group is Kamenable. This generalizes previous results of JulgValette, R. Vergnioux and P. Fima.

CCAP for universal discrete quantum groups (with K. De Commer and M. Yamashita, with an appendix by S. Vaes), Comm. Math. Phys. 331 (2014), n^{o} 2, 677701 : link and pdf (old version).
Abstract
In this paper we show that the discrete duals of the free orthogonal quantum groups have the Haagerup property
and the completely contractive approximation property.
Analogous results also hold for the free unitary quantum groups
and the quantum automorphism groups of finitedimensional C*algebras.
The proof relies on the monoidal equivalence between free orthogonal quantum groups and $SU$_{q}(2) quantum groups,
on the construction of a sufficient supply of bounded central functionals for $SU$_{q}(2) quantum groups,
and on the free product techniques of Ricard and Xu.
Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property,
and due to the second author on the CCAP.

Examples of weakly amenable discrete quantum groups, J. Funct. Anal. 265 (2013), n^{o} 9, 21642187 : link and pdf (old version).
Abstract
In this paper we give a polynomial bound for the completely bounded norm of the projections
on coefficients of a fixed irreducible representation
in free orthogonal quantum groups. This enables us to compute their CowlingHaagerup constant,
which happens to be equal to $1$. We then use an argument of monoidal equivalence to extend our result to other
free orthogonal quantum groups and quantum automorphism groups of finitedimensional C*algebras.
This gives in particular nonunimodular examples of weakly amenable discrete quantum groups which are not amenable.

A note on weak amenability for free products of discrete quantum groups, C. R. Acad. Sci. Paris 350 (2012), n^{o} 78, 403406 : link and pdf (old version).
Abstract
In this paper we prove that if two discrete quantum groups are weakly amenable with CowlingHaagerup constant equal to $1$,
then their free product is also weakly amenable with CowlingHaagerup constant equal to $1$.