Publications (64)27.15 Total impact
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ABSTRACT: Associated to a complex Hadamard matrix $H\in M_N(\mathbb C)$ is the complex probability measure $\mu\in\mathcal P(\mathbb C)$ describing the distribution of $\varphi(a,b)=<a,Hb>$, where $a,b\in\mathbb T^N$ are random. This measure is called "glow" of the matrix, due to the analogy with the GaleBerlekamp switching game, where $H,a,b$ are real. We prove here that: (1) $\mu$ becomes complex Gaussian in the $N\to\infty$ limit, (2) the universality holds as well at order 2, (3) the order 3 term seems to be quite interesting, particularly for the master Hadamard matrices, (4) in the Fourier matrix case, some of the higher order terms control counting problems for circulant Hadamard matrices.10/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest nontrivial case is M=3, and we obtain here several results, notably with a classification at N=7. We discuss as well the potential applications of the M=3 results to various $M=N$ questions.07/2013;  [Show abstract] [Hide abstract]
ABSTRACT: The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. The singularity at a point $H\in C_N$ is described by a filtration of cones $T^\times_HC_N\subset T^\circ_HC_N\subset T_HC_N\subset\widetilde{T}_HC_N$, coming from the trivial, affine, smooth and first order deformations. We study here these cones in the case where $H=F_N$ is the Fourier matrix, $(w^{ij})$ with $w=e^{2\pi i/N}$, our main result being a simple description of $\widetilde{T}_HC_N$. As a consequence, the rationality conjecture $dim_\mathbb R(\widetilde{T}_HC_N)=dim_\mathbb Q(\widetilde{T}_HC_N\cap M_N(\mathbb Q))$ holds at $H=F_N$.02/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $q_0=...=q_{N1}=1$ the quantity $\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the bruteforce minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here these questions, with some results and conjectures.12/2012;  [Show abstract] [Hide abstract]
ABSTRACT: In our previous work, we introduced the following relaxation of the Hadamard property: a square matrix $H\in M_N(\mathbb R)$ is called "almost Hadamard" if $U=H/\sqrt{N}$ is orthogonal, and locally maximizes the 1norm on O(N). We review our previous results, notably with the formulation of a new question, regarding the circulant and symmetric case. We discuss then an extension of the almost Hadamard matrix formalism, by making use of the pnorm on O(N), with $p\in[1,\infty]{2}$, with a number of theoretical results on the subject, and the formulation of some open problems.Discrete Applied Mathematics 11/2012; · 0.72 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. We have $C_N=M_N(\mathbb T)\cap\sqrt{N}U_N$, and following Tadej and \.Zyczkowski we investigate here the computation of the enveloping tangent space $\widetilde{T}_HC_N=T_HM_N(\mathbb T)\cap T_H\sqrt{N}U_N$, and notably of its dimension $d(H)=\dim(\widetilde{T}_HC_N)$, called undephased defect of $H$. Our main result is an explicit formula for the defect of the Fourier matrix $F_G$ associated to an arbitrary finite abelian group $G=\mathbb Z_{N_1}\times...\times\mathbb Z_{N_r}$. We also comment on the general question "does the associated quantum permutation group see the defect", with a probabilistic speculation involving DiaconisShahshahani type variables.Linear Algebra and its Applications. 10/2012; 438(9).  [Show abstract] [Hide abstract]
ABSTRACT: We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1norm on O(N). Our study includes a detailed discussion of the circulant case ($H_{ij}=\gamma_{ji}$) and of the twoentry case ($H_{ij}\in{x,y}$), with the construction of several families of examples, and some 1norm computations.02/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We study the random matrices of type $\tilde{W}=(id\otimes\varphi)W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $\varphi:M_n(\mathbb C)\to M_n(\mathbb C)$ is a selfadjoint linear map. We prove that, under suitable assumptions, we have the $d\to\infty$ eigenvalue distribution formula $\delta m\tilde{W}\sim\pi_{mn\rho}\boxtimes\nu$, where $\rho$ is the law of $\varphi$, viewed as a square matrix, $\pi$ is the free Poisson law, $\nu$ is the law of $D=\varphi(1)$, and $\delta=tr(D)$.01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\hat{\Lambda}\subset G$. In the matrix case $G\subset U_n^+$ the embedding condition is equivalent to having a quotient map $\Gamma_U\to\Lambda$, where $F=\{\Gamma_UU\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon's classification of group dual subgroups $\hat{\Lambda}\subset S_n^+$. These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have nonabelian group dual isometries.01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $\pi:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Ces\`aro limit of the linear functional $\phi=tr\circ\pi$. We discuss then some consequences of this result, notably to inner linearity questions.12/2011;  [Show abstract] [Hide abstract]
ABSTRACT: Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length.09/2011;  [Show abstract] [Hide abstract]
ABSTRACT: Given a quantum subgroup $G\subset U_n$ and a number $k\leq n$ we can form the homogeneous space $X=G/(G\cap U_k)$, and it follows from the StoneWeierstrass theorem that $C(X)$ is the algebra generated by the last $nk$ rows of coordinates on $G$. In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type $X=G/(G\cap U_k^+)$.Journal of Geometry and Physics 09/2011; · 1.06 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.09/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal", where $O_n$ is the usual orthogonal group and $O_n^*$ is the halfliberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.Communications in Algebra 06/2011; · 0.36 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study the partial transposition ${W}^\Gamma=(\mathrm{id}\otimes \mathrm{t})W\in M_{dn}(\mathbb C)$ of a Wishart matrix $W\in M_{dn}(\mathbb C)$ of parameters $(dn,dm)$. Our main result is that, with $d\to\infty$, the law of $m{W}^\Gamma$ is a free difference of free Poisson laws of parameters $m(n\pm 1)/2$. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line.Journal of Theoretical Probability 05/2011; · 0.55 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factorization $\mathbb S_4 = \mathbb Z_4 \mathbb S_3$ and has dimension 24. We show that, in fact, this is the smallest possible dimension that a non quantum permutation group can have.04/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We introduce and study natural twoparameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of nonnegative integers we define and investigate quantum groups O+(p,q), B+(p,q), S+(p,q) and H+(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier. For H+(p,q) the situation is different and we show that , where the latter can be viewed as a liberation of the classical isometry group of the pdimensional torus.Journal of Functional Analysis. 01/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We study the quantum isometry groups of the noncommutative Riemannian manifolds associated to discrete group duals. The basic representation theory problem is to compute the law of the main character of the relevant quantum group, and our main result here is as follows: for the group Z_s^{*n}, with s>4 and n>1, half of the character follows the compound free Poisson law with respect to the measure $\underline{\epsilon}$/2, where $\epsilon$ is the uniform measure on the sth roots of unity, and $\epsilon\to\underline{\epsilon}$ is the canonical projection map from complex to real measures. We discuss as well a number of technical versions of this result, notably with the construction of a new quantum group, which appears as a "representationtheoretic limit", at s equal to infinity.11/2010;  [Show abstract] [Hide abstract]
ABSTRACT: We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case $a\in M_{2\times q}(2\mathbb N)$ to the general case $a\in M_{p\times q}(\mathbb N)$, (2) the construction of the "best algebraic normalization" of $I(a)$, in the case $a\in M_{2\times q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{3\times 3}(\mathbb N)$, (4) a modelling result in the case $a\in M_{1\times 2}(\mathbb N)$, in relation with the EulerRodrigues formula. Most proofs use various combinatorial techniques.11/2010;  [Show abstract] [Hide abstract]
ABSTRACT: We study the Gram matrix determinants for the groups $S_n,O_n,B_n,H_n$, for their free versions $S_n^+,O_n^+,B_n^+,H_n^+$, and for the halfliberated versions $O_n^*,H_n^*$. We first collect all the known computations of such determinants, along with complete and simplified proofs, and with generalizations where needed. We conjecture that all these determinants decompose as $D=\prod_\pi\phi(\pi)$, with product over all associated partitions. Comment: 18 pagesJournal of Mathematical Physics 09/2010; · 1.30 Impact Factor
Publication Stats
782  Citations  
27.15  Total Impact Points  
Top Journals
Institutions

2009–2012

Université de CergyPontoise
95001 CEDEX, IledeFrance, France 
Queen's University
 Department of Mathematics & Statistics
Kingston, Ontario, Canada


2006–2009

University of Toulouse
Tolosa de Llenguadoc, MidiPyrénées, France


2008

University of Ottawa
Ottawa, Ontario, Canada


2005–2008

Paul Sabatier University  Toulouse III
 Département de Mathématiques
Tolosa de Llenguadoc, MidiPyrénées, France


2007

Claude Bernard University Lyon 1
Villeurbanne, RhôneAlpes, France


1998

Institut de Mathématiques de Luminy
Marsiglia, ProvenceAlpesCôte d'Azur, France 
University of California, Berkeley
 Department of Mathematics
Berkeley, California, United States
