arXiv subjects

Robert Coquereaux

Publications and source records attributed to Robert Coquereaux.

At least 19 recordsLinked to original sources

arXiv2024

SU(3) higher roots and their lattices

After recalling the notion of higher roots (or hyper-roots) associated with "quantum modules" of type $(G, k)$, for $G$ a semi-simple Lie group and $k$ a positive integer, following the definition given by A. Ocneanu in 2000, we study the theta series of their lattices. Here we only consider the higher roots associated with quantum modules (aka module-categories over the fusion category defined by the pair $(G,k)$) that are also "quantum subgroups". For $G=SU{2}$ the notion of higher roots coincides with the usual notion of roots for ADE Dynkin diagrams and the self-fusion restriction (the property of being a quantum subgroup) selects the diagrams of type $A_{r}$, $D_{r}$ with $r$ even, $E_6$ and $E_8$; their theta series are well known. In this paper we take $G=SU{3}$, where the same restriction selects the modules ${\mathcal A}_k$, ${\mathcal D}_k$ with $mod(k,3)=0$, and the three exceptional cases ${\mathcal E}_5$, ${\mathcal E}_9$ and ${\mathcal E}_{21}$. The theta series for their associated lattices are expressed in terms of modular forms twisted by appropriate Dirichlet characters.

math.QA↗

arXiv2024

Counting partitions by genus: a compendium of results

We study the enumeration of set partitions, according to their length, number of parts, cyclic type, and genus. We introduce genus-dependent Bell, Stirling numbers, and Faà di Bruno coefficients. Besides attempting to summarize what is already known on the subject, we obtain new generic results (in particular for partitions into two parts, for arbitrary genus), and present computer generated new data extending the number of terms known for sequences or families of such coefficients; this also leads to new conjectures.

math.CO↗

arXiv2023

About integer-valued variants of the theta and 6j symbols

These notes contain essentially a rewriting of several properties of two well-known quantities, the so-called theta symbol (or triangular symbol), which is rational, and the 6j symbol, which is usually irrational, in terms of two related integer-valued functions called gon and tet. Existence of these related integer-valued avatars, sharing most essential properties with their more popular partners, although a known fact, is often overlooked. The properties of gon and tet are easier to obtain, or to formulate, than those of the corresponding theta and 6j symbols, both in the classical and quantum situations. Their evaluation is also simpler (the paper displays a number of explicit formulae and evaluation procedures that may speed up some computer programs). These two integer-valued functions are unusual, in that their properties do not appear to be often discussed in the literature, but their features reflect those of related real-valued functions discussed in many places. Some of the properties that we shall discuss seem however to be new, in particular several relations between the function gon and the inverse Hilbert matrices.

math-ph↗

arXiv2021

About left-invariant geometry and homogeneous pseudo-Riemannian Einstein structures on the Lie group SU(3)

This is a collection of notes on the properties of left-invariant metrics on the eight-dimensional compact Lie group SU(3). Among other topics we investigate the existence of invariant pseudo-Riemannian Einstein metrics on this manifold. We recover the known examples (Killing metric and Jensen metric) in the Riemannian case (signature (8,0)), as well as a Gibbons et al example of signature (6,2), and we describe a new example, which is Lorentzian (ie of signature (7,1). In the latter case the associated metric is left-invariant, with isometry group SU(3) x U(1), and has positive Einstein constant. It seems to be the first example of a Lorentzian homogeneous Einstein metric on this compact manifold. These notes are arranged into a paper that deals with various other subjects unrelated with the quest for Einstein metrics but that may be of independent interest: Among other topics we describe the various groups that may arise as isometry groups of left-invariant metrics on SU(3), provide parametrizations for these metrics, give several explicit results about the curvatures of the corresponding Levi-Civita connections, discuss modified Casimir operators (quadratic, but also cubic) and Laplace-Beltrami operators. In particular we discuss the spectrum of the Laplacian for metrics that are invariant under SU(3) x U(2), a subject that may be of interest in particle physics.

math.DG↗

arXiv2020

On Horn's Problem and its Volume Function

We consider an extended version of Horn's problem: given two orbits $\mathcal{O}_α$ and $\mathcal{O}_β$ of a linear representation of a compact Lie group, let $A\in \mathcal{O}_α$, $B\in \mathcal{O}_β$ be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum $A+B$. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of $\mathrm{SO}(n)$, $\mathrm{SU}(n)$ and $\mathrm{USp}(n)$ respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood--Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of $B_2=\mathfrak{so}(5)$.

math-ph↗

arXiv2020

Multiplicities, pictographs, and volumes

The present contribution is the written counterpart of a talk given in Yerevan at the SQS'2019 International Workshop "Supersymmetries and Quantum Symmetries" (SQS'2019, 26 August - August 31, 2019). After a short presentation of various pictographs (O-blades, metric honeycombs) that one can use in order to calculate SU(n) multiplicities (Littlewood-Richardson coefficients, Kostka numbers), we briefly discuss the semi-classical limit of these multiplicities in relation with the Horn and Schur volume functions and with the so-called Rn-polynomials that enter the expression of volume functions. For n < 7 the decomposition of the Rn-polynomials on Lie group characters is already known, the case n=7 is obtained here.

math-ph↗

arXiv2020

On Schur problem and Kostka numbers

We reconsider the two related problems: distribution of the diagonal elements of a Hermitian n x n matrix of known eigenvalues (Schur) and determination of multiplicities of weights in a given irreducible representation of SU(n) (Kostka). It is well known that the former yields a semi-classical picture of the latter. We present explicit expressions for low values of n that complement those given in the literature, recall some exact (non asymptotic) relation between the two problems, comment on the limiting procedure whereby Kostka numbers are obtained from Littlewood-Richardson coefficients, and finally extend these considerations to the case of the B2 algebra, with a few novel conjectures.

math.RT↗

arXiv2019

Revisiting Horn's Problem

We review recent progress on Horn's problem, which asks for a description of the possible eigenspectra of the sum of two matrices with known eigenvalues. After revisiting the classical case, we consider several generalizations in which the space of matrices under study carries an action of a compact Lie group, and the goal is to describe an associated probability measure on the space of orbits. We review some recent results about the problem of computing the probability density via orbital integrals and about the locus of singularities of the density. We discuss some relations with representation theory, combinatorics, pictographs and symmetric polynomials, and we also include some novel remarks in connection with Schur's problem.

math-ph↗

arXiv2019

The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices

Horn's problem, i.e., the study of the eigenvalues of the sum $C=A+B$ of two matrices, given the spectrum of $A$ and of $B$, is re-examined, comparing the case of real symmetric, complex Hermitian and self-dual quaternionic $3\times 3$ matrices. In particular, what can be said on the probability distribution function (PDF) of the eigenvalues of $C$ if $A$ and $B$ are independently and uniformly distributed on their orbit under the action of, respectively, the orthogonal, unitary and symplectic group? While the two latter cases (Hermitian and quaternionic) may be studied by use of explicit formulae for the relevant orbital integrals, the case of real symmetric matrices is much harder. It is also quite intriguing, since numerical experiments reveal the occurrence of singularities where the PDF of the eigenvalues diverges. Here we show that the computation of the PDF of the symmetric functions of the eigenvalues for traceless $3\times 3$ matrices may be carried out in terms of algebraic functions;- roots of quartic polynomials;- and their integrals. The computation is carried out in detail in a particular case, and reproduces the expected singular patterns. The divergences are of logarithmic or inverse power type. We also relate this PDF to the (rescaled) structure constants of zonal polynomials and introduce a zonal analogue of the Weyl ${\rm SU}(n)$ characters.

math.RT↗

arXiv2018

From orbital measures to Littlewood-Richardson coefficients and hive polytopes

The volume of the hive polytope (or polytope of honeycombs) associated with a Littlewood- Richardson coefficient of SU(n), or with a given admissible triple of highest weights, is expressed, in the generic case, in terms of the Fourier transform of a convolution product of orbital measures. Several properties of this function -- a function of three non-necessarily integral weights or of three multiplets of real eigenvalues for the associated Horn problem-- are already known. In the integral case it can be thought of as a semi-classical approximation of Littlewood-Richardson coefficients. We prove that it may be expressed as a local average of a finite number of such coefficients. We also relate this function to the Littlewood-Richardson polynomials (stretching polynomials) i.e., to the Ehrhart polynomials of the relevant hive polytopes. Several SU(n) examples, for n=2,3,...,6, are explicitly worked out.

math.RT↗

arXiv2017

Theta functions for lattices of SU(3) hyper-roots

We recall the definition of the hyper-roots that can be associated to modules-categories over the fusion categories defined by the choice of a simple Lie group G together with a positive integer k. This definition was proposed in 2000, using another language, by Adrian Ocneanu. If G=SU(2), the obtained hyper-roots coincide with the usual roots for ADE Dynkin diagrams. We consider the associated lattices when G=SU(3) and determine their theta functions in a number of cases; these functions can be expressed as modular forms twisted by appropriate Dirichlet characters.

math.QA↗

arXiv2016

Conjugation Properties of Tensor Product and Fusion Coefficients

We review some recent results on properties of tensor product and fusion coefficients under complex conjugation of one of the factors. Some of these results have been proven, some others are conjectures awaiting a proof, one of them involving hitherto unnoticed observations on ordinary representation theory of finite simple groups of Lie type

math-ph↗

arXiv2016

On some properties of SU(3) Fusion Coefficients

Three aspects of the SU(3) fusion coefficients are revisited: the generating polynomials of fusion coefficients are written explicitly; some curious identities generalizing the classical Freudenthal-de Vries formula are derived; and the properties of the fusion coefficients under conjugation of one of the factors, previously analysed in the classical case, are extended to the affine algebra of su(3) at finite level.

math-ph↗

arXiv2016

Maps, immersions and permutations

We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolourability is no longer automatic and has to be assumed. We thus have two separate countings in non zero genus, that of bicolourable maps and that of general maps. We use a classical method of encoding maps in terms of permutations, on which the constraints of "one-componentness" and of a given genus may be applied. Depending on the orientation issue and on the bicolourability assumption, permutations for a map with n vertices live in S(4n) or in S(2n). In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of S(4n) (resp. S(2n)) under the action of the centralizer of a certain element of S(4n) (resp. S(2n)). This is achieved either by appealing to a formula by Frobenius or by a direct enumeration of these orbits. Applications to knot theory are briefly mentioned.

math.CO↗

arXiv2014

The history of the universe is an elliptic curve

Friedmann-Lemaitre equations with contributions coming from matter, curvature, cosmological constant, and radiation, when written in terms of conformal time u rather than in terms of cosmic time t, can be solved explicitly in terms of standard Weierstrass elliptic functions. The spatial scale factor, the temperature, the densities, the Hubble function, and almost all quantities of cosmological interest (with the exception of t itself) are elliptic functions of u, in particular they are bi-periodic with respect to a lattice of the complex plane, when one takes u complex. After recalling the basics of the theory, we use these explicit expressions, as well as the experimental constraints on the present values of density parameters (we choose for the curvature density a small value in agreement with experimental bounds) to display the evolution of the main cosmological quantities for one real period 2 omega_r of conformal time (the cosmic time t never end but it goes to infinity for a finite value u_f < 2 omega_r of u). A given history of the universe, specified by the measured values of present-day densities, is associated with a lattice in the complex plane, or with an elliptic curve, and therefore with two Weierstrass invariants g2, g3. Using the same experimental data we calculate the values of these invariants, as well as the associated modular parameter and the corresponding Klein j-invariant. If one takes the flat case k=0, the lattice is only defined up to homotheties, and if one, moreover, neglects the radiation contribution, the j-invariant vanishes and the corresponding modular parameter tau can be chosen in one corner of the standard fundamental domain of the modular group (equihanharmonic case: tau = exp(2 i pi/3)). Several exact, ie non-numerical, results of independent interest are obtained in that case.

gr-qc↗

arXiv2014

Conjugation properties of tensor product multiplicities

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product lambda x mu of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for the fusion product of the affine algebra of su(3) at finite levels, but this is not investigated in the present paper.

math-ph↗

arXiv2013

Drinfeld Doubles for Finite Subgroups of SU(2) and SU(3) Lie Groups

Drinfeld doubles of finite subgroups of SU(2) and SU(3) are investigated in detail. Their modular data - S, T and fusion matrices - are computed explicitly, and illustrated by means of fusion graphs. This allows us to reexamine certain identities on these tensor product or fusion multiplicities under conjugation of representations that had been discussed in our recent paper [J. Phys. A: Math. Theor. 44 (2011), 295208, 26 pages, arXiv:1103.2943], proved to hold for simple and affine Lie algebras, and found to be generally wrong for finite groups. It is shown here that these identities fail also in general for Drinfeld doubles, indicating that modularity of the fusion category is not the decisive feature. Along the way, we collect many data on these Drinfeld doubles which are interesting for their own sake and maybe also in a relation with the theory of orbifolds in conformal field theory.

math-ph↗

arXiv2013

Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups

Global dimensions for fusion categories defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum dimensions of simple objects, for the category of integrable modules over an affine Lie algebra at some level. The same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory WZW models. Similar results are also presented for those associated module-categories that can be obtained via conformal embeddings (they are "quantum subgroups" of a particular kind). As a side result, we express the classical (or quantum) Weyl denominator of simple Lie groups in terms of products of classical (or quantum) factorials calculated for the exponents of the group. Some calculations use the correspondence existing between periodic quivers for simply-laced Lie groups and fusion rules for module-categories (alias nimreps) of type SU(2).

math.QA↗