Global Units modulo Circular Units : descent without Iwasawa's Main Conjecture

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a \ZM_p-extension $F_\infty/F$ of a number field $F$, to Iwasawa structural invariants \la and $\mu$ attached to the inverse limit X_\infty=\limpro X_n. It relies on "good" descent properties satisfied by $X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic it is known that the $p$-parts of the orders of the global units modulo circular units $U_n/C_n$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients $U_n/C_n$, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa's Main Conjecture

News from the Federations...

Last November marked the CHA’s first formail participation in the administrative council of the CFH since we joined the organization last summer. At this meeting, the CHA’s representative, Jean-Claude Robert was invited to take part in a new rite known as “Annual Lobby Day”

Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi-Dirac statistics

This paper is devoted to multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems. By considering the case of Fermi-Dirac statistics as a singular perturbation of Maxwell-Boltzmann one, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allows us to describe the set of all solutions by a non-autonomous perturbation of an autonomous dynamical system. This has interesting consequences in terms of bifurcation diagrams, which are illustrated by a some numerical computations. More specifically, we study a model based on the Fermi function as well as a simplified one for which estimates are easier to establish. The main difficulty comes from the fact that the mass enters in the equation as a parameter which makes the whole problem non-local

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.Comment: 32 pages, 4 figures. This version (V4): a few corrected typo

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.Comment: 29 pages, 23 figures. v2: Added references. Corrected typos. Some more explanations and comments have been added : subsections 1.4, 4.2.4 and a last paragraph in section 3.3. To appear in J Phys

Regularity of plurisubharmonic upper envelopes in big cohomology classes

The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge-Ampere operator; applications to geodesics in the space of Kahler metrics are discussed. A similar technique provides a logarithmic modulus of continuity for Tsuji's "supercanonical" metrics, which generalize a well-known construction of Narasimhan-Simha.Comment: 27 pages, no figure

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 typeComment: 8 pages, 2 figure

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.Comment: 46 pages, 18 figures, 9 tables. Version 2: added precisions on the notion used for the equivalence of immersed curves, new references. Version 3: Corrected typos, one array in Appendix B1 was duplicated by mistake, the position of tables and the order of the final sections have been modified, results unchanged. To be published in the Journal of Knot Theory and Its Ramification

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

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.Comment: 33 pages, 10 figures. Contribution to Mathematical Foundations of Quantum Field Theory, special issue in memory of Raymond Stora, Nucl. Phys.