On the Counting of Fully Packed Loop Configurations. Some new conjectures

New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the Temperley-Lieb chain, for periodic boundary conditions and so-called ``identified connectivities'', up to size $2n=22$

Revisiting SU(N) integrals

In this note, I revisit integrals over \SU(N) of the form \int DU\, U_{i_1j_1}\cdots U_{i_pj_p}\Ud_{k_1l_1}\cdots \Ud_{k_nl_n}. While the case $p=n$ is well known, it seems that explicit expressions for $p=n+N$ had not appeared in the literature. Similarities and differences, in particular in the large $N$ limit, between the two cases are discussedComment: 1 figur

Generalized Dynkin diagrams and root systems and their folding

Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter element". The folding of these graphs and groups is also discussed, using the theory of C-algebras. (Proceedings of the Taniguchi Symposium {Topological Field Theory, Primitive Forms and Related Topics}, Kyoto Dec 1996)Comment: plain tex, 7 figure

Invariances in Physics and Group Theory

This is a short review of the heritage of Klein's Erlangen program in modern physics.Comment: Talk given at the Conference "Lie and Klein; the Erlangen program and its impact on mathematics and physics", Strasbourg, Sept. 201

CFT, BCFT, ADE and all that

These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting ``in the bulk'' or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invariance. It is shown why the two problems are intimately connected and how graphs -ADE Dynkin diagrams and their generalizations- appear in a natural way.Comment: Lectures at Bariloche, Argentina, January 2000. 36 pages, 4 figure

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

A-D-E Classification of Conformal Field Theories

The ADE classification scheme is encountered in many areas of mathematics, most notably in the study of Lie algebras. Here such a scheme is shown to describe families of two-dimensional conformal field theories.Comment: 19 pages, 4 figures, 4 tables; review article to appear in Scholarpedia, http://www.scholarpedia.org