Volume 2 (2002), Number 1. Abstracts

R. Agarwal and D. O'Regan. Singular Differential, Integral and Discrete Equations: the Semipositone Case [PDF]

Fixed point methods play a major role in the paper. In particular, we use lower type inequalities together with Krasnoselskii's fixed point theorem in a cone to deduce the existence of positive solutions for a general class of problems. Moreover, the results and technique are applicable also to positone problems.

Keywords. Singular, differential, integral, discrete, semipositone.

2000 Mathematics Subject Classification. 34B15, 47H30, 39A10.

D. Akhiezer and D. Panyushev. Multiplicities in the Branching Rules and the Complexity of Homogeneous Spaces [PDF]

Let $H$ be an algebraic subgroup of a connected algebraic group $G$ defined over an algebraically closed field $\k$ of characteristic zero. For a dominant weight $\lambda$ of $G$, let $V_\lambda$ be a simple $G$-module with highest weight $\lambda$, $d_{\lambda} = \dim V_{\lambda}$, and denote by $\k[G/H]_{(\lambda)}$ the isotypic $\lambda $-component in $k[G/H]$. For $G/H$ quasi-affine, we show that the ratio $\k[G/H]_{(\lambda)}/ d_{\lambda}$ grows no faster than a polynomial in $\Vert\lambda\Vert$ whose degree is the complexity of the homogeneous space $G/H$. If $H$ is reductive and connected, this yields an estimate of branching coefficients for the pair $(G,H)$ in terms of the complexity of $G/B_H$, where $B_H$ is a Borel subgroup of $H$. We classify all affine homogeneous spaces $G/H$ such that $G$ is simple and the comlexity of $G/B_H$ is at most 1. Some explicit descriptions of branching rules are also given.

Keywords. Complexity of a homogeneous space, branching rule, Grosshans subgroup, algebra of covariants.

2000 Mathematics Subject Classification. 14L30, 20G05.

S. Arkhipov. Semiinfinite Cohomology of Tate Lie Algebras [PDF]

In this note we give a definition of semiinfinite cohomology for Tate Lie algebras using the language of differential graded Lie algebroids with curvature (CDG Lie algebroids).

Keywords. Tate Lie algebra, Lie algebroid, DG-algebra.

2000 Mathematics Subject Classification. 17-XX.

R. Coquereaux. Notes on the Quantum Tetrahedron [PDF]

This is a set of notes describing several aspects of the space of paths on ADE Dynkin diagrams, with a particular attention paid to the graph $E_6$. Many results originally due to A.\,Ocneanu are described here in a very elementary way (manipulation of square or rectangular matrices). We recall the concept of essential matrices (intertwiners) for a graph and describe their module properties with respect to right and left actions of fusion algebras. In the case of the graph $E_6$, essential matrices build up a right module with respect to its own fusion algebra, but a left module with respect to the fusion algebra of $A_{11}$. We present two original results: 1) Our first contribution is to show how to recover the Ocneanu graph of quantum symmetries of the Dynkin diagram $E_6$ from the natural multiplication defined in the tensor square of its fusion algebra (the tensor product should be taken over a particular subalgebra); this is the Cayley graph for the two generators of the twelve-dimensional algebra $E_6 \otimes_{A_3} E_6$ (here $A_3$ and $E_6$ refer to the commutative fusion algebras of the corresponding graphs). 2) To every point of the graph of quantum symmetries one can associate a particular matrix describing the ``torus structure'' of the chosen Dynkin diagram; following Ocneanu, one obtains in this way, in the case of $E_6$, twelve such matrices of dimension $11\times 11$, one of them is a modular invariant and encodes the partition function of which corresponding conformal field theory. Our own next contribution is to provide a simple algorithm for the determination of these matrices.

Keywords. ADE, conformal field theory, Platonic bodies, path algebras, subfactors, modular invariance, quantum groups, quantum symmetries, Racah—Wigner bigebra.

2000 Mathematics Subject Classification. 81R50, 81R05, 81T40, 82B20, 46L37.

S. Dougherty and M. Skriganov. MacWilliams Duality and the Rosenbloom—Tsfasman Metric [PDF]

A new non-Hamming metric on linear spaces over finite fields has recently been introduced by Rosenbloom and Tsfasman [8]. We consider orbits of linear groups preserving the metric and show that weight enumerators suitably associated with such orbits satisfy MacWilliams-type identities for mutually dual codes. Furthermore, we show that the corresponding weight spectra of dual codes are related by transformations which involve multi-dimensional generalizations of known Krawtchouk polynomials. The relationships with recent results by Godsil [5] and Martin and Stinson [7] on MacWilliams-type theorems for association schemes and ordered orthogonal arrays are also briefly discussed in the paper.

Keywords. Codes in the Rosenbloom—Tsfasman metric, MacWilliams relations, uniform distributions.

2000 Mathematics Subject Classification. 94B, 11K, 94A.

O. Gelfond and A. Khovanskii. Toric Geometry and Grothendieck Residues [PDF]

We consider a system of $n$ algebraic equations $P_1=\dots=P_n=0$ in the torus $(\C\setminus 0)^n$. It is assumed that the Newton polyhedra of the equations are in a sufficiently general position with respect to one another. Let $\omega$ be any rational $n$-form which is regular on $(\C\setminus0)^n$ outside the hypersurface $P_1\dotsb P_n=0$. Formerly we have announced an explicit formula for the sum of the Grothendieck residues of the form $\omega$ at all roots of the system of equations. In the present paper this formula is proved.

Keywords. Grothendieck residues, Newton polyhedra, toric varieties.

2000 Mathematics Subject Classification. 14M25.

J. Hilgert, A. Pasquale, and E. Vinberg. The Dual Horospherical Radon Transform for Polynomials [PDF]

Let $X=G/K$ be a semisimple symmetric space of non-compact type. A horosphere in $X$ is an orbit of a maximal unipotent subgroup of $G$. The set $\Hor X$ of all horospheres is a homogeneous space of $G$. The horospherical Radom transform suggested by I. M. Gelfand and M. I. Graev in 1959 takes any function $\varphi$ on $X$ to a function on $\Hor X$ obtained by integrating $\varphi$ over horospheres. We explicitly describe the dual transform in terms of its action on polynomial functions on $\Hor X$.

Keywords. Symmetric space, horosphere, Radon transform, Harish-Chandra $\mathbf c$-function.

2000 Mathematics Subject Classification. 14M17, 53C65, 22E45, 43A90.

G. Jones and A. Zvonkin. Orbits of Braid Groups on Cacti [PDF]

One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to $S_n$ or $A_n$, where $n$ is the degree. Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on ``cacti'' (ordered sets of monodromy permutations). Applying these techniques, we provide a complete topological classification for the three exceptional cases mentioned above.

Keywords. Topological classification of polynomials, mono\-dromy groups, Braid group actions.

2000 Mathematics Subject Classification. Primary: 30C10; Secondary: 57M12, 05B25, 57M60, 20B15.

E. Materov. The Bott Formula for Toric Varieties [PDF]

The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $\Omega_{\P}^p(D)= \Omega_{\P}^p\otimes {\O_\P}(D)$ of $p$-th differential forms Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for ${\O_\P}(D)$. Comparison of two versions of the Bott formula gives some elegant corollaries in the combinatorics of simple polytopes. Also, we obtain a generalization of the reciprocity law. Some applications of the Bott formula are discussed.

Keywords. $p$-th Hilbert—Ehrhart polynomial, Zariski forms.

2000 Mathematics Subject Classification. Primary 14M25; Secondary 52B20, 52B11, 32L10, 58A10.

S. Tabachnikov. Ellipsoids, Complete Integrability and Hyperbolic Geometry [PDF]

We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (the Caley—Klein model of hyperbolic space).

Keywords. Riemannian and Finsler metrics, symplectic and contact structures, geodesic flow, mathematical billiard, hyperbolic metric, Caley—Klein model, exact transverse line fields.

2000 Mathematics Subject Classification. 53A15, 53A20, 53D25.

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