Compositio Mathematica 

About: Compositio Mathematica is an academic journal published by Cambridge University Press. The journal publishes majorly in the area(s): Conjecture & Cohomology. It has an ISSN identifier of 0010-437X. Over the lifetime, 3425 publications have been published receiving 102139 citations.

A combinatorial problem in geometry

Abstract: Our present problem has been suggested by Miss Esther Klein in connection with the following proposition.

First order interpolation inequalities with weights

Abstract: Il existe une constante positive C telle que l'inegalite suivante est vraie pour tout u∈C 0 ∞ (R n ): ||X| G u| L r≤C||x|α|Du|| L p 2 ||x|βu| L q 1−a si et seulement si on a 1/r+γ/n=a(1/p+(α−1)/n)+(1−a)(1/q+β/u), 0≤α−σ si a>0 et α−σ≤1 si a>0 et 1/p+(α−1)/n=1/r+γ/n

Cluster algebras IV: Coefficients

Abstract: We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of ‘principal’ coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parameterizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of Zamolodchikov's -systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical ‘universal’ choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

Conjugacy classes in the Weyl group

Abstract: Although both the conjugacy classes and irreducible characters of all the individual Weyl groups have been determined, no unified approach has been obtained which makes use of the common structure of the groups as reflection groups. We outline here a unified approach to the problem of finding the conjugacy classes. Let V be a Euclidean space of dimension i and ~ be a finite subset which spans V such that ~ is the set of roots of some simple Lie algebra. For each root a e ~, r denotes the reflection a in the hyperplane orthogonal to a, and the Weyl group W is the group generated by the reflections r for all a c r a The following are elementary lemmas concerning elements of W. 1. Each element w ~ W can be expressed in the form