International Journal of Algebra and Computation

We describe a simple algorithm for computing Kashiwara's global crystal basis of a finite-dimensional irreducible representation of U_q(sl_n).

Résumé: Nous décrivons un algorithme simple pour calculer la base cristalline globale de Kashiwara d'une représentation irréductible de dimension finie de U_q(sl_n).

A language L is called thin if there exists an integer n₀ such that for all n≥n₀L contains at most one word of length n. We show that thinness is decidable for exponential D0L languages. We show also that Siegel's result concerning integral points on algebraic curves of positive genus can often be used to prove that a polynomially bounded HD0L language is thin.

We prove the theorem of Schutzenberger that a language is star-free if and only if it is recognized by a finite monoid with trivial subgroups. The forward direction follows the lines of the proof of Kleene's Theorem which characterizes regular languages as those recognized by finite automata. The converse is a relatively short induction.

Denote by Φ₂ the automorphism group of the free group on two generators. We classify the indecomposable complex linear representations of Φ₂ of degree at most 5, and determine which ones have an infinite image. As a side-effect, our classification incorporates all the indecomposable representations of GL₂(Z) in these degrees.

Recent results related to hyperdecidability and their applications to decidability of semidirect products are reviewed. Hyperdecidability is studied for pseudovarieties of completely simple semigroups as a means of illustrating the kind of techniques that may be used in this area.