Marcel Paul Schützenberger

Sur une variante des fonctions sequentielles

NOUS faisons reference aux Chapitres XI et XII du Trait6 de Eilenberg [l] pour la Gfinition et la Thkorie des fonctions (partielles) rationnelles et sequentielles gZnhalis&es d’un monoi’de libre dans un autre. Nous nous proposons de montrer ici qu’au prix d’une perte de simplicitk dans leur formulation certaines parties des theorZ?rnes classiques de Ginsburg et Rose [3] et ck Eilenberg s’appliquefit 5 we amille un peu iffkente que nous appellerons famlllc des fonctions sowskquentielles (sSq: . Dans ce qui suit A * et B * sont les monoi’des libres engendres par les ensembles A et B ; B * est consid& comme un sous-monoi’de du groupe libre B(*) (engendrk par B) et 0 est le zero de I’algkbre de B(*) sur 2%. Si X est une partie de A *, on peliera pour abreger function de X toute application de X dans l’union de B * et n’importe quelle function dont I’image est 0. a 0 un entier nature1 fixe. Pour chaque n 2 0 OIJ dksigne par l’ensemble A ‘M * \ A k+‘?4 * des mots de A * dont la longueur est comprise k:ntre k et k + n. LTI famille ssq est I’union sur tous les n finis des sous-familles sSq(n) des fonctions sous-skquentielles de dimension au plus n dont la definition est la suivante:

Minimization of rational word functions

Rational functions from a free monoid into another are characterized by the finiteness of the index of some congruence naturally associated with the function. A sequential bimachine is constructed computing the function, which is completely canonical, and in some sense minimal. This generalizes the Nerode criterion and the minimal automaton of a rational language, and similar results for sequential functions.

A conjecture on sets of differences of integer pairs

Abstract We propose a conjecture on the set of differences of integer pairs taken out of a sufficiently dense subset of the plane.

Varieties and rational functions

Abstract We say that a rational (resp. a subsequential) function α from a free monoid into another one is in the variety of monoids V if it may realized by some unambiguous (resp. subsequential) transducer whose monoid of transitions is in V . We characterize these functions when V is the variety of aperiodic monoids, and the variety of groups. In the first case, the period of α −1 ( L ) divides that of L , for each rational language L on the outputs. In the second case, α −1 ( L ) is a group-language for each group language L ; equivalently, α is continuous for the pro-finite topology. Examples of such functions are: the multiplication by a given number in a given basis, which is aperiodic; the division, which is a group-function.We say that a rational (resp. a subsequential) function tl from a free monoid into another one is in the variety of monoids V if it may realized by some unambiguous (resp. subsequential) transducer whose monoid of transitions is in V. We characterize these functions when V is the variety of aperiodic monoids, and the variety of groups. In the first case, the period of am ’ (15) divides that of L, for each rational language L on the outputs. In the second case, a-‘(L) is a group-language for each group language L; equivalently, a is continuous for the pro-finite topology. Examples of such functions are: the multiplication by a given number in a given basis, which is aperiodic; the division, which is a group-function.

Decompositions in divided difference algebra

Resume The group algebra of the symmetric group on the ring of rational functions has, apart from its canonical basis of permutations, several bases of symmetrizing operators , among which the classical Newtons divided differences . We deal here with the explicitation of the matrices of change of bases. Our main result is that, in the case of Gl( n ), the components of the matrices associated to the classical bases are just specializations of Schubert or Grothendieck polynomials . We refer to the work of Arabia, Bernstein-Gelfand-Gelfand, Kac, Kostant, Kumar, Rossmann for the interpretation of these matrices in terms of (equivariant) cohomology and K -theory rings of the flag manifold, or equivariant singularities of Schubert varieties. Our main technical tool is the simple observation that all but one specialization of the maximal twofold Schubert polynomial H vanish.Lascoux, A. et M.-P. Schiitzenberger, Decompositions dans I’algtbre des differences divistes, Discrete Mathematics 99 (1992) 165-179. The group algebra of the symmetric group on the ring of rational functions has, apart from its canonical basis of permutations, several bases of symmetrizing operators, among which the classical Newton’s divided differences. We deal here with the exphcitation of the matrices of change of bases. Our main result is that, in the case of Gl(n), the components of the matrices associated to the classical bases are just specializations of Schubert or Grothendieck polynomials. We refer to the work of Arabia, Bernstein-Gelfand-Gelfand, Kac, Kostant,. Kumar, Rossmann for the interpretation of these matrices in terms of (equivariant) cohomology and K-theory rings of the flag manifold, or equivariant singularities of Schubert varieties. Our main technical tool is the simple observation that all but one specialization of the maximal twofold Schubert polynomial Z? vanish.