Décompositions dans l'algèbre des différences divisées

✍ Scribed by Alain Lascoux; Marcel-Paul Schützenberger

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 958 KB
• Volume: 99
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)90372-m

✦ Synopsis

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.