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Décompositions dans l’algèbre des différences divisées. (Decompositions in the algebra of divided differences). (French) Zbl 0784.05060

Let \(A\) be a finite alphabet (independent variables), \({\mathcal R}(A)\) the ring of rational functions on \(A\), \({\mathcal G}(A)\) the symmetric group of \(A\), and \(\mathcal E\) the group algebra of \({\mathcal G}(A)\) on the field \({\mathcal R}(A)\). The elements of \(\mathcal E\) are linear operators on \({\mathcal R}(A)\), and \(\mathcal E\) is an \({\mathcal R}(A)\)-module the “canonical” basis of which is formed by the permutations of \({\mathcal G}(A)\). Here, several other classical bases of \(\mathcal E\) consisting of symmetrizing operators are considered, namely Newton’s divided differences, the convex symmetrizers, and the concave symmetrizers. The authors deal with the problem to determine the matrices of change of such bases explicitly. Their main result is that the elements of these transformation matrices are just specializations of Schubert or Grothendieck polynomials. An important tool is the fact that all but one specialization of the maximal (twofold) Schubert polynomial vanish. (Unfortunately, there are misprints in some formulas.) Moreover, the paper contains instructive remarks on references to algebraic geometry regarding the interpretation of the matrices (cohomology and \(K\)-theory rings of flag manifolds, Schubert varieties).

MSC:

05E05 Symmetric functions and generalizations
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
14M15 Grassmannians, Schubert varieties, flag manifolds
19D55 \(K\)-theory and homology; cyclic homology and cohomology
19D50 Computations of higher \(K\)-theory of rings
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