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Key polynomials and a flagged Littlewood-Richardson rule. (English) Zbl 0819.05058
The authors study a family of polynomials \(k_ \alpha\) called key polynomials introduced by Demazure [see M. Demazure, Une nouvelle formule des caractères, Bull. Sci. Math., II. Sér. 98(1974), 163-172 (1974; Zbl 0365.17005)] for Weyl groups and investigated by Lascoux and Schützenberger (who called them standard bases) [see A. Lascoux and M.-P. Schützenberger, Keys & standard bases, Invariant theory and tableaux, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 19, 125-144 (1990; Zbl 0815.20013)] in the case of symmetric groups. They gave several combinatorial descriptions of the key polynomials as well as an explicit formula for the expansion of Schubert polynomials as a positive sum of key polynomials. The authors show that a variant of the recent expression of Stanley [S. Fomin and R. P. Stanley, Schubert polynomials and the nilCoxeter algebra, Adv. Math. 103, No. 2, 196-207 (1994; Zbl 0809.05091)] for the Schubert polynomials yields a key polynomial, thus obtaining an independent proof of the aforementioned expansion. The preprint [W. Kraskiewicz and P. Pragacz, Schubert functors and Schubert polynomials] shows that Schubert polynomials may be viewed as characters of certain \(B\)-modules, where \(B\) is Borel subgroup of the general linear group. The authors give a similar interpretation for key polynomials, using one of the combinatorial descriptions of the key polynomials \(k_ \alpha\) to produce a basis for the corresponding \(B\)-module. This provides an independent and purely combinatorial proof of the Demazure character formula for Weyl groups of type A. They also give a flagged Littlewood-Richardson rule for expanding a flagged skew Schur function as a nonnegative sum of key polynomials. Moreover, they discuss conditions when key polynomials, Schubert polynomials, and flagged skew Schur functions coincide.

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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