# zbMATH — the first resource for mathematics

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:
 500000 Symmetric functions and generalizations 5e+10 Combinatorial aspects of representation theory
Full Text:
##### References:
 [1] Billey, S; Jockusch, W; Stanley, R.P, Some combinatorial properties of Schubert polynomials, J. algebraic combin., 2, 345-374, (1993) · Zbl 0790.05093 [2] Demazure, M, Une nouvelle formule des caractères, Bull. sci. math., 98, 163-172, (1974) · Zbl 0365.17005 [3] Edelman, P.H; Greene, C, Balanced tableaux, Adv. math., 63, 42-99, (1987) · Zbl 0616.05005 [4] Fomin, S; Stanley, R.P, Schubert polynomials and the nilcoxeter algebra, Adv. math., 103, 196-207, (1994) · Zbl 0809.05091 [5] \scI. Gessel and G. X. Viennot, Determinants and plane partitions, preprint. · Zbl 0579.05004 [6] Greene, C, An extension of Schensted’s theorem, Adv. math., 14, 254-265, (1974) · Zbl 0303.05006 [7] Hillman, A.P; Grassl, R.M, Skew tableaux and the insertion algorithm, J. combin. inform. systems sci., 5, 305-316, (1980) · Zbl 0474.05007 [8] James, G.D, The representation theory of symmetric groups, () · Zbl 0393.20009 [9] Kohnert, A, Weintrauben, polynome, tableaux, Bayreuth math. schrift., 38, 1-97, (1990) · Zbl 0755.05095 [10] \scW. Kraśkiewicz and P. Pragacz, Schubert functors and Schubert polynomials, preprint. [11] Lascoux, A; Schützenberger, M.-P, Keys and standard bases, (), 125-144 · Zbl 0815.20013 [12] Lascoux, A; Schützenberger, M.-P, Tableaux and non-commutative Schubert polynomials, Funct. anal. appl., 23, 63-64, (1989) [13] Lascoux, A; Schützenberger, M.-P, Le monoïde plaxique, Ann. discrete math., 6, 251-255, (1980) [14] Lascoux, A; Schützenberger, M.-P, Schubert polynomials and the Littlewood-Richardson rule, Lett. math. phys., 10, 111-124, (1985) · Zbl 0586.20007 [15] MacDonald, I.G, Notes on Schubert polynomials, () · Zbl 0784.05061 [16] Remmel, J.B; Whitney, R, Multiplying Schur functions, J. algorithms, 5, 471-487, (1984) · Zbl 0557.20008 [17] Shimozono, M, Littlewood-Richardson rules for ordinary and projective representations of symmetric groups, () [18] Thomas, G.P, On Schensted’s construction and the multiplication of Schur functions, Adv. math., 30, 8-32, (1978) · Zbl 0408.05004 [19] Wachs, M.L, Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. combin. theory ser. A, 40, 276-289, (1985) · Zbl 0579.05001 [20] White, D, Some connections between the Littlewood-Richardson rule and the construction of Schensted, J. combin. theory ser. A, 30, 237-247, (1981) · Zbl 0472.05005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.