rc-graphs and Schubert polynomials. (English) Zbl 0803.05054

Bergeron, Nantel; Billey, Sara

Exp. Math. 2, No. 4, 257-269 (1993).

Using a formula of S. C. Billey, W. Jockusch and R. P. Stanley [Some combinatorial properties of Schubert polynomials, J. Algebr. Comb. 2, No. 4, 345-374 (1993; Zbl 0790.05093)], S. Fomin and A. N. Kirillov [Yang-Baxter equation, symmetric functions, and Schubert polynomials, Proceedings of the conference on power series and algebraic combinatorics, Firenze (1993)] have introduced a new set of diagrams that encode the Schubert polynomials. In this paper, these objects are called rc-graphs. Here, two variants of an algorithm for constructing the set of all rc-graphs for a given permutation are defined and proved. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we find a new proof of Monk’s rule using an insertion algorithm on rc- graphs. This insertion rule is a generalization of the Schensted insertion for tableaux. We find two conjectures of analogs of Pieri’s rule for multiplying Schubert polynomials. The authors also extend the algorithm to generate the double Schubert polynomials.

Reviewer: N.Bergeron (North York)

MSC:

05E05	Symmetric functions and generalizations
05E15	Combinatorial aspects of groups and algebras (MSC2010)
05A99	Enumerative combinatorics
05A19	Combinatorial identities, bijective combinatorics
05A05	Permutations, words, matrices
14M15	Grassmannians, Schubert varieties, flag manifolds

Keywords:

Schubert polynomials; rc-graphs; Monk’s rule; Pieri’s rule

Citations:

Zbl 0790.05093

References:

[1]	Bergeron N., J. Comb. Theory 60 pp 168– (1992) · Zbl 0771.05097 · doi:10.1016/0097-3165(92)90002-C
[2]	Bernstein I. N., Russian Math. Surveys 28 (1973)
[3]	Billey S., J. Alg. Comb. (2) pp 345– (1993) · Zbl 0790.05093 · doi:10.1023/A:1022419800503
[4]	Demazure M., Ann. Sci. Éc. Norm. Sup. (Paris) pp 53– (1974)
[5]	Fomin, S. and Kirillov, A. N. ”Yang-Baxter equation, symmetric functions, and Schubert polynomials”. Proceedings of the Conference on Power Series and Algebraic Combinatorics. Firenze. [Fomin and Kirillov 1993] · Zbl 0852.05078
[6]	Fomin S., Adv. Math. 103 pp 196– (1994) · Zbl 0809.05091 · doi:10.1006/aima.1994.1009
[7]	Kohnert A., Bayreuth. Math. Schrift. 38 (1990)
[8]	Lascoux A., C. R. Acad. Sci. Paris 294 pp 447– (1982)
[9]	Lascoux A., Let. Math. Phys. 10 pp 111– (1985) · Zbl 0586.20007 · doi:10.1007/BF00398147
[10]	Macdonald I. G., Notes on Schubert Polynomials (1991) · Zbl 0784.05061
[11]	Schubert H., Kalkül der Abzädhlenden Geometrie (1879)