## Experimental Mathematics

- Experiment. Math.
- Volume 2, Issue 4 (1993), 257-269.

### RC-graphs and Schubert polynomials

Nantel Bergeron and Sara Billey

#### Abstract

Using a formula of Billey, Jockusch and Stanley, Fomin and Kirillov have introduced a new set of diagrams that encode the Schubert polynomials. We call these objects rc-graphs. We define and prove two variants of an algorithm for constructing the set of all rc-graphs for a given permutation. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we give a new proof of Monk's rule using an insertion algorithm on rc-graphs. We conjecture two analogs of Pieri's rule for multiplying Schubert polynomials. We also extend the algorithm to generate the double Schubert polynomials.

#### Article information

**Source**

Experiment. Math., Volume 2, Issue 4 (1993), 257-269.

**Dates**

First available in Project Euclid: 24 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1048516036

**Mathematical Reviews number (MathSciNet)**

MR1281474

**Zentralblatt MATH identifier**

0803.05054

**Subjects**

Primary: 05E99: None of the above, but in this section

Secondary: 05E05: Symmetric functions and generalizations 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 20C30: Representations of finite symmetric groups

#### Citation

Bergeron, Nantel; Billey, Sara. RC-graphs and Schubert polynomials. Experiment. Math. 2 (1993), no. 4, 257--269. https://projecteuclid.org/euclid.em/1048516036