Schubert Polynomials and the Nilcoxeter Algebra

✍ Scribed by S. Fomin; R.P. Stanley

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 310 KB
• Volume: 103
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1994.1009

✦ Synopsis

Schubert polynomials were introduced and extensively developed by Lascoux and Schützenberger, after an earlier less combinatorial version had been considered by Bernstein, Gelfand and Gelfand and Demazure. We give a new development of the theory of Schubert polynomials based on formal computations in the algebra of operators (u_{1}, u_{2}, \ldots) satisfying the relations (u_{i}^{2}=0, u_{i} u_{j}=u_{j} u_{i}) if (|i-j| \geqslant 2), and (u_{i} u_{i+1} u_{i}=u_{i+1} u_{i} u_{i+1}). We call this algebra the nilCoxeter algebra of the symmetric group (\mathscr{S}{n}). Our development leads to simple proofs of many standard results, in particular, (a) symmetry of the "stable Schubert polynomials" (F{w}), (b) an explicit combinatorial formula for Schubert polynomials due to Billey, Jockusch and Stanley, (c) the "Cauchy formula" for Schubert polynomials, and (d) a formula of Macdonald for (\Xi_{w}(1,1, \ldots)). Our main new result is a proof of a conjectured (q)-analogue of (d), due to Macdonald which gives a formula for (\Xi_{n}\left(1, q, q^{2}, \ldots\right.) ). c. 1994 Academic Press, Inc.