Schubert Polynomials and the Nilcoxeter Algebra

Abstract

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, ... satisfying the relations u 2 i =0, u iu j = u ju i if | i− j| ≥ 2, and u iu i+1 u i = u i + 1 u i u i + 1 . We call this algebra the nilCoxeter algebra of the symmetric group 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 S w ,(1, 1, ...). Our main new result is a proof of a conjectured q-analogue of (d), due to Macdonald which gives a formula for S w (1, q, q 2,...).