# Schubert polynomials

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Polynomials introduced by A. Lascoux and M.-P. Schützenberger [a18] as distinguished polynomial representatives of Schubert cycles (cf. also Schubert cycle) in the cohomology ring of the manifold of complete flags in . This extended work by I.N. Bernshtein, I.M. Gel'fand and S.I. Gel'fand [a1] and M. Demazure [a8], who gave algorithms for computing representatives of Schubert cycles in the co-invariant algebra, which is isomorphic to the cohomology ring of [a6]: Here, is the ideal generated by the non-constant polynomials that are symmetric in . See [a19] for an elegant algebraic treatment of Schubert polynomials, and [a13] and [a20] for a more geometric treatment.

For each , let be the transposition in the symmetric group , which acts on . The divided difference operator is defined by These satisfy (a1)

If is a representative of the Schubert cycle , then where is the length of a permutation and represents the Schubert cycle . Given a fixed polynomial representative of the Schubert cycle (the class of a point as is the longest element), successive application of divided difference operators gives polynomial representatives of all Schubert cycles, which are independent of the choices involved, by (a1).

The choice of the representative for gives the Schubert polynomials. Since , Schubert polynomials are independent of and give polynomials for . These form a basis for this polynomial ring, and every Schur polynomial is also a Schubert polynomial.

The transition formula gives another recursive construction of Schubert polynomials. For , let be the last descent of ( ) and define by . Set , where is the transposition. Then the sum over all with . This formula gives an algorithm to compute as the permutations that appear on the right-hand side are either shorter than or precede it in reverse lexicographic order, and the minimal such permutation of length has .

The transition formula shows that the Schubert polynomial is a sum of monomials with non-negative integral coefficients. There are several explicit formulas for the coefficient of a monomial in a Schubert polynomial, either in terms of the weak order of the symmetric group [a3], [a5], [a12], an intersection number [a15] or the Bruhat order [a4]. An elegant conjectural formula of A. Kohnert [a16] remains unproven (as of 2000). The Schubert polynomial for is also the normal form reduction of any polynomial representative of the Schubert cycle with respect to the degree-reverse lexicographic term order on with .

The above-mentioned results of [a6], [a1], [a8] are valid more generally for any flag manifold with a semi-simple reductive group and a Borel subgroup. When is an orthogonal or symplectic group, there are competing theories of Schubert polynomials [a2], [a10], [a17], each with own merits. There are also double Schubert polynomials suited for computations of degeneracy loci [a11], quantum Schubert polynomials [a9], [a7] and universal Schubert polynomials [a14].

How to Cite This Entry:
Schubert polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schubert_polynomials&oldid=14636
This article was adapted from an original article by Frank Sottile (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article