# 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 $\mathcal{F}_n$ of complete flags in $\CC^{n}$. 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 $\mathcal{F}_n$ [a6]:

\begin{equation*} H ^ { * } ( \mathcal{F}_n , \ZZ ) \simeq \ZZ [ x _ { 1 } , \dots , x _ { n } ] / \ZZ^ { + } [ x _ { 1 } , \dots , x _ { n } ] ^ { \mathcal{S} _ { n } }. \end{equation*}

Here, $\ZZ ^ { + } [ x _ { 1 } , \ldots , x _ { n } ] ^ { \mathcal{S} _ { n } }$ is the ideal generated by the non-constant polynomials that are symmetric in $x _ { 1 } , \ldots , x _ { n }$. See [a19] for an elegant algebraic treatment of Schubert polynomials, and [a13] and [a20] for a more geometric treatment.

For each $i = 1 , \dots , n - 1$, let $s_i$ be the transposition $( i , i + 1 )$ in the symmetric group $\mathcal{S} _ { n }$, which acts on $\ZZ [ x _ { 1 } , \ldots , x _ { n } ]$. The divided difference operator $\partial_{i}$ is defined by

\begin{equation*} \partial _ { i } f = \frac { f - s _ { i } f } { x _ { i } - x _ { i + 1} }. \end{equation*}

These satisfy

\begin{equation} \tag{a1} \left\{ \begin{array} { l } { \partial _ { i } ^ { 2 } = 0, } \\ { \partial _ { i } \partial _ { j } = \partial _ { j } \partial _ { i } \text { if } | i - j | > 1, } \\ { \partial _ { i } \partial _ { i + 1 } \partial _ { i } = \partial _ { i + 1 } \partial _ { i } \partial _ { i + 1 }. } \end{array} \right. \end{equation}

If $f _ { w } \in \ZZ [ x _ { 1 } , \dots , x _ { n } ]$ is a representative of the Schubert cycle $\sigma_w$, then

\begin{equation*} \partial _ { i }\, f _ { w } = \left\{ \begin{array} { l l } { 0 } & { \text{if} \ \ell ( s _ { i } w ) > \ell ( w ), } \\ { f _ { s _ { i } w } } & { \text{if} \ \ell( s _ { i } w ) < \ell( w ), } \end{array} \right. \end{equation*}

where $\ell ( w )$ is the length of a permutation $w$ and $f _ { s _ { i } w }$ represents the Schubert cycle $\sigma _ { s _ { i } w}$. Given a fixed polynomial representative of the Schubert cycle $\sigma_{w_n}$ (the class of a point as $w_n \in \mathcal{S}_n$ 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 $\mathfrak{S}_{w_n}=x_1^{n-1} x_2^{n-2} \cdots x_{n-1}$ for $w_n$ gives the Schubert polynomials. Since $\partial _ { n } \ldots \partial _ { 1 } \mathfrak { S } _ { w _ { n + 1 } } = \mathfrak { S } _ { w _ { n } }$, Schubert polynomials are independent of $n$ and give polynomials $\mathfrak { S } _ { w } \in \ZZ [ x _ { 1 } , x _ { 2 } , \ldots ]$ for $w \in \mathcal{S} _ { \infty } = \cup \mathcal{S} _ { n }$. 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 $w \in \mathcal{S} _ { \infty }$, let $r$ be the last descent of $w$ ($w ( r ) > w ( r + 1 ) < w ( r + 2 ) <\dots$) and define $s > r$ by $w ( s ) < w ( r ) < w ( s + 1 )$. Set $v = w ( r , s )$, where $( r , s )$ is the transposition. Then

\begin{equation*} \mathfrak { S } _ { w } = x _ { r } \mathfrak { S } _ { v } + \sum \mathfrak { S } _ { v ( q , r ) }, \end{equation*}

the sum over all $q < r$ with $\ell(v(q,r)=\ell(v)+1=\ell(w)$. This formula gives an algorithm to compute $\mathfrak { S } _ { w }$ as the permutations that appear on the right-hand side are either shorter than $w$ or precede it in reverse lexicographic order, and the minimal such permutation $u$ of length $m$ has $\mathfrak { S } _ { u } = x _ {1 } ^ {m }$.

The transition formula shows that the Schubert polynomial $\mathfrak { S } _ { w }$ 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 $\mathfrak { S } _ { w }$ for $w \in \mathcal{S} _ { n }$ is also the normal form reduction of any polynomial representative of the Schubert cycle $\sigma_w$ with respect to the degree-reverse lexicographic term order on $\ZZ [ x _ { 1 } , \ldots , x _ { n } ]$ with $x _ { 1 } < \ldots < x _ { n }$.

The above-mentioned results of [a6], [a1], [a8] are valid more generally for any flag manifold $G / B$ with $G$ a semi-simple reductive group and $B$ a Borel subgroup. When $G$ 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=53570
This article was adapted from an original article by Frank Sottile (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article