View source for Schubert polynomials

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 58 formulas, 50 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|part}}
Polynomials introduced by A. Lascoux and M.-P. Schützenberger [[#References|[a18]]] as distinguished polynomial representatives of Schubert cycles (cf. also [[Schubert cycle|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 [[#References|[a1]]] and M. Demazure [[#References|[a8]]], who gave algorithms for computing representatives of Schubert cycles in the co-invariant algebra, which is isomorphic to the cohomology ring of $\mathcal{F}_n$ [[#References|[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 [[#References|[a19]]] for an elegant algebraic treatment of Schubert polynomials, and [[#References|[a13]]] and [[#References|[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|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|Schubert cycle]] $\sigma_w$, then

\begin{equation*}
\partial _ { i }\, f _ { w } = \left\{ \begin{array} { l l } { 0 } &amp; { \text{if} \ \ell ( s _ { i } w ) > \ell ( w ), } \\
 { f _ { s _ { i } w } } &amp; { \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|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 [[#References|[a3]]], [[#References|[a5]]], [[#References|[a12]]], an intersection number [[#References|[a15]]] or the Bruhat order [[#References|[a4]]]. An elegant conjectural formula of A. Kohnert [[#References|[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 [[#References|[a6]]], [[#References|[a1]]], [[#References|[a8]]] are valid more generally for any flag manifold $G / B$ with $G$ a semi-simple [[Reductive group|reductive group]] and $B$ a [[Borel subgroup|Borel subgroup]]. When $G$ is an orthogonal or [[Symplectic group|symplectic group]], there are competing theories of Schubert polynomials [[#References|[a2]]], [[#References|[a10]]], [[#References|[a17]]], each with own merits. There are also double Schubert polynomials suited for computations of degeneracy loci [[#References|[a11]]], quantum Schubert polynomials [[#References|[a9]]], [[#References|[a7]]] and universal Schubert polynomials [[#References|[a14]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  I.N. Bernshtein,   I.M. Gel'fand,   S.I. Gel'fand,   "Schubert cells and cohomology of the spaces $G / P$"  ''Russian Math. Surveys'' , '''28''' :  3  (1973)  pp. 1–26</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Billey,   M. Haiman,   "Schubert polynomials for the classical groups"  ''J. Amer. Math. Soc.'' , '''8''' :  2  (1995)  pp. 443–482</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. Billey,   W. Jockush,   R. Stanley,   "Some combinatorial properties of Schubert polynomials"  ''J. Algebraic Combin.'' , '''2''' :  4  (1993)  pp. 345–374</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Bergeron,   F. Sottile,   "Skew Schubert functions and the Pieri formula for flag manifolds"  ''Trans. Amer. Math. Soc.''  (to appear)</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top">  N. Bergeron,   "A combinatorial construction of the Schubert polynomials"  ''J. Combin. Th. A'' , '''60'''  (1992)  pp. 168–182 {{ZBL|0771.05097}}</td></tr>
<tr><td valign="top">[a6]</td> <td valign="top">  A. Borel,   "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts"  ''Ann. Math.'' , '''57'''  (1953)  pp. 115–207</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  I. Ciocan–Fontanine,   "On quantum cohomology rings of partial flag varieties"  ''Duke Math. J.'' , '''98''' :  3  (1999)  pp. 485–524</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Demazure,   "Désingularization des variétés de Schubert généralisées"  ''Ann. Sci. École Norm. Sup. (4)'' , '''7'''  (1974)  pp. 53–88</td></tr>
<tr><td valign="top">[a9]</td> <td valign="top">  S. Fomin,   S. Gelfand,   A. Postnikov,   "Quantum Schubert polynomials"  ''J. Amer. Math. Soc.'' , '''10'''  (1997)  pp. 565–596</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  S. Fomin,   A.N. Kirillov,   "Combinatorial $B _ { n }$-analogs of Schubert polynomials"  ''Trans. Amer. Math. Soc.'' , '''348'''  (1996)  pp. 3591–3620</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  W. Fulton,   P. Pragacz,   "Schubert varieties and degeneracy loci" , ''Lecture Notes in Mathematics'' , '''1689''' , Springer  (1998)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  S. Fomin,   R. Stanley,   "Schubert polynomials and the nilCoxeter algebra"  ''Adv. Math.'' , '''103'''  (1994)  pp. 196–207</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  W. Fulton,   "Young tableaux" , Cambridge Univ. Press  (1997)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  W. Fulton,   "Universal Schubert polynomials"  ''Duke Math. J.'' , '''96''' :  3  (1999)  pp. 575–594</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  A. Kirillov,   T. Maeno,   "Quantum double Schubert polynomials, quantum Schubert polynomials, and the Vafa–Intriligator formula"  ''Discr. Math.'' , '''217''' :  1–3  (2000)  pp. 191–223  (Formal Power Series and Algebraic Combinatorics (Vienna, 1997))</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  A. Kohnert,   "Weintrauben, polynome, tableaux"  ''Bayreuth Math. Schrift.'' , '''38'''  (1990)  pp. 1–97</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  A. Lascoux,   P. Pragacz,   J. Ratajski,   "Symplectic Schubert polynomials à la polonaise, appendix to operator calculus for $\widetilde{Q}$-polynomials and Schubert polynomials"  ''Adv. Math.'' , '''140'''  (1998)  pp. 1–43</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  A. Lascoux,   M.-P. Schützenberger,   "Polynômes de Schubert"  ''C.R. Acad. Sci. Paris'' , '''294'''  (1982)  pp. 447–450</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  I.G. Macdonald,   "Notes on Schubert polynomials"  ''Lab. Combin. et d'Inform. Math. (LACIM) Univ. Québec''  (1991)</td></tr>
<tr><td valign="top">[a20]</td> <td valign="top">  L. Manivel,   "Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence"  ''Cours Spécialisés Soc. Math. France'' , '''3'''  (1998)</td></tr>
</table>