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These satisfy
 
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}
+
\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
 
If $f _ { w } \in \ZZ [ x _ { 1 } , \dots , x _ { n } ]$ is a representative of the [[Schubert cycle|Schubert cycle]] $\sigma_w$, then
  
 
\begin{equation*}
 
\begin{equation*}
\partial _ { i }\, f _ { w } = \left\{ \begin{array} { l l } { 0 } & { \text{if} \ \ell ( s _ { i } w ) > \ell ( w ), } \\
+
\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.
+
  { f _ { s _ { i } w } } &amp; { \text{if} \ \ell( s _ { i } w ) < \ell( w ), } \end{array} \right.
 
\end{equation*}
 
\end{equation*}
  
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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 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 $w$ ($w ( r ) &gt; w ( r + 1 ) &lt; w ( r + 2 ) &lt;\dots$) and define $s &gt; r$ by $w ( s ) &lt; w ( r ) &lt; w ( s + 1 )$. Set $v = w ( r , s )$, where $( r , s )$ is the transposition. Then
+
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*}
 
\begin{equation*} \mathfrak { S } _ { w } = x _ { r } \mathfrak { S } _ { v } + \sum \mathfrak { S } _ { v ( q , r ) }, \end{equation*}
  
the sum over all $q &lt; r$ 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 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 $x _ { 1 } &lt; \ldots &lt; x _ { n }$.
+
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]]].
 
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====
 
====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>
+
<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">[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>
+
<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>

Latest revision as of 09:23, 10 November 2023

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].

References

[a1] 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
[a2] S. Billey, M. Haiman, "Schubert polynomials for the classical groups" J. Amer. Math. Soc. , 8 : 2 (1995) pp. 443–482
[a3] S. Billey, W. Jockush, R. Stanley, "Some combinatorial properties of Schubert polynomials" J. Algebraic Combin. , 2 : 4 (1993) pp. 345–374
[a4] N. Bergeron, F. Sottile, "Skew Schubert functions and the Pieri formula for flag manifolds" Trans. Amer. Math. Soc. (to appear)
[a5] N. Bergeron, "A combinatorial construction of the Schubert polynomials" J. Combin. Th. A , 60 (1992) pp. 168–182 Zbl 0771.05097
[a6] 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
[a7] I. Ciocan–Fontanine, "On quantum cohomology rings of partial flag varieties" Duke Math. J. , 98 : 3 (1999) pp. 485–524
[a8] M. Demazure, "Désingularization des variétés de Schubert généralisées" Ann. Sci. École Norm. Sup. (4) , 7 (1974) pp. 53–88
[a9] S. Fomin, S. Gelfand, A. Postnikov, "Quantum Schubert polynomials" J. Amer. Math. Soc. , 10 (1997) pp. 565–596
[a10] S. Fomin, A.N. Kirillov, "Combinatorial $B _ { n }$-analogs of Schubert polynomials" Trans. Amer. Math. Soc. , 348 (1996) pp. 3591–3620
[a11] W. Fulton, P. Pragacz, "Schubert varieties and degeneracy loci" , Lecture Notes in Mathematics , 1689 , Springer (1998)
[a12] S. Fomin, R. Stanley, "Schubert polynomials and the nilCoxeter algebra" Adv. Math. , 103 (1994) pp. 196–207
[a13] W. Fulton, "Young tableaux" , Cambridge Univ. Press (1997)
[a14] W. Fulton, "Universal Schubert polynomials" Duke Math. J. , 96 : 3 (1999) pp. 575–594
[a15] 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))
[a16] A. Kohnert, "Weintrauben, polynome, tableaux" Bayreuth Math. Schrift. , 38 (1990) pp. 1–97
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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