Schubert polynomials

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
[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
[a17]	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
[a18]	A. Lascoux, M.-P. Schützenberger, "Polynômes de Schubert" C.R. Acad. Sci. Paris , 294 (1982) pp. 447–450
[a19]	I.G. Macdonald, "Notes on Schubert polynomials" Lab. Combin. et d'Inform. Math. (LACIM) Univ. Québec (1991)
[a20]	L. Manivel, "Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence" Cours Spécialisés Soc. Math. France , 3 (1998)