Last update: 3 July 2013
Let $V$ be a vector space of dimension $n$ over a field $K,$ and let $({e}_{1},\dots ,{e}_{n})$ be a basis of $V,$ fixed once and for all. A flag in $V$ is a sequence $U={\left({U}_{i}\right)}_{0\le i\le n}$ of subspaces of $V$ such that
$$0={U}_{0}\subset {U}_{1}\subset \dots \subset {U}_{n}=V$$with strict inclusions at each stage, so that $\text{dim}\hspace{0.17em}{U}_{i}=i$ for each $i\text{.}$ In particular, if ${V}_{i}$ is the subspace of $V$ spanned by ${e}_{1},\dots ,{e}_{i},$ then $V={\left({V}_{i}\right)}_{0\le i\le n}$ is a flag in $V,$ called the standard flag.
The set $F=F\left(V\right)$ of flags in $V$ is called the flag manifold of $V\text{.}$
Let $G$ be the group of all automorphisims of the vector space $V\text{.}$ Since we have fixed a basis of $V,$ we may identify $G$ with the general linear group ${GL}_{n}\left(k\right)\text{:}$ if $g\in G$ and
$$g{e}_{j}=\sum _{i=1}^{n}{g}_{ij}{e}_{i}\phantom{\rule{2em}{0ex}}(1\le j\le n)$$then $g$ is identified with the matrix $\left({g}_{ij}\right)\text{.}$
The group $G$ acts on $F\text{:}$ if $U=\left({U}_{i}\right)$ and $g\in G,$ then $gU$ is the flag $\left(g{U}_{i}\right)\text{.}$ Let $B$ be the subgroup of $G$ that fixes the standard flag $V\text{.}$ Then $g\in B$ if and only if $g{e}_{j}$ is a linear combination of ${e}_{1},\dots ,{e}_{j},$ for $1\le j\le n,$ that is to say if and only if ${g}_{ij}=0$ whenever $i>j,$ so that $B$ is the group of upper triangular matrices in ${GL}_{n}\left(k\right)\text{.}$
A basis of a flag $U=\left({U}_{i}\right)$ is a sequence $({u}_{1},\dots ,{u}_{n})$ in $V$ such that ${u}_{i}\in {U}_{i}{U}_{i1}$ $1\le i\le n,$ or equivalently such that ${u}_{1},\dots ,{u}_{i}$ is a basis of ${U}_{i}$ for each $i\text{.}$ Given such a basis of $U,$ there is a unique $g\in G$ such that $g{e}_{i}={u}_{i}$ for each $i,$ and we have $U=gV\text{.}$ Hence $G$ acts transitively on the flag manifold $F,$ and the mapping $gV\mapsto gB$ is a bijection of $F$ onto the coset space $G/B\text{.}$
For a flag $U=\left({U}_{i}\right),$ let
$${E}_{i}={E}_{i}\left(U\right)=\{j:1\le j\le n\hspace{0.17em}\text{and}\hspace{0.17em}{U}_{i}\cap {V}_{j}\ne {U}_{i}\cap {V}_{j1}\}$$for $0\le i\le n\text{.}$ Then $({E}_{0},\dots ,{E}_{n})$ is a 'flag of sets', i.e. we have
(A.1) 

Proof.  
(i) Fix $i$ and let ${d}_{j}=\text{dim}\hspace{0.17em}({U}_{i}\cap {V}_{j})\text{.}$ Since $$\frac{{V}_{i}\cap {V}_{j}}{{U}_{i}\cap {V}_{j1}}=\frac{{U}_{i}\cap {V}_{j}}{({U}_{i}\cap {V}_{j})\cap {V}_{j1}}\cong \frac{({U}_{i}\cap {V}_{j})+{V}_{j1}}{{V}_{j1}}\subset \frac{{V}_{j}}{{V}_{j1}}$$it follows that ${d}_{j}{d}_{j1}=0$ or $1\text{.}$ Since ${d}_{0}=0$ and ${d}_{n}=i,$ there are therefore $i$ jumps in the sequence $({d}_{0},{d}_{1},\dots ,{d}_{n}),$ which proves (i). (ii) Suppose that $j\notin {E}_{i},$ so that ${U}_{i}\cap {V}_{j}={U}_{i}\cap {V}_{j1}\text{.}$ Intersecting with ${U}_{i1},$ we see that $j\notin {E}_{i1}\text{.}$ Hence ${E}_{i1}\subset {E}_{i}\text{.}$ $\square $ 
From (A.1) it follows that that each $U\in F$ determines a permutation $w\in {S}_{n}$ as follows: $w\left(i\right)$ is the unique element of ${E}_{i}{E}_{i1},$ for $i=1,2,\dots ,n\text{.}$ Let $\varphi :F\to {S}_{n}$ denote the mapping so defined.
The symmetric group acts on $V$ by permuting the basis elements ${e}_{i}:$
$$w\left({e}_{i}\right)={e}_{w\left(i\right)}$$for $w\in {S}_{n}$ and $1\le i\le n\text{.}$ Hence we may regard ${S}_{n}$ as a subgroup of $G\text{.}$
(A.2) Let $U\in F,$ $w\in {S}_{n}\text{.}$ Then $\varphi \left(U\right)=w$ if and only if $U=bwV$ for some $b\in B\text{.}$
Proof.  
Suppose $\varphi \left(U\right)=w\text{.}$ Then for $i1,\dots ,n$ we have $$\begin{array}{cc}\text{(1)}& {U}_{i}\cap {V}_{w\left(i\right)}\supset {U}_{i}\cap {V}_{w\left(i\right)1}\end{array}$$and $$\begin{array}{cc}\text{(2)}& {U}_{i1}\cap {V}_{w\left(i\right)}={U}_{i1}\cap {V}_{w\left(i\right)1}\end{array}$$By virtue of (1) we can choose ${u}_{i}\in {U}_{i}$ of the form $$\begin{array}{cc}\text{(3)}& {u}_{i}={e}_{w\left(i\right)}+\text{lower terms}\end{array}$$where by 'lower terms' is meant a linear combination of ${e}_{1},\dots ,{e}_{w\left(i\right)1}\text{;}$ and ${u}_{i}\notin {U}_{i1}$ by virtue of (2). By rewriting (3) in the form $${u}_{{w}^{1}\left(j\right)}={e}_{j}+\text{lower terms}\phantom{\rule{2em}{0ex}}(1\le j\le n)$$we see that there exists $b\in B$ such that ${u}_{{w}^{1}\left(j\right)}=b{e}_{j}$ for all $j,$ or equivalently $${u}_{i}=b{e}_{w\left(i\right)}=bw{e}_{i}\text{.}$$Hence $UbwV$ as required. For the converse it is enough to show that (i) $\varphi \left(wV\right)=w$ and (ii) $\varphi \left(bU\right)=\varphi \left(U\right)$ for all $b\in B$ and $U\in F\text{.}$ As to (i), $w{V}_{i}\cap {V}_{j}$ is spanned by the basis vectors ${e}_{w\left(k\right)}$ such that $k\le i$ and $w\left(k\right)\le j,$ and therefore $w{V}_{i}\cap {V}_{j}\ne w{V}_{i}\cap {V}_{j1}$ if and only if $j=w\left(k\right)$ for some $k\le i\text{.}$ Thus the set ${E}_{i}\left(wV\right)$ consists of $w\left(1\right),\dots ,w\left(i\right),$ which establishes (i). Finally as to (ii), we have $b{U}_{i}\cap {V}_{j}=b({U}_{i}\cap {V}_{j})$ if $b\in B,$ so that ${E}_{i}\left(bU\right)={E}_{i}\left(U\right)$ and hence $\varphi \left(bV\right)=\varphi \left(U\right)$ as required. $\square $ 
From (A2) we have immediately
(A3) (Bruhat decomposition) $G$ is the disjoint union of the double cosets $BwB,$ $w\in {S}_{n}\text{.}$
For each $w\in {S}_{n},$ let
$${C}_{w}=\left(BwB\right)/B\subset G/B=F\text{.}$$The subsets ${C}_{w}$ are the Schubert cells in the flag manifold $F\text{.}$ By (A.3), $F$ is the disjoint union of the ${C}_{w}\text{.}$
Let $U\in F\text{.}$ Then $U\in {C}_{w}$ if and only if $U$ has a basis $({u}_{1},\dots ,{u}_{n})$ such that ${u}_{i}\in {V}_{w\left(i\right)}{V}_{w\left(i\right)1}$ for each $i\text{.}$ We may normalize the ${u}_{i}$ by taking
$${u}_{i}={e}_{w\left(i\right)}+\text{lower terms.}$$We can then subtract from ${u}_{i}$ suitable multiples of the ${u}_{k}$ for which $k<i$ and $w\left(k\right)<w\left(i\right),$ so as to make the coefficient of ${e}_{w\left(k\right)}$ in ${u}_{i}$ zero for each such $k\text{.}$ Then ${u}_{i}$ is replaced by a vector of the form
$${e}_{w\left(i\right)}+\sum _{j}{a}_{ij}{e}_{j}$$where the sum is over $j<w\left(i\right)$ such that $j\ne w\left(k\right)$ for any $k<i,$ i.e., such that $j<w\left(i\right)$ and ${w}^{1}\left(j\right)>i,$ or equivalently $(i,j)\in D\left(w\right),$ the diagram of $w\text{.}$
(A.4) Let $U\in F\text{.}$ Then $U\in {C}_{w}$ if and only if $U$ has a basis $({u}_{1},\dots ,{u}_{n})$ of the form
$${u}_{i}={e}_{w\left(i\right)}+\sum _{j}{a}_{ij}{e}_{j}$$where the sum is over all $j$ in the ${i}^{\text{th}}$ row of the diagram of $w,$ and the coefficients ${a}_{ij}$ are arbitrary elements of the field $K\text{.}$ Moreover, the ${a}_{ij}$ are uniquely determined by the flag $U,$ and the mapping ${C}_{w}\to {K}^{D\left(w\right)}$ so defined is a bijection.
Proof.  
Clearly each "matrix" $a=\left({a}_{ij}\right)$ of shape $D\left(w\right)$ determines a basis $({u}_{1},\dots ,{u}_{n})$ of $V$ as above, and hence a flag $U\in {C}_{w}\text{.}$ If ${a}^{*}=\left({a}_{ij}^{*}\right)$ determines $({u}_{1}^{*},\dots ,{u}_{n}^{*})$ and the same flag $U,$ then each ${u}_{i}^{*}$ must be expressible as $${u}_{i}^{*}={u}_{i}+\sum _{j<i}{c}_{ij}{u}_{j},$$and from the form of ${u}_{i}^{*}$ and the ${u}_{j}$ it follows that ${u}_{i}^{*}={u}_{i}$ for each $i,$ and hence ${a}^{*}=a\text{.}$ $\square $ 
Since $\text{Card}\hspace{0.17em}D\left(w\right)=\ell \left(w\right)$ it follows from (A.4) that the Schubert cell ${C}_{w}$ is isomorphic to affine space of dimension $\ell \left(w\right)\text{.}$
Let $U\in F$ and let $({u}_{1},\dots ,{u}_{n})$ be any basis of $U\text{.}$ Since ${u}_{1},\dots ,{u}_{i}$ is a basis for ${U}_{i}$ for each $i=1,\dots ,n1,$ the flag $U$ determines each of the exterior products ${u}_{1}\wedge \dots \wedge {u}_{i}\in {\Lambda}^{i}\left(V\right)$ up to a nonzero scalar multiple, and hence $U$ determines the vector
$$\begin{array}{cc}\text{(1)}& {u}_{1}\otimes ({u}_{1}\wedge {u}_{2})\otimes \dots \otimes ({u}_{1}\wedge \dots \wedge {u}_{n1})\in E\end{array}$$up to a nonzero scalar multiple, where $E=V\otimes {\Lambda}^{2}V\otimes \dots \otimes {\Lambda}^{n1}V\text{.}$ If $P\left(E\right)$ denotes the projective space of $E$ (i.e. the space whose points are the lines in $E\text{),}$ we have an injective mapping
$$\pi :F\mapsto P\left(E\right)$$(the Plücker embedding) for which $\pi \left(U\right)$ is the line in $E$ generated by the vector (1).
Assume from now on that the field $K$ is the field of complex numbers. Then the embedding $\pi $ realizes the flag manifold $F$ as a complex projective algebraic variety, which is smooth because $F$ has a transitive group of automorphisms (namely $G\text{).}$ Each Schubert cell ${C}_{w}$ is a locally closed subvariety of $F,$ isomorphic to affine space of dimension $\ell \left(w\right)\text{.}$
For each $w\in {S}_{n}$ let
$${X}_{w}={\stackrel{\u203e}{C}}_{w}$$be the closure of ${C}_{w}$ in $F\text{.}$ The ${X}_{w}$ are the Schubert varieties in $F,$ and a flag $U$ lies in ${X}_{w}$ if and only if $U$ has a basis $({u}_{1},\dots ,{u}_{n})$ such that ${u}_{i}\in {V}_{w\left(i\right)}$ for each $i\text{.}$ Each ${X}_{w}$ is in fact a union of Schubert cells ${C}_{v}\text{:}$ if $({a}_{1},\dots ,{a}_{p})$ is a reduced word for $w,$ then ${C}_{v}\subset {X}_{w}$ if and only if $v$ is of the form ${s}_{{b}_{1}}\dots {s}_{{b}_{q}}$ where $({b}_{1},\dots ,{b}_{q})$ is a subsequence of $({a}_{1},\dots ,{a}_{p}),$ that is to say if and only if $v\le w$ in the Bruhat order. In particular, ${X}_{1}={C}_{1}$ is the single point $V\in F\text{.}$ At the other extreme, if ${w}_{0}$ is the longest element of ${S}_{n},$ then ${X}_{{w}_{0}}$ is the whole of $F,$ and the dimension of $F$ is $\ell \left({w}_{0}\right)=\frac{1}{2}n(n1)\text{.}$
Let ${H}^{*}(F;\mathbb{Z})$ be the cohomology ring (with integral coefficients) of the projective variety $F\text{.}$ Each closed subvariety $X$ of $F$ determines an element $\left[X\right]\in {H}^{*}(F;\mathbb{Z}),$ and cupproduct in ${H}^{*}(F;\mathbb{Z})$ corresponds, roughly speaking, to intersection of subvarieties. In particular, for each $w\in {S}_{n},$ we have a cohomology class $\left[{X}_{w}\right]\in {H}^{*}(F;\mathbb{Z}),$ and it is a consequence of the cell decomposition (A.3) of $F$ that the $\left[{X}_{w}\right]$ form a $\mathbb{Z}\text{basis}$ of ${H}^{*}(F;\mathbb{Z})\text{.}$ In particular, $\left[{X}_{{w}_{0}}\right]$ is the identity element.
The connection between the classes $\left[{X}_{w}\right]$ and the Schubert polynomials ${\U0001d516}_{w}(w\in {S}_{n})$ is given by
(A.5) There is a surjective ring homomorphism
$$\alpha :\mathbb{Z}[{x}_{1},\dots ,{x}_{n}]\to {H}^{*}(F;\mathbb{Z})$$such that
$$\alpha \left({\U0001d516}_{w}\right)=\left[{X}_{{w}_{0}w}\right]$$for each $w\in {S}_{n}\text{.}$
Proof.  
Let us temporarily write $${\sigma}_{w}=\left[{X}_{{w}_{0}w}\right]$$for $w\in {S}_{n}\text{.}$ Monk [Mon1959] proved that for all $w\in {S}_{n}$ and $r=1,\dots ,n1$ $$\begin{array}{cc}\text{(1)}& {\sigma}_{w}\xb7{\sigma}_{{s}_{r}}=\sum _{t}{\sigma}_{wt}\end{array}$$where the sum on the right hand side is over all transpositions $t={t}_{ij}$ such that $i\le r<j\le n$ and $\ell \left(wt\right)=\ell \left(w\right)+1,$ as in (4.15''), Define ${\xi}_{1},\dots ,{\xi}_{n}\in {H}^{*}(F,\mathbb{Z})$ by $$\begin{array}{ccc}{\xi}_{1}& =& {\sigma}_{1}\\ {\xi}_{i}& =& {\sigma}_{i}{\sigma}_{i1}\phantom{\rule{2em}{0ex}}(2\le i\le n1)\\ {\xi}_{n}& =& {\sigma}_{n1}\end{array}$$From (1) we deduce the counterpart of (4.16): if $r$ is the last descent of $w$ (so that $r\le n1\text{),}$ then we have $$\begin{array}{cc}\text{(2)}& {\sigma}_{w}={\sigma}_{v}{\xi}_{r}+\sum _{w\prime}{\sigma}_{w\prime}\end{array}$$where $v,w\prime $ are as in (4.16). Now iteration of (4.16) will ultimately express ${\U0001d516}_{w}$ as a sum of monomiais, i.e. as a polynomial in ${x}_{1},\dots ,{x}_{n1}\text{;}$ and iteration of (2) will express ${\sigma}_{w}$ as the same polynomial in ${\xi}_{1},\dots ,{\xi}_{n1}\text{.}$ Hence if we define $\alpha :{P}_{n}\mapsto {H}^{*}(F;\mathbb{Z})$ by $\alpha \left({x}_{i}\right)={\xi}_{i}$ $(1\le i\le n),$ we have ${\sigma}_{w}=\alpha \left({\U0001d516}_{w}\right)$ for all $w\in {S}_{n},$ and the proof of (A.5) is complete. $\square $ 
In fact the kernel of the homomorphism $\alpha $ is generated by the elementary symmetric functions ${e}_{1},\dots ,{e}_{n}$ of the $x\text{'s.}$
We shall draw one consequence of (A.5) that we have not succeeded in deriving directly from the definition (4.1) of the Schubert polynomials. Since the ${\sigma}_{w},w\in {S}_{n},$ form a $\mathbb{Z}\text{basis}$ of ${H}^{*}(F;\mathbb{Z}),$ any product ${\sigma}_{u}{\sigma}_{v}(u,v\in {S}_{n})$ is uniquely a linear combination of the ${\sigma}_{w},$ and it follows from intersection theory on $F$ that the coefficient of ${\sigma}_{w}$ in ${\sigma}_{u}{\sigma}_{v}$ is a nonnegative integer. From this we deduce
(A.6) Let $u,v$ be permutations, and write ${\U0001d516}_{u}{\U0001d516}_{v}$ as an integral linear combination of the ${\U0001d516}_{w},$ say
$$\begin{array}{cc}\text{(1)}& {\U0001d516}_{u}{\U0001d516}_{v}=\sum _{w}{c}_{uv}^{w}{\U0001d516}_{w}\text{.}\end{array}$$Then the coefficients ${c}_{uv}^{w}$ are nonnegative.
We have only to choose $n$ sufficiently large so that $u,v$ and all the permutations $w$ such that ${c}_{uv}^{w}\ne 0$ lie in ${S}_{n},$ and then apply the homomorphism $\alpha $ of (A,5).
Remark. The coefficients ${c}_{uv}^{w}$ in (A.6) are zero unless
(a)  $\ell \left(w\right)=\ell \left(u\right)+\ell \left(v\right),$ 
(b)  $u\le w$ and $v\le w\text{.}$ 
For ${\U0001d516}_{u}{\U0001d516}_{v}$ is homogeneous of degree $\ell \left(u\right)+\ell \left(v\right),$ which gives condition (a). Also we have
$$\begin{array}{ccc}{c}_{uv}^{w}& =& {\partial}_{w}\left({\U0001d516}_{u}{\U0001d516}_{v}\right)\\ & =& \sum _{{v}_{1}\le w}{v}_{1}{\partial}_{w/{v}_{1}}\left({\U0001d516}_{u}\right){\partial}_{{v}_{1}}\left({\U0001d516}_{v}\right)\end{array}$$by (2.17), and the only possible nonzero term in this sum is that corresponding to ${v}_{1}=v\text{.}$ Hence if ${c}_{uv}^{w}\ne 0$ we must have $v\le w,$ and by symmetry also $u\le w\text{.}$
This is a typed excerpt of the book Notes on Schubert Polynomials by I. G. Macdonald.