Schubert polynomials have an important geometrical interpretation as
Schubert cycles. One now works in the quotient ring
(Flag )
of the ring
of polynomials modulo the ideal 
generated by the symmetric polynomials
without constant term, which is the so-called ring of coinvariants of the
symmetric group. This ring can be interpreted as the cohomology ring
, or Chow ring,
of the variety of complete flags in 
. With this interpretation, the class
of the Schubert polynomial 
is the Poincaré dual of the Schubert cycle
.
The flag manifold is equipped with tautological line bundles
, whose respective first Chern classes are equal
to the classes 
of 
modulo the ideal 
.
The double Schubert polynomials also have a geometrical
interpretation.
When the variables 
and 
are interpreted as Chern classes of certain
line bundles over a variety M, they give the cohomology classes corresponding
to various degeneracy loci of maps of flagged bundles over M.
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