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DOUBLE SCHUBERT POLYNOMIALS AND DOUBLE SCHUBERT VARIETIES
 

Summary: DOUBLE SCHUBERT POLYNOMIALS AND DOUBLE
SCHUBERT VARIETIES
DAVE ANDERSON
The purpose of this note is to explain the geometry underlying a certain
identity of Schubert polynomials, namely,
Sw(x; y) =
v-1u=w
Su(x) Sv(-y),(1)
where the sum is over those u, v Sn such that v-1u = w and (u) + (v) =
(w). (See [LS],[M, (6.3)].) This implies an identity in the cohomology
ring of a product of two flag varieties, when the variables are specialized to
appropriate Chern classes of universal bundles on Fl Fl:
[w] =
v-1u=w
[u] [w0 v w0 ].(2)
Here w is a certain degeneracy locus in Fl Fl, which we will call a
double Schubert variety, since it describes pairs of flags in special position
with respect to one another; the degeneracy locus formula of [F1] gives
[w] = Sw(x; y).
The geometric formula (2) is a priori a weaker statement than (1), since

  

Source: Anderson, Dave - Department of Mathematics, University of Washington at Seattle

 

Collections: Mathematics