Identities of Littlewood-Richardson coefficients for Schubert
           polynomials and orders on Weyl groups

  Frank Sottile
  MSRI workshop on symmetric functions and representation theory
  15 April 1997

For Schubert polynomials, the analogues of Littlewood-Richardson
coefficients are expected to be related to the enumeration of chains in
the Bruhat order on  S_n. We give a number of new identities
among these coefficients, primarily among those which arise in the
multiplication of a Schubert polynomial by a Schur symmetric polynomial.
For many of these identities, there is a companion result about the
Bruhat order which we expect would imply the identity, were it known how
to express these coefficients in terms of the Bruhat order.  This
analysis leads to a new graded partial order on the symmetric group,
results on the enumeration of chains in the Bruhat order, an upper bound
on these coefficients, and a determination of many of them.

Considering the analogous problem for the multiplication of type C
Schubert polynomials by Schur Q symmetric polynomials, we obtain similar
results, this time for chains in the Bruhat order on the group of signed
permutations.  We also give a Pieri-type formula in this setting,
expressed in terms of chains.


This represents joint work with Nantel Bergeron, and the results of the
first paragraph are contained in a recent preprint:   MSRI # 1996 - 083.