| Abstract: In this talk we discuss certain
noncommutative analogs of Schubert polynomials. Our results represent an
extension of the work of S. Fomin and C. Greene on non-commutative Schur
functions. If the variables satisfy certain relations (essentially the
same as those needed in the theory of noncommutative Schur functions),
we prove a Pieri-type formula and a Cauchy identity for our non-commutative
polynomials. We sketch the proof of these results, which is based on the
combinatorics of certain (0,1)-tableaux of staircase shape. Our results
have applications to the K-theory of flag varieties, namely to the expansion
of Grothendieck polynomials (which represent Schubert classes in K-theory)
in the basis of Schubert polynomials. We conclude with a brief discussion
of the geometrical significance of this expansion and some open problems
related to it. |
alexsuciu@neu.edu