Combinatorial constructions of Schubert polynomials: a unified approach
refreshments at 3:45pm
In this talk I investigate the connections not yet understood between the
existing combinatorial structures for the construction of Schubert
polynomials. These structures are: certain strand diagrams called
rc-graphs, increasing labeled chains in the Bruhat order on the symmetric
group, balanced labelings of the diagram of a permutation, Kohnert
diagrams (which are certain diagrams obtained from the diagram of a
permutation by specified moves), and semistandard Young tableaux. Recent
research on Schubert polynomials has shown the importance of rc-graphs.
Thus, we relate the other structures to rc-graphs by constructing new
bijections and by studying the properties of existing bijections. For
instance, the construction of Schubert polynomials in terms of increasing
labeled chains in the Bruhat order, due to N. Bergeron and F. Sottile,
and the construction in terms of Kohnert diagrams, due to A. Kohnert and
R. Winkel, were done independently of rc-graphs. In this case, our goal
is to construct bijections from rc-graphs to the mentioned structures,
and, hence, to give more transparent proofs to the corresponding
formulas. On the other hand, S. Billey, W. Jockusch, and R. Stanley, in
their work which initiated the study of rc-graphs, constructed a map from
rc-graphs to a multiset of semistandard Young tableaux. It is known that
the latter are endowed with a crystal graph structure, given by the
action of certain operators on tableaux, which are relevant to the
representation theory of GL_n(C). We discuss the way in which these
operators can be lifted to rc-graphs, as well as some consequences.
Speaker's Contact Info: lenart(at-sign)csc.albany.edu
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