Combinatorial constructions of Schubert polynomials: a unified approach

Cristian Lenart

SUNY Albany

April 6,
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)

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