Schur positivity of the Stanley Symmetric Functions
Without symmetric Functions.
Without Representation Theory
UC San Diego
Thursday May 10,
3:00pm (note unusual time and day)
In the summer 1982 R. Stanley initiated the study of reduced
factorizations of elements of $S_n$. Central to this work
was the introduction of a family of symmetric functions indexed by
permutations. He conjectured these functions to be Schur positive
and proved a number of their interesting properties
including the enumeration of certain classes of reduced
factorizations. At the same time, in a completely independent
development, Lascoux and Schützenberger developed the
theory of Schubert polynomials. One of the byproducts of this
theory was the first proof (nov 1982) of the Schur positivity of
the Stanley symmetric functions. This proof as well as several that
appeared in the sequel requires the introduction of a substantial
amount of extraneous machinery. In this lecture we show how some
the Lascoux-Schützenberger results on Schubert polynomials lead to a
completely elementary purely combinatorial proof which uses only reduced
decompositions. The crucial ingredient in this proof is a remarkable
bijection discovered by David Little. An applet yielding this bijection
will be the highlight of the talk.
Speaker's Contact Info: garsia(at-sign)schur.ucsd.edu
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