Schur positivity of the Stanley Symmetric Functions
Without symmetric Functions.
Without Schensted.
Without Representation Theory

Adriano Garsia

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)

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