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Reduced words and a formula of MacdonaldAlgebra & Discrete Mathematics
Macdonald gave a remarkable formula connecting a weighted sum of
reduced words for a permutation with the number of terms in a Schubert
polynomial. We will review some of the fascinating results on the set
of reduced words in order to put our main results in context. The
main result is a new bijective proof of Macdonald's reduced word
identity using pipe dreams and Little's bumping algorithm. This proof
extends to a principal specialization due to Fomin and Stanley. This
approach has been sought for over 20 years. Our bijective tools also
allow us to address a problem posed by Fomin and Kirillov from 1997
using work of Wachs, Lenart, Serrano and Stump. This project extends
earlier work by the third author on a Markov process for reduced words
of the longest permutation.
This is joint work with Ben Young and Alexander Holroyd.
Prof Billey is around all Monday and much of Tues. Email Monica or her to set up some time to talk research!