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A bijective proof of Macdonald’s reduced word formula. (English) Zbl 1409.05024
Summary: We give a 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 S. Fomin and R. P. Stanley [Adv. Math. 103, No. 2, 196–207 (1994; Zbl 0809.05091)]. Such a proof has been sought for over 20 years. Our bijective tools also allow us to solve a problem posed by S. Fomin and A. N. Kirillov [J. Algebr. Comb. 6, No. 4, 311–319 (1997; Zbl 0882.05010)] using work of Wachs, Lenart, Serrano and Stump. These results extend earlier work by the third author [“A Markov growth process for Macdonald’s distribution on reduced words”, Preprint, arXiv:1409.7714] on a Markov process for reduced words of the longest permutation.

##### MSC:
 05A17 Combinatorial aspects of partitions of integers 05A15 Exact enumeration problems, generating functions 05E10 Combinatorial aspects of representation theory 05E05 Symmetric functions and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds 60J99 Markov processes
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