an:07049524
Zbl 1409.05024
Billey, Sara C.; Holroyd, Alexander E.; Young, Benjamin J.
A bijective proof of Macdonald's reduced word formula
EN
Algebr. Comb. 2, No. 2, 217-248 (2019).
00432277
2019
j
05A17 05A15 05E10 05E05 14M15 60J99
Young tableaux; Ferrers shape; reduced words; identity; Stanley's formula; Macdonald's formula; Schubert polynomials; enumeration of plane partitions; permutations; shapes
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 \textit{S. Fomin} and \textit{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 \textit{S. Fomin} and \textit{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, \url{arXiv:1409.7714}] on a Markov process for reduced words of the longest permutation.
Zbl 0809.05091; Zbl 0882.05010