# A bijective proof of Macdonald's reduced word formula

## TL;DR

This paper provides a new bijective proof of Macdonald's reduced word formula using pipe dreams and bumping algorithms, extending to special cases and solving longstanding open problems in algebraic combinatorics.

## Contribution

It introduces a novel bijective approach to Macdonald's identity, extending previous work and addressing open problems posed over two decades ago.

## Key findings

- Bijective proof of Macdonald's reduced word identity
- Extension to principal specialization by Fomin and Stanley
- Solution to a problem posed by Fomin and Kirillov in 1997

## Abstract

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 Fomin and Stanley. Such a proof has been sought for over 20 years. Our bijective tools also allow us to solve a problem posed by Fomin and Kirillov from 1997 using work of Wachs, Lenart, Serrano and Stump. These results extend earlier work by the third author on a Markov process for reduced words of the longest permutation.