Richard Stanley through a crystal lens and from a random angle

TL;DR

This paper explores Stanley's work on symmetric functions and reduced words of permutations, offering a crystal theoretic interpretation and analyzing a Markov chain on these words with potential generalizations.

Contribution

It introduces a novel crystal theoretic perspective on Stanley's symmetric functions and reduced words, connecting them with decreasing factorizations and Markov chains.

Findings

Crystal operators on decreasing factorizations relate to Edelman-Greene insertion.

A Markov chain on reduced words is studied with potential for generalization.

New insights into the combinatorial structure of permutation factorizations.

Abstract

We review Stanley's seminal work on the number of reduced words of the longest element of the symmetric group and his Stanley symmetric functions. We shed new light on this by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion. We also view this from a random perspective and study a Markov chain on reduced words of the longest element in a finite Coxeter group, in particular the symmetric group, and mention a generalization to a poset setting.