Mitosis recursion for coefficients of Schubert polynomials

Abstract

Mitosis is a rule introduced by Knutson and Miller for manipulating subsets of the n× n grid. It provides an algorithm that lists the reduced pipe dreams (also known as rc-graphs) of Fomin and Kirillov for a permutation w∈ S n by downward induction on weak Bruhat order, thereby generating the coefficients of Schubert polynomials of Lascoux and Schützenberger inductively. This note provides a short and purely combinatorial proof of these properties of mitosis.