Monk's formula explained

In mathematics, Monk's formula, found by, is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold.

Write t_ij for the transposition (i j), and s_i = t_i,i+1. Then _sr = x₁ + ⋯ + x_r, and Monk's formula states that for a permutation w,

ak{S} _{s _{r}} ak{S} _{w} = \sum _{{i} \atop {\ell(wt _{ij} ) = \ell(w) +1 }} ak{S} _{wt _{ij}} ,

where

\ell(w) is the length of w. The pairs (i, j) appearing in the sum are exactly those such that i ≤ r < j, w_i < w_j, and there is no i < k < j with w_i < w_k < w_j; each wt_ij is a cover of w in Bruhat order