Monk's formula
# Monk's formula

In mathematics, Monk's formula, found by Monk (1959), 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 $𝔖_{sr}$ = $x_1$ + ⋯ + $x_r,$ and Monk's formula states that for a permutation w,

${\mathfrak {S}}_{s_{r}}{\mathfrak {S}}_{w}=\sum _{{i\leq r<j} \atop {\ell (wt_{ij})=\ell (w)+1}}{\mathfrak {S}}_{wt_{ij}},$

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.

