Monk's formula

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 tij for the transposition (i j), and si = ti,i+1. Then 𝔖s r = x1 + ⋯ + xr, 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 &le; r < j, wi < wj, and there is no i < k < j with wi < wk < wj; each wtij is a cover of w in Bruhat order.