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 = x₁ + ⋯ + 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.

References

• Monk, D. (1959), "The geometry of flag manifolds", Proceedings of the London Mathematical Society, Third Series, 9 (2): 253–286, CiteSeerX 10.1.1.1033.7188, doi:10.1112/plms/s3-9.2.253, ISSN 0024-6115, MR 0106911