Monk's formula
In mathematics, Monk's formula, found by
flag manifold
.
Write tij for the
transposition
(i j), and si = ti,i+1. Then 𝔖sr = x1 + ⋯ + xr, and Monk's formula states that for a permutation w,
where is the length of w. The pairs (i, j) appearing in the sum are exactly those such that i ≤ 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.
References
- Monk, D. (1959), "The geometry of flag manifolds", Proceedings of the London Mathematical Society, Third Series, 9 (2): 253–286, MR 0106911