In mathematics, Monk's formula, found by , is an analogue of

Pieri's formula In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function. In terms of Schur func ...

that describes the product of a linear

Schubert polynomial In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by and are named after Hermann Schubert. Background described the history ...

by a Schubert polynomial. Equivalently, it describes the product of a special

Schubert cycle In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using li ...

by a Schubert cycle in the

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

of a

flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smoot ...

. Write ''t''_ij for the transposition ''(i j)'', and ''s''_i = ''t''_i,i+1. Then 𝔖_{s_r} = ''x''₁ + ⋯ + ''x''_r, and Monk's formula states that for a permutation ''w'',

\mathfrak_ \mathfrak_w = \sum_ \mathfrak_,

where

\ell(w)

is the

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

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 In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion or ...

References

*{{Citation , last1=Monk , first1=D. , title=The geometry of flag manifolds , doi=10.1112/plms/s3-9.2.253 , mr=0106911 , year=1959 , journal=Proceedings of the London Mathematical Society , series=Third Series , issn=0024-6115 , volume=9 , pages=253–286 , issue=2, citeseerx=10.1.1.1033.7188 Symmetric functions