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 fu ...
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
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 viewed ...
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 sm ...
.
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'',
where
is the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
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
*{{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