Oxygenation (environmental): Difference between revisions

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In mathematics, '''Monk's formula''', found by {{harvtxt|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''<sub>ij</sub> for the [[Transposition (mathematics)|transposition]] ''(i j)'', and ''s''<sub>i</sub> = ''t''<sub>i,i+1</sub>. Then 𝔖<sub>s<sub>r</sub></sub> = ''x''<sub>1</sub> + ⋯ + ''x''<sub>r</sub>, and Monk's formula states that for a permutation ''w'',


<math>
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\mathfrak{S}_{s_r} \mathfrak{S}_w = \sum_{{i \leq r < j} \atop {\ell(wt_{ij}) = \ell(w)+1}} \mathfrak{S}_{wt_{ij}},
</math>
 
where <math>\ell(w)</math> is the [[length]] of ''w''. The pairs (''i'', ''j'') appearing in the sum are exactly those such that ''i'' &le; ''r'' < ''j'', ''w''<sub>i</sub> < ''w''<sub>j</sub>, and there is no ''i'' < ''k'' < ''j'' with ''w''<sub>i</sub> < ''w''<sub>k</sub> < ''w''<sub>j</sub>; each ''wt''<sub>ij</sub> 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 | id={{MathSciNet | id = 0106911}} | year=1959 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=9 | pages=253–286 | issue=2}}
 
[[Category:Symmetric functions]]

Latest revision as of 06:19, 16 August 2014


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