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 functions ''s''_λ indexed by partitions λ, it states that
:$\displaystyle s_\mu h_r=\sum_\lambda s_\lambda$
where ''h''_''r'' is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding ''r'' elements, no two in the same column.
By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule
for multiplying an elementary symmetric polynomial with a Schur polynomial:
:$\displaystyle s_\mu e_r=\sum_\lambda s_\lambda$
The sum is now taken over all partitions λ obtained from μ by adding ''r'' elements, no two in the same ''row''.

Pieri's formula implies Giambelli's formula. The Littlewood–Richardson rule is a generalization of Pieri's formula
giving the product of any two Schur functions. Monk's formula is an analogue of Pieri's formula for flag manifolds.

References

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*{{eom, title=Schubert calculus, first=Frank, last= Sottile

Symmetric functions