Quasi-homogeneous polynomial

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.

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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