Pieri's formula explained

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} _{λ} s _{λ} 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 rulefor multiplying an elementary symmetric polynomial with a Schur polynomial:\displaystyle s _{\mu} e _{r=\sum} _{λ} s _{λ} 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.