Symmetric functions

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 sλ indexed by partitions λ, it states that where hr 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: 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. (Wikipedia).

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Related pages

Elementary symmetric polynomial | Giambelli's formula | Mario Pieri | Partition (number theory) | Complete homogeneous symmetric polynomial | Schubert calculus | Schur polynomial | Littlewood–Richardson rule | Monk's formula