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
s μ h r = ∑ λ s λ {\displaystyle \displaystyle s_{\mu }h_{r}=\sum _{\lambda }s_{\lambda }}
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 rule for multiplying an elementary symmetric polynomial with a Schur polynomial:
s μ e r = ∑ λ s λ {\displaystyle \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 Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144{{citation}}: CS1 maint: deprecated archival service (link) Sottile, Frank (2001) [1994], "Schubert calculus", Encyclopedia of Mathematics, EMS Press
