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

• 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, archived from the original on 2012-12-11
• Sottile, Frank (2001) [1994], "Schubert calculus", Encyclopedia of Mathematics, EMS Press