# 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

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:

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 - Sottile, Frank (2001) [1994], "S/s130080", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4