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
- [math]\displaystyle{ \displaystyle s_\mu h_r=\sum_\lambda s_\lambda }[/math]
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:
- [math]\displaystyle{ \displaystyle s_\mu e_r=\sum_\lambda s_\lambda }[/math]
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, http://www.oup.com/uk/catalogue/?ci=9780198504504
- Hazewinkel, Michiel, ed. (2001), "Schubert calculus", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
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