In mathematics, Giambelli's formula, named after
Giovanni Giambelli, expresses
Schubert class
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of ...
es in terms of special Schubert classes, or
Schur functions in terms of
complete symmetric function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\l ...
s.
It states
:
where σ
λ is the Schubert class of a
partition λ.
Giambelli's formula is a consequence of
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 fu ...
. The
Porteous formula In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern ...
is a generalization to morphisms of vector bundles over a variety.
See also
*
Schubert calculus - includes examples
References
*
*
Symmetric functions
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