In mathematics, Giambelli's formula, named after
Giovanni Giambelli, expresses
Schubert classes as determinants in terms of special Schubert classes.
It states
:
where σ
λ is the Schubert class of a
partition λ.
Giambelli's formula may be derived as 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 is a generalization to morphisms of vector bundles over a variety.
In the theory of symmetric functions, the same identity, known as the
first Jacobi-Trudi identity expresses
Schur functions as determinants in terms of
complete symmetric functions. There is also the dual
second Jacobi-Trudi identity which expresses
Schur functions as determinants in terms of
elementary symmetric functions. The corresponding identity also holds for Schubert classes.
There is another
Giambelli identity, expressing
Schur functions as determinants of matrices whose entries are Schur functions corresponding to ''hook partitions'' contained within the same
Young diagram
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...
. This too is valid for Schubert classes, as are all Schur function identities. For instance, hook partition Schur functions can be expressed bilinearly in terms of elementary and complete symmetric functions, and Schubert classes satisfy these same relations.
See also
*
Schubert calculus
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 and, as such, is viewed as part of enumerative geometr ...
- includes examples
References
*
*
Symmetric functions
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