In mathematics, Pieri's formula, named after
Mario Pieri
Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry.
Biography
Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pie ...
, describes the product of a
Schubert cycle In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, \mathbf_k(V) of k-dimensional subspaces of a vector space V, usually with singular points. Like the Grassmannian, it is a kind of moduli space, whose elements sati ...
by a special Schubert cycle in the
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 ...
, or the product of a
Schur polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In ...
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
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in ...
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
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
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
In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes as determinants in terms of special Schubert classes.
It states
:\displaystyle \sigma_\lambda= \det(\sigma_)_
where σλ is the Schubert ...
. The
Littlewood–Richardson rule In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural number ...
is a generalization of Pieri's formula
giving the product of any two Schur functions.
Monk's formula In mathematics, Monk's formula, found by , is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cyc ...
is an analogue of Pieri's formula for flag manifolds.
References
*
*{{eom, title=Schubert calculus, first=Frank, last= Sottile
Symmetric functions