Schur Polynomial - Properties

Properties

The degree d Schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetric polynomials in n variables.

For a partition λ, the Schur polynomial is a sum of monomials:

where the summation is over all semistandard Young tableaux T of shape λ; the exponents t₁, ..., t_n give the weight of T, in other words each t_i counts the occurrences of the number i in T. This can be shown to be equivalent to the definition from the first Giambelli formula using the Lindström–Gessel–Viennot lemma (as outlined on that page).

The first Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the complete homogeneous symmetric polynomials:

where

.

The second Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the elementary symmetric polynomials:

,

where

where is the dual partition to .

These two formulae are known as "determinantal identities". Another such identity is the Giambelli formula, which expresses the Schur function for an arbitrary partition in terms of those for "hook partitions. In Frobenius notation, a partition is determined by the "arm" and "leg" lengths of the Young diagram for the partition; i.e., for each diagonal element in position "ii", the number of boxes in the same row to the right is denoted and the number of elements in the same column beneath it is denoted . In Frobenius notation, the partitation is denoted . the Giambelli identity expresses the partition as the determinant

Schur polynomials s_λ can be expressed as linear combinations of monomial symmetric functions m_μ with non-negative integer coefficients K_λμ called Kostka numbers:

Evaluating the Schur polynomial s_λ in (1,1,...,1) gives the number of semi-standard Young tableaux of shape λ with entries in 1,2,...n. One can show, by using the Weyl character formula for example, that