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 representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.
Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions
are alternating polynomials by properties of the determinant. A polynomial is alternating if it changes sign under any transposition of the variables.
Since they are alternating, they are all divisible by the Vandermonde determinant
The Schur polynomials are defined as the ratio
This is known as the bialternant formula of Jacobi. It is a special case of the Weyl character formula.
This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant.
Properties
The degree Schur polynomials in variables are a linear basis for the space of homogeneous degree symmetric polynomials in variables. For a partition , the Schur polynomial is a sum of monomials,
where the summation is over all semistandard Young tableaux of shape . The exponents give the weight of , in other words each counts the occurrences of the number in . 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).
Schur polynomials can be expressed as linear combinations of monomial symmetric functions with non-negative integer coefficients called Kostka numbers,
The Kostka numbers are given by the number of semi-standard Young tableaux of shape λ and weight μ.
Jacobi−Trudi identities
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
and is the conjugate partition to .
In both identities, functions with negative subscripts are defined to be zero.
The Giambelli identity
Another determinantal identity is Giambelli's formula, which expresses the Schur function for an arbitrary partition in terms of those for the hook partitions contained within the Young diagram. In Frobenius' notation, the partition is denoted
where, for each diagonal element in position , denotes the number of boxes to the right in the same row and denotes the number of boxes beneath it in the same column (the arm and leg lengths, respectively).
The Giambelli identity expresses the Schur function corresponding to this partition as the determinant
of those for hook partitions.
The Cauchy identity
The Cauchy identity for Schur functions (now in infinitely many variables), and its dual state that
and
where the sum is taken over all partitions λ, and , denote the complete symmetric functions and elementary symmetric functions, respectively. If the sum is taken over products of Schur polynomials in variables , the sum includes only partitions of length since otherwise the Schur polynomials vanish.
There are many generalizations of these identities to other families of symmetric functions. For example, Macdonald polynomials, Schubert polynomials and Grothendieck polynomials admit Cauchy-like identities.
Further identities
The Schur polynomial can also be computed via a specialization of a formula for Hall–Littlewood polynomials,
where is the subgroup of permutations such that
for all i, and w acts on variables by permuting indices.
The Murnaghan−Nakayama rule
The Murnaghan–Nakayama rule expresses a product of a power-sum symmetric function with a Schur polynomial, in terms of Schur polynomials:
where the sum is over all partitions μ such that μ/λ is a rim-hook of size r and ht(μ/λ) is the number of rows in the diagram μ/λ.