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 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 functions 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.

Schur polynomials are indexed by integer partitions. Given a partition $λ = (λ 1, λ 2, \dots, λ n)$ , where $λ 1 \geq λ 2 \geq \dots \geq λ n$ , and each $λ j$ 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,

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