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Concept
Schur polynomial
Summary
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.
Schur polynomials are indexed by integer partitions. Given a partition λ = (λ1, λ2, ...,λn),
where λ1 ≥ λ2 ≥ ... ≥ λn, and each λj 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.
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 = (λ1, λ2, ..., λn), the Schur polynomial is a sum of monomials,
where the summation is over all semistandard Young tableaux T of shape λ. The exponents t1, ..., tn give the weight of T, in other words each ti 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).
Official source
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