The homogeneous polynomials of this kind of any given degree form a basis for the space of all homogeneous symmetric polynomials of that degree
This construct is associated with, or parametrized by, an unordered integer partition
- 1 Definition
- 2 Particular cases
- 3 Facts
- 4 Relation with other polynomials
Direct definition in terms of content of tableaux
Let denote an unordered partition of the integer into at most parts (or equivalently a Young diagram with cells and at most rows). The Schur polynomial associated with is the sum of all the following monomials:
For each semistandard tableau whose shape is that Young diagram, set the corresponding monomial such that the exponent of is the number of occurrences of in the tableau. In other words, the monomial is where is the content of the given tableau.
As images from the tableau ring
The Schur polynomial can also be viewed as the image via the canonical map from the tableau ring on variables, denoted , to the polynomial ring, of the Schur element corresponding to a tableau shape . The Schur element is defined as the sum of all semistandard tableaux of the given shape . These elements are called Schur elements.
Th Schur polynomial for a shape and variables is denoted as .
As a quotient of antisymmetric polynomials
Define, for any partition , the antisymmetric polynomial corresponding to as follows:
where denotes the sign of (it is +1 if is an even permutation, and -1 if is an odd permutation).
The the Schur polynomial for a partition is defined as:
where is the column partition with all distinct entries from 1 to . This needs to be clarified/explained properly
Nature of the polynomials
The Schur polynomials are all symmetric polynomials in the s. In other words, the number of semistandard tableaux with the exponents forming an ordered integer partition of (with parts) is independent of the ordering within .
In fact, the Schur polynomials of degree form a basis for the space of all homogeneous symmetric polynomials of degree in variables. Thus, the set of all Schur polynomials in variables form a graded basis for the space of all symmetric functions in the variables.
Rule for multiplication
The structure constants for the multiplication for these basis elements are: Littlewood-Richardson numbers
The structure constants for the ring of symmetric polynomials, taking the Schur polynomials as basis, are the Littlewood-Richardson numbers. In fact, the Littlewood-Richardson rule tells us something much stronger: the Schur elements (viz symmetric sums of tableau elements of a given shape) also have the Littlewood-Richardson numbers as their structure constants.
Relation with other polynomials
Complete symmetric polynomials
Another basis for the space of symmetric polynomials are the complete symmetric polynomials associated with partitions. First, define the complete symmetric polynomial associated with an integer as the sum of all monomials of degree in the variables. Then, the complete symmetric polynomial for a partition isthe product of the complete symmetric polynomials for each part.
This is denoted as .
The complete symmetric polynoimals are related to the Schur polynomials by the matrix of Kostka numbers, viz:
Elementary symmetric polynomials
Another basis for the space of symmetric polynomials are the elementary symmetric polynomials associated with partitions. The elementary symmetric polynomial for a partition is defined as the product of elementary symmetric polynomials corresponding to each part, where the elementary symmetric polynomial of degree is the sum of all multilinear monomials of degree .
The elementary symmetric polynomial for a partition and variables is defined as .
The elementary symmetric polynomials are related to the Schur polynomials by the matrix of Kostka numbers, with one of the partitions conjugated:
where denotes the conjugate partition to .
Another basis for the space of symmetric polynomials are the Newton polynomials associated with partitions. The Newton polynomial associated with a partition is the product of Newton polynomials associated with each part, where the Newton polynomial of degree is the sum of the powers of all the variables.
The Newton polynomials are related to the Schur polynomials by the character table, viz:
Here equals the character of the representation corresponding to on the conjugacy class corresponding to .