Schur polynomial

Direct definition in terms of content of tableaux
Let $$\lambda$$ denote an unordered partition of the integer $$n$$ into at most $$m$$ parts (or equivalently a Young diagram with $$n$$ cells and at most $$m$$ rows). The Schur polynomial associated with $$\lambda$$ 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 $$x_i$$ is the number of occurrences of $$i$$ in the tableau. In other words, the monomial is $$x^{\mu}$$ where $$\mu$$ 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 $$m$$ variables, denoted $$R_m$$, to the polynomial ring, of the Schur element corresponding to a tableau shape $$\lambda$$. The Schur element is defined as the sum of all semistandard tableaux of the given shape $$\lambda$$. These elements are called Schur elements.

Th Schur polynomial for a shape $$\lambda$$ and variables $$x_1,x_2,\ldots,x_m$$ is denoted as $$s_\lambda(x_1,x_2,\ldots,x_m)$$.

As a quotient of antisymmetric polynomials
Define, for any partition $$\mu$$, the antisymmetric polynomial corresponding to $$\mu$$ as follows:

$$a_\mu = \sum_{g \in S_n} \varepsilon(g)x^{w(\mu)}$$

where $$\varepsilon(g)$$ denotes the sign of $$g$$ (it is +1 if $$g$$ is an even permutation, and -1 if $$g$$ is an odd permutation).

The the Schur polynomial for a partition $$\lambda$$ is defined as:

$$s_\lambda = \frac{a_{\lambda + \delta}}{a_\delta}$$

where $$\delta$$ is the column partition with all distinct entries from 1 to $$n$$.

Nature of the polynomials
The Schur polynomials are all symmetric polynomials in the $$x_i$$s. In other words, the number of semistandard tableaux with the exponents forming an ordered integer partition $$\mu$$ of $$n$$ (with $$m$$ parts) is independent of the ordering within $$\mu$$.

In fact, the Schur polynomials of degree $$n$$ form a basis for the space of all homogeneous symmetric polynomials of degree $$n$$ in $$m$$ variables. Thus, the set of all Schur polynomials in $$m$$ variables form a graded basis for the space of all symmetric functions in the $$m$$ variables.

Rule for multiplication
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.

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 $$r$$ as the sum of all monomials of degree $$r$$ in the $$m$$ variables. Then, the complete symmetric polynomial for a partition isthe product of the complete symmetric polynomials for each part.

This is denoted as $$h_\lambda(x_1,x_2,x_3,\ldots,x_m)$$.

The complete symmetric polynoimals are related to the Schur polynomials by the matrix of Kostka numbers, viz:

$$h_{\mu}(x_1,x_2,\ldots,x_m) = \sum_\lambda K_{\lambda\mu} s_\lambda(x_1,x_2,\ldots,x_m)$$

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 $$r$$ is the sum of all multilinear monomials of degree $$r$$.

The elementary symmetric polynomial for a partition $$\lambda$$ and variables $$x_1,x_2,x_3,\ldots,x_m$$ is defined as $$e_\lambda(x_1,x_2,\ldots,x_m)$$.

The elementary symmetric polynomials are related to the Schur polynomials by the matrix of Kostka numbers, with one of the partitions conjugated:

$$e_{\mu}(x_1,x_2,\ldots,x_m) = \sum_\lambda K_{\tilde{\lambda}\mu} s_\lambda(x_1,x_2,\ldots,x_m)$$

where $$\tilde{\lambda}$$ denotes the conjugate partition to $$\lambda$$.

Newton polynomials
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 $$r$$ is the sum of the $$r^{th}$$ powers of all the variables.

The Newton polynomials are related to the Schur polynomials by the character table, viz:

$$p_\mu(x_1,x_2,\ldots,x_m) = \sum_\lambda \chi_\mu^\lambda s_\lambda (x_1,x_2,\ldots,x_m)$$

Here $$\chi_\mu^\lambda$$ equals the character of the representation corresponding to $$\lambda$$ on the conjugacy class corresponding to $$\mu$$.