# Schur polynomial

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

## Definition

### 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 $n$. This needs to be clarified/explained properly

## Particular cases

$\! n$ $\! m$ $\! \lambda$ $s_\lambda(x_1,\dots,x_m)$
$\! n$ $\! 1$ $\! n$ $\! x_1^n$
$\! n$ $\! 2$ $\! n$ $\! x_1^n + x_1^{n-1}x_2 + \dots x_1x_2^{n-1} + x_2^n$
$\! n$ $\! n$ $1 + 1 + \dots + 1$ $\! x_1x_2 \dots x_n$
$\! 1$ $\! 1$ $\! 1$ $\! x_1$
$\! 1$ $\! m$ $\! 1$ $\! x_1 + \dots + x_m$
$\! 2$ $\! 2$ $\! 1 + 1$ $\! x_1x_2$
$\! 2$ $\! m$ $\! 1 + 1$ $\! \sum_{i < j} x_ix_j$
$\! 2$ $\! 1$ $\! 2$ $\! x_1^2$
$\! 2$ $\! 2$ $\! 2$ $\! x_1^2 + x_2^2 + x_1x_2$
$\! 2$ $\! m$ $\! 2$ $\! \sum_i x_i^2 + \sum_{i < j} x_ix_j$
$\! 3$ $\! 1$ $\! 3$ $\! x_1^3$
$\! 3$ $\! 2$ $\! 3$ $\! x_1^3 + x_1^2x_2 + x_1x_2^2 + x_2^3$
$\! 3$ $\! 2$ $\! 2 + 1$ $\! x_1^2x_2 + x_1x_2^2$
$\! 4$ $\! 1$ $\! 4$ $\! x_1^4$
$\! 4$ $\! 2$ $\! 4$ $\! x_1^4 + x_1^3x_2 + x_1^2x_2^2 + x_1x_2^3 + x_2^4$
$\! 4$ $\! 2$ $\! 3 + 1$ $\! x_1^3x_2 + x_1^2x_2^2 + x_1x_2^3$
$\! 4$ $\! 2$ $\! 2 + 2$ $\! x_1^2x_2^2$
$\! 4$ $\! 3$ $\! 2 + 1 + 1$ $\! x_1^2x_2x_3 + x_1x_2^2x_3 + x_1x_2x_3^2$

## Facts

### 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 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

The matrix for changing basis from complete symmetric polynomials to Schur polynomials is the matrix of: Kostka numbers

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

The matrix for changing basis from elementary symmetric polynomials to Schur polynomials is the matrix of: Kostka numbers for the conjugate partition

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

The matrix for changing basis from Newton polynomials to Schur polynomials is the matrix of: character table of the symmetric group

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 $\mu$.