Complete homogeneous symmetric polynomial

In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials.

The complete homogeneous symmetric polynomial of degree $k$ in $n$ variables $X 1, \dots, X n$ , written $h k$ for $k = 0, 1, 2, \dots$ , is the sum of all monomials of total degree $k$ in the variables. Formally,

The formula can also be written as:

Indeed, $l p$ is just the multiplicity of $p$ in the sequence $i k$ .

The first few of these polynomials are

Thus, for each nonnegative integer $k$ , there exists exactly one complete homogeneous symmetric polynomial of degree $k$ in $n$ variables.

Another way of rewriting the definition is to take summation over all sequences $i k$ , without condition of ordering $i p \leq i p + 1$ :

here $m p$ is the multiplicity of number $p$ in the sequence $i k$ .

For example

The polynomial ring formed by taking all integral linear combinations of products of the complete homogeneous symmetric polynomials is a commutative ring.

The following lists the $n$ basic (as explained below) complete homogeneous symmetric polynomials for the first three positive values of $n$ .

For $n = 1$ :

For $n = 2$ :

For $n = 3$ :

The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in $t$ :

(this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials). Here each fraction in the final expression is the usual way to represent the formal geometric series that is a factor in the middle expression. The identity can be justified by considering how the product of those geometric series is formed: each factor in the product is obtained by multiplying together one term chosen from each geometric series, and every monomial in the variables $X i$ is obtained for exactly one such choice of terms, and comes multiplied by a power of $t$ equal to the degree of the monomial.

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