Complete homogeneous symmetric polynomial

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In mathematics, specifically in commutative algebra, the complete homogeneous symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial can be expressed as a sum and difference of products of complete homogeneous symmetric polynomials.

1 Definition
2 Examples
3 Properties
4 References
5 See also

[edit] Definition

The complete homogeneous symmetric polynomial of degree k in $n$ variables x₁, ..., x_n, written h_k for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables; that is, the sum of the exponents of all the variables is k. Formally,

$h_k (x_1, x_2, \dots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq n} x_{i_1} x_{i_2} \cdots x_{i_k}.$

The first few of these polynomials are

$h_0 (x_1, x_2, \dots,x_n) = 1,$

$h_1 (x_1, x_2, \dots,x_n) = \sum_{1 \leq j \leq n} x_j,$

$h_2 (x_1, x_2, \dots,x_n) = \sum_{1 \leq j \leq k \leq n} x_j x_k,$

$h_3 (x_1, x_2, \dots,x_n) = \sum_{1 \leq j \leq k \leq l \leq n} x_j x_k x_l.$

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

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

[edit] Examples

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

For n = 1:

$h_1 = x_1.\,$

For n = 2:

$h_1 = x_1 + x_2\,$ and

$h_2 = x_1^2 + x_1x_2 + x_2^2.\,$

For n = 3:

$h_1 = x_1 + x_2 + x_3,\,$

$h_2 = x_1^2 + x_2^2 + x_3^2 + x_1x_2 + x_1x_3 + x_2x_3,\,$ and

$h_3 = x_1^3+x_2^3+x_3^3 + x_1^2x_2+x_1^2x_3+x_2^2x_1+x_2^2x_3+x_3^2x_1+x_3^2x_2 + x_1x_2x_3.\,$

[edit] Properties

The set of complete homogeneous symmetric polynomials of degree1 to n in n variables generates the ring of symmetric polynomials in n variables. More specifically, the ring of symmetric polynomials with integer coefficients equals the integral polynomial ring $\mathbb Z[h_1,\ldots,h_n].$ We say that $h_1,\ldots,h_n$ form an algebraic basis of the ring of symmetric functions with integral coefficients. A similar statement holds if the ring of integers is replaced by any other commutative ring.

For other systems of symmetric polynomials with similar properties see elementary symmetric polynomials and power sum symmetric polynomials.

[edit] References

Macdonald, I.G. (1979), Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford: Clarendon Press.
Macdonald, I.G. (1995), Symmetric Functions and Hall Polynomials, second ed. Oxford: Clarendon Press. ISBN 0-19-850450-0 (paperback, 1998).
Richard P. Stanley (1999), Enumerative Combinatorics, Vol. 2. Camridge: Cambridge University Press. ISBN 0-521-56069-1