Symmetric Polynomial - Special Kinds of Symmetric Polynomials - Complete Homogeneous Symmetric Polynomials

Complete Homogeneous Symmetric Polynomials

For each nonnegative integer k, the complete homogeneous symmetric polynomial hk(X1, …, Xn) is the sum of all distinct monomials of degree k in the variables X1, …, Xn. For instance

The polynomial hk(X1, …, Xn) is also the sum of all distinct monomial symmetric polynomials of degree k in X1, …, Xn, for instance for the given example

\begin{align} h_3(X_1,X_2,X_3)&=m_{(3)}(X_1,X_2,X_3)+m_{(2,1)}(X_1,X_2,X_3)+m_{(1,1,1)}(X_1,X_2,X_3)\\ &=(X_1^3+X_2^3+X_3^3)+(X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_3^2+X_2^2X_3+X_2X_3^2)+(X_1X_2X_3).\\ \end{align}

All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X1, …, Xn can be obtained from the complete homogeneous symmetric polynomials h1(X1, …, Xn), …, hn(X1, …, Xn) via multiplications and additions. More precisely:

Any symmetric polynomial P in X1, …, Xn can be written as a polynomial expression in the polynomials hk(X1, …, Xn) with 1 ≤ k ≤ n.
If P has integral coefficients, then the polynomial expression also has integral coefficients.

For example, for, the relevant complete homogeneous symmetric polynomials are h1(X1,X2) = X1+X2), and h2(X1,X2) = X12+X1X2+X22. The first polynomial in the list of examples above can then be written as

Like in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond hn(X1, …, Xn), allowing them to be expressed in the ones up to that point; again the resulting identities become invalid when the number of variables is increased.

An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be given as the identities

, for all k > 0, and any number of variables n.

Since e0(X1, …, Xn) and h0(X1, …, Xn) are both equal to 1, one can isolate either the first or the last terms of these summations; the former gives a set of equations that allows to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the hk(X1, …, Xn) with 1 ≤ k ≤ n: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.

Read more about this topic:  Symmetric Polynomial, Special Kinds of Symmetric Polynomials

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