Complete Homogeneous Symmetric Polynomial - Properties - Relation With The Elementary Symmetric Polynomials

Relation With The Elementary Symmetric Polynomials

There is a fundamental relation between the elementary symmetric polynomials and the complete homogeneous ones:

which is valid for all m > 0, and any number of variables n. The easiest way to see that it holds is from an identity of formal power series in t for the elementary symmetric polynomials, analogous to the one given above for the complete homogeneous ones:

(this is actually an identity of polynomials in t, because after en(X1,…Xn) the elementary symmetric polynomials become zero). Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1, and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of tm. A somewhat more direct way to understand that relation, is to consider the contributions in the summation involving a fixed monomial Xα of degree m. For any subset S of the variables appearing with nonzero exponent in the monomial, there is a contribution involving the product XS of those variables as term from es(X1,…,Xn), where s = #S, and the monomial Xα / XS from hm−s(X1,…,Xn); this contribution has coefficient (−1)s. The relation then follows from the fact that

by the binomial formula, where l ≤ m denotes the number of distinct variables occurring (with nonzero exponent) in Xα. Since e0(X1, …, Xn) and h0(X1, …, Xn) are both equal to 1, one can isolate from the relation either the first or the last terms of the summation. The former gives a sequence of equations

\begin{align} h_1(X_1,\ldots,X_n)&=e_1(X_1,\ldots,X_n),\\ h_2(X_1,\ldots,X_n)&=h_1(X_1,\ldots,X_n)e_1(X_1,\ldots,X_n)-e_2(X_1,\ldots,X_n),\\ h_3(X_1,\ldots,X_n)&=h_2(X_1,\ldots,X_n)e_1(X_1,\ldots,X_n)-h_1(X_1,\ldots,X_n)e_2(X_1,\ldots,X_n)+e_3(X_1,\ldots,X_n),\\ \end{align}

and so on, that allows to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials; the latter gives a set of equations

\begin{align} e_1(X_1,\ldots,X_n)&=h_1(X_1,\ldots,X_n),\\ e_2(X_1,\ldots,X_n)&=h_1(X_1,\ldots,X_n)e_1(X_1,\ldots,X_n)-h_2(X_1,\ldots,X_n),\\ e_3(X_1,\ldots,X_n)&=h_1(X_1,\ldots,X_n)e_2(X_1,\ldots,X_n)-h_2(X_1,\ldots,X_n)e_1(X_1,\ldots,X_n)+h_3(X_1,\ldots,X_n),\\ \end{align}

and so forth, that allows doing the inverse. The first n elementary and complete homogeneous symmetric polynomials play perfectly similar roles in these relations, even though the former polynomials then become zero, whereas the latter do not. This phenomenon can be understood in the setting of the ring of symmetric functions. It has a ring automorphism that interchanges the sequences of the n elementary and first n complete homogeneous symmetric functions.

The set of complete homogeneous symmetric polynomials of degree 1 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 . This can be formulated by saying that form an algebraic basis of the ring of symmetric polynomials in X1, … Xn with integral coefficients (as is also true for the elementary symmetric polynomials). The same is true with the ring Z of integers replaced by any other commutative ring. These statements follow from analogous statements for the elementary symmetric polynomials, due to the indicated possibility of expressing either kind of symmetric polynomials in terms of the other kind.

Read more about this topic:  Complete Homogeneous Symmetric Polynomial, Properties

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