140
where a = (a, a, .., o) is a permutation of the integers (1,2, **,n) is said to be symmet-
ric with respect to the given permutation. If this property applies for all n! permutations
of the xk's, then V is a symmetric polynomial [MoSt64]. Some examples of symmetric
polynomials are
V(x,, x2, 3) = X + x2 +X3
V(x1, x2, x3) = XIX2 + XIx3 + x2X3
and (xi,X2, .,x + kXk ...+
It is straightforward to show that any sum, difference or product of symmetric polyno-
mials is a symmetric polynomial. In fact, the symmetric polynomials form a ring.
If the symmetric polynomial ((x) = V(x1,x2, 2, Xn) includes the monomial
Pl P2 Pn
ax x2 ...xn
among its terms, then it includes also the monomial
PI P2 Pn
ax x x .x..
where a = (c,, c2, -, cn) is an arbitrary permutation of the integers (1,2, -, n). If for a
given p = (p,, p2, ",Pn) the sum of all distinct monomials of this form is designated by
(X)= yx xI P.x P Eq. C.6
then p(x) is symmetric, and further, the arbitrary symmetric polynomial V(x) can be
written as a linear combination of a finite number of such polynomials. That is
V(x) = ap+x).
P
A transposition of the ordered list of n variables xk, I 5 k < n is a permutation which
exchanges the positions of any two of the xk's. Every permutation of the ordered list can
be written as a composition of transpositions applied to (1,2,- --, n), and for any specified
permutation, if any such composition includes an odd number of transpositions, then
every such composition includes an odd number of transpositions. Similarly, if any such