Thus, for each positive integer k less than or equal to n there exists exactly one elementary symmetric polynomial of degree k in n variables.
Given an integer partition (that is, a finite non-increasing sequence of positive integers) λ = (λ1, ..., λm), one defines the symmetric polynomial eλ(X1, ..., Xn), also called an elementary symmetric polynomial, by Sometimes the notation σk is used instead of ek.
Similarly, the determinant is – up to the sign – the constant term of the characteristic polynomial, i.e. the value of en.
Here the doubly indexed σj,n − 1 denote the elementary symmetric polynomials in n − 1 variables.
In other words, the coefficient of R before each monomial which contains only the variables X1, ..., Xn − 1 equals the corresponding coefficient of P. As we know, this shows that the lacunary part of R coincides with that of the original polynomial P. Therefore the difference P − R has no lacunary part, and is therefore divisible by the product X1···Xn of all variables, which equals the elementary symmetric polynomial σn,n.
The fact that the polynomial representation is unique implies that A[X1, ..., Xn]Sn is isomorphic to A[Y1, ..., Yn].
Furthermore parametrize all products of elementary symmetric polynomials that have degree d (they are in fact homogeneous) as follows by partitions of d. Order the individual elementary symmetric polynomials ei(X1, ..., Xn) in the product so that those with larger indices i come first, then build for each such factor a column of i boxes, and arrange those columns from left to right to form a Young diagram containing d boxes in all.
The essential ingredient of the proof is the following simple property, which uses multi-index notation for monomials in the variables Xi.
Since P is symmetric, its leading monomial has weakly decreasing exponents, so it is some X λ with λ a partition of d. Let the coefficient of this term be c, then P − ceλt (X1, ..., Xn) is either zero or a symmetric polynomial with a strictly smaller leading monomial.
The lemma shows that all these products have different leading monomials, and this suffices: if a nontrivial linear combination of the eλt (X1, ..., Xn) were zero, one focuses on the contribution in the linear combination with nonzero coefficient and with (as polynomial in the variables Xi) the largest leading monomial; the leading term of this contribution cannot be cancelled by any other contribution of the linear combination, which gives a contradiction.