A method for computing symmetric and related polynomials (Q1841853)

There are many algorithms which express any symmetric polynomial in \(n\) variables \(x_1,\ldots,x_n\) over an integral domain \(A\) as a polynomial of the elementary symmetric functions \(s_1,\ldots,s_n\). The main purpose of the paper under review is to describe another method to do this. Unlike most algorithms of this type, the proposed one is not a recursive procedure. As a by-product of his approach the author obtains an algorithmic proof of the theorem of E. Artin saying that every polynomial in \(A[x_1,\ldots,x_n]\) is an \(A[s_1,\ldots,s_n]\)-linear combination of the \(n!\) elements \(x_1^{k_1}\ldots x_n^{k_n}\), \(0\leq k_i\leq i-1\), and such an expression is unique up to a rearangement of terms. This result is used to construct generators of the algebra of invariants and to express the invariants in terms of these generators for the Weyl groups \(W(B_n)\), \(W(D_n)\), \(W(G_2)\) and \(W(F_4)\).