The expressions for the elementary symmetric functions have coefficients with the same absolute value, but a sign equal to the sign of ?, namely ( " 1 ) " m " 2 + " m " 4 + . . ..
2.
The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of & Lambda; " R " already shows the existence of an automorphism & omega; sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3.
3.
(where " e " " n " is the " n " th elementary symmetric function ) uniquely extends to a ring homomorphism and the images of the basis elements u _ \ lambda may be interpreted via the Schur functions, which are thus closely connected with the theory of Hall polynomials.
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