By Sheppard L.

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1) n p{t) := Y[(l + txi) := 1 + ^ e , r ' . , Jc„) is the sum of (") terms: the products, over all subsets of {1, 2 , . . , Az} with / elements of the variables with indices in that subset. 2) Ci = ^ l

36 2 Symmetric Functions Proof. (1) and (4) are clearly equivalent, taking as the matrix conjugate to A the one of the same linear transformation in the basis A' u, / = 0 , . . , n — 1. (2) and (5) are easily seen to be equivalent and also (5) and (1). We do not prove (3) since we have not yet developed enough geometry of orbits. 7 showing that the dimension of an orbit equals the dimension of the group minus the dimension of the stabilizer and then one has to compute the centralizer of a regular matrix and prove that it has dimension n.

One often develops formal identities in this ring with the idea that, in order to verify an identity which is homogeneous of some degree m, it is enough to do it for symmetric functions in m variables. In the same way the reader may understand the following fact. Consider the n\ monomials 4' '"4-1^ 0