A note on symmetric functions in Formanek polynomials

Abstract:

Formanek’s proof of the existence of central identities has the following form: one constructs $n$ polynomials $G(X,{Y_1},{Y_2}, \cdots ,{Y_n}),G(X,{Y_2}, \cdots ,{Y_n},{Y_1})$, etc., whose sum is the desired central identity. The variables $X,{Y_i}$ are generic matrices. These $G$’s commute pairwise which raises the question whether all symmetric functions in them also give central identities. Here we show that this is not so for $n > 2$, and connect this question with Amitsur’s solution of the general crossed product problem.