Automorphism of \(\Sigma_{n+1}\)-invariant trilinear forms (Q2467081)

\textit{Y. Egawa} and \textit{H. Suzuki} [Hokkaido Math. J. 11, 39--47 (1985; Zbl 0571.20003)] showed that \(\Sigma_{n+1}\)- invariant trilinear form in \(n\) variables is similar to a certain trilinear form \(\Theta_n\) and \(\Aut(\Theta_n)\cong \mu_3\times\Sigma_{n+1}.\) In the paper under review the reader finds a very short proof of that result for \(n\geq 4.\) The key idea is to use the results of \textit{A. Chlebowicz} and the second author [Tatra Mt. Math. Publ. 32, 33--39 (2005; Zbl 1150.11417)] and show that the form \(f_{\Theta_n}\) of degree 3 associated with \(\Theta_n\) has a unique representation as a sum of \(n+1\) cubes of linear forms.