\(A\)-identities for the \(2\times 2\) matrix algebra. (Q284102)

\textit{A. Henke} and \textit{A. Regev} [Isr. J. Math. 133, 339-355 (2003; Zbl 1026.16006)] suggested to study \(A\)-identities, i.e., multilinear polynomial identities of the form \[ f(x_1,\ldots,x_n)=\sum_{\sigma\in A_n}\alpha_\sigma x_{\sigma(1)}\cdots x_{\sigma(n)}, \] where the summation runs on the alternating group \(A_n\) instead of on the whole symmetric group \(S_n\). They conjectured that the minimal degree of the \(A\)-identities of the \(n\times n\) matrix algebra \(M_n(K)\) over a field \(K\) of characteristic 0 is equal to \(2n+2\). This was disproved for \(n=6\) [in \textit{D. J. Gonçalves} and \textit{P. Koshlukov}, Isr. J. Math. 186, 407-426 (2011; Zbl 1264.16023)]. In the paper under review the authors show that \(M_2(K)\) does not satisfy \(A\)-identities of degree 5 and construct an explicit \(A\)-identity of degree 6.