New polynomial identities for \(2\times 2\) generic matrices in characteristic \(2\)
For a commutative ring \(K\) let \(R_m(K)\) denote the \(K\)-algebra generated by \(m\) generic \(2\times 2\) matrices. The kernel of the canonical homomorphism \(\vartheta_m\colon R_m(\mathbb{Z})\to R_m(\mathbb{Z}/2\mathbb{Z})\) is studied, where \(\mathbb{Z}\) denotes the ring of integers. Answering Procesi's question \textit{W. F. Schelter} [J. Algebra 96, 48-53 (1985; Zbl 0577.16009)] has shown the existence of an element in the kernel of \(\vartheta_5\) which is not contained in \(2R_5(\mathbb{Z})\). This \(5\)-variable element is multihomogeneous of degree \(6\). In the present paper it is proved that there exists a multilinear element of degree \(5\) in \(\text{Ker}(\vartheta_5)\setminus 2R_5(\mathbb{Z})\). This result is obtained by comparing the dimensions of the multilinear polynomial identities of degree \(5\) for the \(2\times 2\) matrix algebras over the rationals and \(\mathbb{Z}/2\mathbb{Z}\).
scientific article; zbMATH DE number 1308270