10.43 (1986)

Let $R$ and $S$ be associative rings with identity such that 2 is invertible in $S$. Let $\Lambda_I : GL_n(R) \to GL_n(R/I)$ be the homomorphism corresponding to an ideal $I$ of $R$ and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by elementary transvections $t_{ij}(x)$. Let $a_{ij} = t_{ij}(1)t_{ji}(-1)t_{ij}(1)$, let the bar denote images in the factor-group of $GL_n(R)$ by the centre and let $PG = \overline{G}$ for $G \leqslant GL_n(R)$. A homomorphism
$$\Lambda : E_n(R) \to GL(W) = GL_m(S)$$ is called standard if $S^m = P \oplus \dots \oplus P \oplus Q$ (a direct sum of $S$-modules in which there are $n$ summands $P$) and
$$\Lambda x = g^{-1}\tau(\delta^*(x)f + ({}^t\delta^*(x)^\nu)^{-1}(1 - f))g, \quad x \in E_n(R),$$ where $\delta^* : GL_n(R) \to GL_n(\operatorname{End} P)$ is the homomorphism induced by a ring homomorphism $\delta : R \to \operatorname{End} P$ taking identity to identity, $g$ is an isomorphism of the module $W$ onto $S^m$, $\tau : GL_n(\operatorname{End} P) \to GL(gW)$ is an embedding, $f$ is a central idempotent of $\delta R$, $t$ denotes transposition and $\nu$ is an antiisomorphism of $\delta R$. Let $n \geqslant 3$, $m \geqslant 2$. One can show that the homomorphism
$$\Lambda_0 : PE_n(R) \to GL(W) = GL_m(S)$$ is induced by a standard homomorphism $\Lambda$ if $\Lambda_0 \bar{a}_{ij} = g^{-1}\tau(a_{ij}^*)g$ for some $g$ and $\tau$ and for any $i \neq j$ where $a_{ij}^*$ denotes the matrix obtained from $a_{ij}$ by replacing 0 and 1 from $R$ by 0 and 1 from $\operatorname{End} P$. Find (at least in particular cases) the form of a homomorphism $\Lambda_0$ that does not satisfy the last condition.