Semigroups of linear transformations whose restrictions belong to a general linear group (Q6636355)

Let \(L(V)\) denote the semigroup of all linear transformations on a vector space \(V\). For a fixed subspace \(U\) of \(V\), let \(\mathrm{GL}(U)\) be the general linear group and let \(L_{\mathrm{GL}(U)}(V)\) denote the semigroup of all linear transformations on \(V\) whose restrictions belong to the general linear group \(\mathrm{GL}(U)\), i.e. \N\[\NL_{\mathrm{GL}(U)}(V)=\{ \alpha \in L(V): \alpha_{\mid_{U}}\in \mathrm{GL}(U)\}.\N\]\NThe author describes Green's relations and ideals of \(L_{\mathrm{GL}(U)}(V)\). In particular, it is shown in Theorem 3.6. that the proper ideals of \(L_{\mathrm{GL}(U)}(V)\) are precisely the sets \N\[\NQ(k)=\{\alpha \in L_{\mathrm{GL}(U)}(V) : \dim(V\alpha/U) < k\}\N\]\Nwhere \(1\leq k \leq \dim(V\alpha/U)\) and so \(Q(1)\) is the minimal ideal of \(L_{\mathrm{GL}(U)}(V)\). It is shown in Theorem 4.2. that the set of all minimal idempotents in \(L_{\mathrm{GL}(U)}(V)\) is \(E(Q(1))=\{ \alpha \in E(L_{\mathrm{GL}(U)}(V)) : V\alpha =U\}\). For two finite dimensional vector spaces over a finite field \(V_{1}\) and \(V_{2}\), if \(U_{1}\) and \(U_{2}\) are subspaces of \(V_{1}\) and \(V_{2}\), respectively, then it is shown in Theorem 4.4 that \(L_{\mathrm{GL}(U_{1})}(V_{1})\) and \(L_{\mathrm{GL}(U_{2})}(V_{2})\) are isomorphic if and only if there exists an isomorphism \(\psi : V_{1}\rightarrow V_{2}\) such that \(U_{1}\psi =U_{2}\), as expected.\N\NFor each \(0\leq k \leq \dim(V\alpha/U)\), let \N\[\NJ(k)=\{\alpha \in L_{\mathrm{GL}(U)}(V) : \dim(V\alpha/U) = k\}.\N\]\N If \(\dim(V)=n\) and \(\dim(U)=r\), then for any \(\alpha\in J(n-r-1)\), it is shown in Lemma 5.4. and Corollary 5.5 that \(J(n-r)\cup \{\alpha \}\) is a minimal generating set of \(L_{\mathrm{GL}(U)}(V)\). Moreover, it is shown that \(J(n-r)\) is a subgroup of \(L_{\mathrm{GL}(U)}(V)\) and that \(J(n-r)\) is a semidirect product of \(\mathrm{Fix}(U)\trianglelefteq J(n-r)\) and \(\mathrm{Fix}(W)\leq J(n-r)\) where \(W\) is any complement of \(U\) in \(V\).