On transposed translation planes (Q2890873)

Let \(V\) be a vector space of dimension \(2(t+1)\) over \(GF(q)\), with \(q\) a prime power. A set \(\mathcal{S}=\{V_{i} \mid 0\leq i \leq q^{i+1}\}\) of \((t+1)\)-dimensional subspaces of \(V\) is a \textit{spread} in \(V\) if \(V_i \cap V_j=\{0\}\) for all \(0\leq i \neq j \leq q^{t+1}\). A \textit{translation plane} of order \(q^{t+1}\) is an incident structure with the vectors of \(V\) as points and the subspaces \(V_{i} \in \mathcal{S}\) together with their cosets in \((V,+)\) as lines.NEWLINENEWLINE A collection \(\mathcal{C}\) of \((t+1)\times (t+1)\) matrices over \(GF(q)\), of size \(q^{t+1}\), containing the zero and the identity matrix and such that \(\forall X\neq Y \in \mathcal{C}\) implies \(\det(X-Y)\neq 0\) is called \(t\)-\textit{spread} set over \(GF(q)\). To this set of matrices it is possible to associate the spread \(\mathcal{S}(\mathcal{C})=\{V(M) \mid M \in \mathcal{C}\} \cup \{V(\infty)\}\), where \(V(M)=\{(x,y)\mid x,y \in GF(q)^{t+1}, y=xM\}\) and \(V(\infty)=\{(0,y) \mid y \in GF(q)^{t+1}\}\).NEWLINENEWLINE The translation plane \(\pi(\mathcal{C})\) associated to this spread is called \textit{translation plane} associated with the \(t\)-spread set \(\mathcal{C}\). A \textit{transposed translation plane} is the translation plane \(\pi(\mathcal{C}^{t})\) associated with the \(t\)-spread \(\mathcal{C}^{t}=\{M^{t} \mid M \in \mathcal{C}\}\), where \(\mathcal{C}\) is a \(t\)-spread set over \(GF(q)\).NEWLINENEWLINE A translation plane \(\pi(\mathcal{C})\) is \textit{flag-transitive} if there exists a collineation group which permutes the subspaces of the spread \(\mathcal{S}(\mathcal{C})\).NEWLINENEWLINE In this paper the authors give an explicit matrix form of the inverse of a fixed isomorphism between two translation planes. The one-one correspondence between the set of all the isomorphisms from \(\pi(\mathcal{C}_{1})\) to \(\pi(\mathcal{C}_{2})\) and the set of all the isomorphisms from \(\pi(\mathcal{C}^{t}_{1})\) to \(\pi(\mathcal{C}^{t}_{2})\) is derived and explicitly given; the particular case \(\mathcal{C}_{1}=\mathcal{C}_{2}\) is studied.NEWLINENEWLINE Moreover the authors show that the transpose of a flag-transitive plane is a flag-transitive plane and they determine a necessary and sufficient condition for \(\pi(\mathcal{C}^{t})\) to be isomorphic with \(\pi(\mathcal{C})\) under a given set of particular conditions.