Irreducible carpets of Lie type \(B_l\), \(C_l\) and \(F_4\) over fields (Q6587368)

Let \( \Phi \) be a reduced indecomposable root system of rank \(l\), \( \Phi ( F ) \) an elementary Chevalley group of type \( \Phi \) over a field \(F\). Then \( \Phi ( F ) \) is generated by its root subgroups \[ x_{r} ( F ) = \{ x_{r} ( t ) \mid t\in F \}, r\in\Phi \] each of which is abelian and \[ x_{r} ( t ) x_{r} ( u ) = x_{r} ( t + u ) \] for every \( r\in \Phi \) and all \( t, u\in F \). For the purpose of the present paper, one can consider either the universal or the adjoint Chevalley group of the type indicated. \N\NA carpet of type \( \Phi \) of rank \(l\) over \(F\) is the collection \( \mathfrak{A} = \{ \mathfrak{A}_{r}\mid r\in \Phi \}\) of additive subgroups \(\mathfrak{A}_{r}\) of the field \(F\) satisfying the condition \[ C_{ij, rs}\mathfrak{A}_{r}^{i}\mathfrak{A}_{s}^{j}\subseteq\mathfrak{A}_{ir + js} \] where \[ r, s, ir + js\in\Phi,\; i> 0, j > 0,\mathfrak{A}_{r}^{i} = \{ a^{i} \mid a\in\mathfrak{A}_{r} \} \] and the constants \[ C_{ij, rs} = \pm 1, \pm 2, \pm 3 \] are determined by the Chevalley commutator formula \[ [ x_{s} ( u ), x_{r} ( t ) ] = \prod_{i, j > 0}x_{ir + js} ( C_{ij, rs} ( -t )^{i} u^{j} ),\; r, s, ir + js\in\Phi .\] \N\NEvery carpet \(\mathfrak{A}\) of type \( \Phi \) over \(F\) determines the carpet subgroup \[ \Phi ( \mathfrak{A} ) = \langle x_{r} ( \mathfrak{A}_{r} ) \mid r\in\Phi\rangle \] of the group \( \Phi ( F ) \), where \(\langle M \rangle \) is the subgroup generated by the set \(M\). A carpet \(\mathfrak{A}\) is called closed if its carpet subgroup \( \Phi ( \mathfrak{A} ) \) does not have any new root elements, that is, if \( \Phi ( F )\cap x_{r} ( F ) = x_{r} ( \mathfrak{A}_{r} ) \) for all \(r\in\Phi \). The carpet \(\mathfrak{A}\) is called irreducible if all of its additive subgroups are nonzero.\N\NThe main result of the paper is the following.\N\NTheorem. Let \( \mathfrak{A} = \{ \mathfrak{A}_{r}\mid r\in \Phi \}\) be an irreducible carpet of type \( B_{l}\) (\(l\ge 2 \)), or of type \( C_{l}\) (\(l\ge 2\)), or of type \( F_{4} \) over a field \(F\). Suppose that at least one of the additive subgroups \( \mathfrak{A}_{r} \) is an \(R\)-module, where \(F\) is an algebraic field extension of a field \(R\). Then up to conjugation by a diagonal element either \( \mathfrak{A}_{r} = P \) for all \(r\in\Phi \) and for a subfield \(P\) of the field \(F\) or the characteristic of \(F\) is \(2\), there exists an imperfect subfield \(K\) of the field \(F\) and\N\[\N\mathfrak{A}_{r}= \begin{cases} P & \mbox{if \(r\) is a short root}, \\\NQ & \mbox{if \(r\) is a long root} \end{cases}\N\]\Nfor two distinct infinite additive subgroups \(P\) and \(Q\) of the field \(K\) satisfying the relations \( K^{2}\le Q<P\le K \) and the equalities \( P = P^{-1}, Q = Q^{-1} \). In addition, the carpet \( \mathfrak{A} = \{ \mathfrak{A}_{r}\mid r\in \Phi \}\) is closed.