Symmetry groups of Rosenbloom-Tsfasman spaces
Let \(V:=\mathbb{F}_q^n\) be the \(n\)-dimensional vector space over the finite field with \(q\) elements. Every Poset (partially ordered set) \(P:=([n],\leq )\) induces a poset metric, \(d_P(.,.)\) on \(V\). The pair \((V,d_P)\) is a metric space or a poset space. A symmetry of \((V,d_P)\) is a bijection \(T:V\rightarrow V\) preserving the distances, i.e., \(d_P(u,v)=d_P(T(u),T(v))\). The symmetries of \((V,d_P)\) are specially interesting for coding theory. Two codes \(C,D\in V\) are equivalent if and only if there exists a symmetry \(T\) such that \(T(C)=D\). The group of linear symmetries of a poset space is given by \textit{L. Panek, M. Firer, H. Kim} and \textit{J. Hyum} [Discrete Math. 308, No. 18, 4116--4123 (2008; Zbl 1211.94054)]. An \(m\cdot n\)-Rosenbloom-Tsfasman space over \(\mathbb{F}_q\) is the pair \((\mathbb{F}_q^{m\cdot n},d_P)\) where \(d_P\) is the metric induced by a poset \(P=([m\cdot n],\leq)\) consisting of the union of \(m\) disjoin chains of order \(n\). The main theorem in this work states that the symmetry group of \(V\) is isomorphic to \(G_n^m \rtimes S_m\), where \(G_n\) is the symmetry group of a Rosenbloom-Tsfasman space over \(\mathbb{F}_q\) induced by a chain of length \(n\) and \(S_m\) is the symmetric group of permutations of a set with \(m\) elements. A generalization of this result is applied to a product of Rosenbloom-Tsfasman spaces. Let \(P_i\) be the poset consisting of the union of \(m_i\) chains of the same length \(n_i\), for \(i=1,\dots,k\) and let \(P=\bigcup_{i=1}^k P_i\). Hence, the symmetric group of the poset space \((\mathbb{F}_q^{|P|},d_P)\) is isomorphic to the group \(G_{m_1\cdot n_1}\times\cdots\times G_{m_k\cdot n_k}\) where \(G_{m_i\cdot n_i}\) is the symmetric group of the poset space \((\mathbb{F}_q^{m_i\cdot n_i},d_{P_i})\), \(\forall i\in\{1,\dots k\}\).
scientific article; zbMATH DE number 5542372