Realizing tiled orders
A \(Z_ p\)-order \(R\) in a central division \(Q\)-algebra \(D\) of degree \(n\) is called realizing if \(R \cong \text{End}(G)\) for a full \(Z_ p\)- submodule \(G\) of \(D\). In the present paper the authors' earlier approach [Houston J. Math. 17, 207-226 (1991; Zbl 0760.16006)] is applied to the study of tiled realizing orders whose imbedding into the corresponding algebra \(\widehat{D}\) over the \(p\)-adic field \(\widehat{Q}_ p\) is a direct sum of \(n\) non-zero left ideals. Due to the one-to-one correspondence between tiled orders \(\widehat{R}\) in \(\widehat{D}\) and a certain set of matrices \(\lambda \in M_ n(Z)\) called decomposition matrices established in the paper, the realizability of \(R\) turns out to be equivalent to the existence of a kind of representation of \(\lambda\) by subgroups of a \(p\)-group. Basing on this result the authors adduce examples of \(Z_ p\)-orders in \(D\) which are not realizable and, in case \(n \geq 3\), which are realizing but not maximal. The incidence tiled orders whose decomposition matrices \(\lambda = (\lambda(i,j))\) satisfy \(\lambda(i,j) = 0\) or \(1\) for all \(i\), \(j\) and \(\lambda(i,j) + \lambda(j,i) = 1\) for \(i \neq j\) are classified up to an isomorphism by certain classes of partial orderings of the set \(I = \{1,2,\dots,n\}\). The last section of the paper, dedicated to the study of representations of finite partially ordered sets, gives a criterion of the realizability of an incidence tiled order in related terms.
representations of finite partially ordered sets
scientific article; zbMATH DE number 563128