Realizing tiled orders
type: Item
instance of: scholarly article
title: Realizing tiled orders
zbMATH Open document ID: 0814.16013
author: R. S. Pierce
author: Charles Vinsonhaler
published in: Journal of Algebra
publication date: 1994-11-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 563128
zbMATH Keywords: central division \(Q\)-algebra
zbMATH Keywords: tiled realizing orders
zbMATH Keywords: imbedding
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: direct sum
zbMATH Keywords: tiled orders
zbMATH Keywords: decomposition matrices
zbMATH Keywords: realizability
zbMATH Keywords: \(p\)-group
zbMATH Keywords: \(Z_ p\)-orders
zbMATH Keywords: incidence tiled orders
zbMATH Keywords: representations of finite partially ordered sets
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
full work available at URL: jabr.1994.1052
OpenAlex ID: W2000649071
DOI: 10.1006/JABR.1994.1052
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description: scientific article; zbMATH DE number 563128