Representation fields for orders of small rank

A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several...

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Autor principal:	Arenas-Carmona,Luis
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2017
Materias:	central simple algebras maximal orders representations of orders spinor genera
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000100008
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spelling	oai:scielo:S0716-091720170001000082017-02-13Representation fields for orders of small rankArenas-Carmona,Luis central simple algebras maximal orders representations of orders spinor genera A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.36 n.1 20172017-03-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000100008en10.4067/S0716-09172017000100008
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	central simple algebras maximal orders representations of orders spinor genera
spellingShingle	central simple algebras maximal orders representations of orders spinor genera Arenas-Carmona,Luis Representation fields for orders of small rank
description	A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.
author	Arenas-Carmona,Luis
author_facet	Arenas-Carmona,Luis
author_sort	Arenas-Carmona,Luis
title	Representation fields for orders of small rank
title_short	Representation fields for orders of small rank
title_full	Representation fields for orders of small rank
title_fullStr	Representation fields for orders of small rank
title_full_unstemmed	Representation fields for orders of small rank
title_sort	representation fields for orders of small rank
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2017
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000100008
work_keys_str_mv	AT arenascarmonaluis representationfieldsforordersofsmallrank
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Representation fields for orders of small rank

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