Generalized permutation matrix: Difference between revisions

Revision as of 09:57, 7 December 2010

Revision as of 02:23, 16 June 2011

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===Group structure===

The set of ''n''×''n'' generalized permutation matrices with entries in a [[field (mathematics)|field]] ''F'' forms a [[subgroup]] of the [[general linear group]] GL(''n'',''F''), in which the group of nonsingular diagonal matrices ΔΔ(''n'', ''F'') forms a [[normal subgroup]]. Indeed, the generalized permutation matrices are the [[normalizer]] of the diagonal matrices, meaning that the generalized permutation matrices are the ''largest'' subgroup of GL in which diagonal matrices are normal.

The abstract group of generalized permutation matrices is the [[wreath product]] of ''F''^× and ''S''_''n''. Concretely, this means that it is the [[semidirect product]] of ΔΔ(''n'', ''F'') by the [[symmetric group]] ''S''_''n'':

:Δ(''n'', ''F'') {{unicode|⋉}} ''S''_''n'',

where ''S''_''n'' acts by permuting coordinates and the diagonal matrices ΔΔ(''n'', ''F'') are isomorphic to the ''n''-fold product (''F''^×)^''n''.

To be precise, the generalized permutation matrices are a (faithful) [[linear representation]] of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.

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==Signed permutation group==

{{see|Hyperoctahedral group}}

A '''signed permutation matrix''' is a generalized permutation matrix whose nonzero entries are ±±1, and are the integer generalized permutation matrices with integer inverse.

===Properties===

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===Monomial representations===

{{main|Monomial representation}}

Monomial matrices occur in [[representation theory]] in the context of [[monomial representation]]s. A monomial representation of a group ''G'' is a linear representation ''ρ'' : ''G'' →→ GL(''n'', ''F'') of ''G'' (here ''F'' is the defining field of the representation) such that the image ''ρ''(''G'') is a subgroup of the group of monomial matrices.

[[Category:Matrices]]