Rational-representation group - Revision history

Vipul at 23:56, 5 July 2019

2019-07-05T23:56:31Z

← Older revision		Revision as of 23:56, 5 July 2019
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	==Definition==		==Definition==

	A '''rational-representation group''' is a [[finite group]] satisfying the following properties:		A '''rational-representation group''' is a [[finite group]] satisfying the following equivalent properties:

	# The [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.		# The [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.

Vipul: /* Facts */

2011-05-19T23:37:17Z

Facts

← Older revision		Revision as of 23:37, 19 May 2011
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	==Facts==		==Facts==

	* [[Symmetric group]]s on finite sets are rational-representation groups. {{proofat\|[[Symmetric groups are rational-representation ~~groups~~]]}}		* [[Symmetric group]]s on finite sets are rational-representation groups. {{proofat\|[[Symmetric groups are rational-representation]]}}
	* [[Dihedral group:D8]] is a rational-representation group.		* [[Dihedral group:D8]] is a rational-representation group.
	* [[Trivial group]], [[cyclic group:Z2]], and [[dihedral group:D8]] are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. {{proofat\|[[Classification of groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers]]}}		* [[Trivial group]], [[cyclic group:Z2]], and [[dihedral group:D8]] are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. {{proofat\|[[Classification of finite groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers]]}}

	==Relation with other properties==		==Relation with other properties==

Vipul: moved Rational representation group to Rational-representation group over redirect

2010-04-06T01:11:39Z

moved Rational representation group to Rational-representation group over redirect

← Older revision	Revision as of 01:11, 6 April 2010
(No difference)

Vipul: /* Definition */

2010-04-06T00:46:59Z

Definition

← Older revision		Revision as of 00:46, 6 April 2010
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	# The [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.		# The [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.
	# Every irreducible representation in characteristic zero can be realized over <math>\mathbb{Z}</math>.		# Every irreducible representation in characteristic zero can be realized over <math>\mathbb{Z}</math>.
	~~# All its characters are rational-valued.~~
	~~# All its characters are integer-valued.~~

	===Equivalence of definitions===		===Equivalence of definitions===

Vipul: moved Rational-representation group to Rational representation group

2010-04-06T00:46:28Z

moved Rational-representation group to Rational representation group

← Older revision	Revision as of 00:46, 6 April 2010
(No difference)

Vipul at 22:09, 5 April 2010

2010-04-05T22:09:25Z

@@ Line 15: / Line 15: @@
 Definitions (1) and (2) are equivalent because [[linear representation is realizable over principal ideal domain iff it is realizable over field of fractions]].
-==Relation with other properties==
+==Examples==
-===Weaker properties===
+Here are some small examples, with relevant contextual information.
-* [[Stronger than::Rational group]]: A rational group is a finite group such that all its characters are rational-valued (hence integer-valued). The [[quaternion group]] is an example of a rational group that is not a rational-representation group.
+{| class="wikitable" border="1"
-* [[Stronger than::Ambivalent group]]: A finite group in which every character is real-valued.
+! Group !! Order !! List of degrees of irreducible representations !! List of entries of matrices arising across all irreducible representations when chosen with simplest entries
 ==Facts==
@@ Line 27: / Line 50: @@
 * [[Dihedral group:D8]] is a rational-representation group.
 * [[Trivial group]], [[cyclic group:Z2]], and [[dihedral group:D8]] are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. {{proofat|[[Classification of groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers]]}}

Vipul at 21:43, 5 April 2010

2010-04-05T21:43:00Z

← Older revision		Revision as of 21:43, 5 April 2010
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	# All its characters are rational-valued.		# All its characters are rational-valued.
	# All its characters are integer-valued.		# All its characters are integer-valued.

			===Equivalence of definitions===

			Definitions (1) and (2) are equivalent because [[linear representation is realizable over principal ideal domain iff it is realizable over field of fractions]].

	==Relation with other properties==		==Relation with other properties==

Vipul at 21:24, 5 April 2010

2010-04-05T21:24:27Z

← Older revision		Revision as of 21:24, 5 April 2010
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	==Definition==		==Definition==

	A '''rational-representation group''' is a [[finite group]] ~~for which~~ the [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.		A '''rational-representation group''' is a [[finite group]] satisfying the following properties:

			# The [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.
			# Every irreducible representation in characteristic zero can be realized over <math>\mathbb{Z}</math>.
			# All its characters are rational-valued.
			# All its characters are integer-valued.

	==Relation with other properties==		==Relation with other properties==

Vipul: Created page with '{{wikilocal}} {{group property}} ==Definition== A '''rational-representation group''' is a finite group for which the field of rational numbers $\mathbb{Q}$ ...'

2009-06-26T16:20:52Z

Created page with '{{wikilocal}} {{group property}} ==Definition== A '''rational-representation group''' is a finite group for which the field of rational numbers <math>\mathbb{Q}</math> ...'

New page

{{wikilocal}}
{{group property}}

==Definition==

A '''rational-representation group''' is a [[finite group]] for which the [[field of rational numbers]] <math>\mathbb{Q}</math> is a [[splitting field]], i.e., every [[irreducible representation]] in characteristic zero is realizable over the rational numbers.

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Rational group]]: A rational group is a finite group such that all its characters are rational-valued (hence integer-valued). The [[quaternion group]] is an example of a rational group that is not a rational-representation group.
* [[Stronger than::Ambivalent group]]: A finite group in which every character is real-valued.

==Facts==

* [[Symmetric group]]s on finite sets are rational-representation groups. {{proofat|[[Symmetric groups are rational-representation groups]]}}
* [[Dihedral group:D8]] is a rational-representation group.
* [[Trivial group]], [[cyclic group:Z2]], and [[dihedral group:D8]] are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. {{proofat|[[Classification of groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers]]}}