Linear representation theory of dihedral group:D8 - Revision history

Vipul: /* Grouping by restriction to center */

2013-02-09T02:47:27Z

Grouping by restriction to center

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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Restriction to center as a representation of [[center of dihedral group:D8]] (this determines, essentially, the cohomology class of the projective representation for the inner automorphism group) !! List of irreducible projective representations of the inner automorphism group (which is [[Klein four-group]]) !! List of corresponding linear representations of [[dihedral group:D8]] !! List of degrees !! Sum of squares of degrees (should equal order of inner automorphism group, which is 4)		! Restriction to center as a representation of [[center of dihedral group:D8]] which is isomorphic to [[cyclic group:Z2]] (this determines, essentially, the cohomology class of the projective representation for the inner automorphism group) !! List of irreducible projective representations of the inner automorphism group (which is [[Klein four-group]]) !! List of corresponding linear representations of [[dihedral group:D8]] !! List of degrees !! Sum of squares of degrees (should equal order of inner automorphism group, which is 4)
	\|-		\|-
	\| trivial representation \|\| trivial representation \|\| trivial representation, and the three one-dimensional representations with kernels the subgroups of order four \|\| 1,1,1,1 \|\| 4		\| trivial representation \|\| trivial representation \|\| trivial representation, and the three one-dimensional representations with kernels the subgroups of order four \|\| 1,1,1,1 \|\| 4

Vipul: /* Action of automorphism group */

2013-02-09T02:46:56Z

Action of automorphism group

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	The automorphism group of the dihedral group preserves the trivial representation, the two-dimensional representation, and the sign representation whose kernel is the cyclic group <math>\langle a \rangle</math>. The two sign representations with kernels <math>\langle a^2, x \rangle</math> and <math>\langle a^2,ax\rangle</math> are exchanged by an outer automorphism.		The automorphism group of the dihedral group preserves the trivial representation, the two-dimensional representation, and the sign representation whose kernel is the cyclic group <math>\langle a \rangle</math>. The two sign representations with kernels <math>\langle a^2, x \rangle</math> and <math>\langle a^2,ax\rangle</math> are exchanged by an outer automorphism.

			==Isoclinism and projective representations==

			{{irrep isoclinism information facts to check against}}

			Please compare this with [[projective representation theory of Klein four-group]].

			===Grouping by restriction to center===

			{\| class="sortable" border="1"
			! Restriction to center as a representation of [[center of dihedral group:D8]] (this determines, essentially, the cohomology class of the projective representation for the inner automorphism group) !! List of irreducible projective representations of the inner automorphism group (which is [[Klein four-group]]) !! List of corresponding linear representations of [[dihedral group:D8]] !! List of degrees !! Sum of squares of degrees (should equal order of inner automorphism group, which is 4)
			\|-
			\| trivial representation \|\| trivial representation \|\| trivial representation, and the three one-dimensional representations with kernels the subgroups of order four \|\| 1,1,1,1 \|\| 4
			\|-
			\| sign representation \|\| nontrivial two-dimensional projective representation \|\| [[faithful irreducible representation of dihedral group:D8]] \|\| 2 \|\| 4
			\|}

			===Grouping by projective representation===

			{\| class="sortable" border="1"
			! Irreducible projective representation of inner automorphism group (which is [[Klein four-group]]) !! Degree !! Size of stabilizer under action of one-dimensional representations of [[dihedral group:D8]] !! Size of orbit (equals order of abelianization divided by size of stabilizer) !! List of irreducible representations of [[dihedral group:D8]]
			\|-
			\| trivial representation \|\| 1 \|\| 1 \|\| 4 \|\| trivial representation, and the three one-dimensional representations with kernels the subgroups of order four
			\|-
			\| nontrivial two-dimensional representation \|\| 2 \|\| 4 \|\| 1 \|\| [[faithful irreducible representation of dihedral group:D8]]
			\|}

	==Group ring interpretation==		==Group ring interpretation==

Vipul: /* Two-dimensional irreducible representation */

2013-01-20T16:37:45Z

Two-dimensional irreducible representation

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	===Two-dimensional irreducible representation===		===Two-dimensional irreducible representation===
	{{further\|[[faithful irreducible representation of dihedral group:D8]]}}		{{further\|[[faithful irreducible representation of dihedral group:D8]]}}
			{{higher-dimensional representation facts to check against}}
	{{#lst:faithful irreducible representation of dihedral group:D8\|representation table}}		{{#lst:faithful irreducible representation of dihedral group:D8\|representation table}}

Vipul: /* Sign representations with kernels as the maximal normal subgroups */

2013-01-20T16:37:08Z

Sign representations with kernels as the maximal normal subgroups

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	===Sign representations with kernels as the maximal normal subgroups===		===Sign representations with kernels as the maximal normal subgroups===

			{{one-dimensional representation facts to check against}}

	The dihedral group has three normal subgroups of index two: the subgroup <math>\langle a \rangle</math>, the subgroup <math>\langle a^2, x \rangle</math>, and the subgroup <math>\langle a^2, ax \rangle</math>. For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to <math>1</math> and elements outside that subgroup to <math>-1</math>.		The dihedral group has three normal subgroups of index two: the subgroup <math>\langle a \rangle</math>, the subgroup <math>\langle a^2, x \rangle</math>, and the subgroup <math>\langle a^2, ax \rangle</math>. For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to <math>1</math> and elements outside that subgroup to <math>-1</math>.

Vipul: /* Trivial representation */

2013-01-20T16:35:12Z

Trivial representation

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	===Trivial representation===		===Trivial representation===

			{{one-dimensional representation facts to check against}}

	The trivial or principal representation is a one-dimensional representation sending every element of the group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements:		The trivial or principal representation is a one-dimensional representation sending every element of the group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements:

Vipul at 01:25, 13 September 2012

2012-09-13T01:25:10Z

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	connective = of}}		connective = of}}

	We shall use the dihedral group with the following presentation:		We shall use the dihedral group with the following presentation (here, <math>e</math> is used to denote the identity element):

	<math>\langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle</math>.		<math>\langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle</math>.

Vipul: /* =Degrees of irreducible representations */

2012-06-07T05:30:53Z

=Degrees of irreducible representations

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	==GAP implementation==		==GAP implementation==

	====Degrees of irreducible representations===		===Degrees of irreducible representations===

	These can be computed using the [[GAP:CharacterDegrees\|CharacterDegrees]] function:		These can be computed using the [[GAP:CharacterDegrees\|CharacterDegrees]] function:

Vipul: /* Relation with representations of subgroups */

2012-06-07T05:30:26Z

Relation with representations of subgroups

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	[[Artin's induction theorem]] states that the characters induced from characters on cyclic subgroups span the space of class functions. Points (2) and (5) cover the case of the two-dimensional irreducible representation. {{fillin}}		[[Artin's induction theorem]] states that the characters induced from characters on cyclic subgroups span the space of class functions. Points (2) and (5) cover the case of the two-dimensional irreducible representation. {{fillin}}

			==GAP implementation==

			====Degrees of irreducible representations===

			These can be computed using the [[GAP:CharacterDegrees\|CharacterDegrees]] function:

			<pre>gap> CharacterDegrees(DihedralGroup(8));
			[ [ 1, 4 ], [ 2, 1 ] ]</pre>

			===Character table===

			The characters of the irreducible representations can be computed using [[GAP:Irr\|Irr]] and [[GAP:CharacterTable\|CharacterTable]]:

			<pre>gap> Irr(CharacterTable(DihedralGroup(8)));
			[ Character( CharacterTable( <pc group of size 8 with 3 generators> ),
			[ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size 8 with
			3 generators> ), [ 1, -1, 1, 1, -1 ] ),
			Character( CharacterTable( <pc group of size 8 with 3 generators> ),
			[ 1, 1, -1, 1, -1 ] ), Character( CharacterTable( <pc group of size
			8 with 3 generators> ), [ 1, -1, -1, 1, 1 ] ),
			Character( CharacterTable( <pc group of size 8 with 3 generators> ),
			[ 2, 0, 0, -2, 0 ] ) ]</pre>

			The character table can be displayed more nicely using the Display function:

			<pre>gap> Display(CharacterTable(DihedralGroup(8)));
			CT1

			2 3 2 2 3 2

			1a 2a 4a 2b 2c

			X.1 1 1 1 1 1
			X.2 1 -1 1 1 -1
			X.3 1 1 -1 1 -1
			X.4 1 -1 -1 1 1
			X.5 2 . . -2 .</pre>

			===Irreducible representations===

			The irreducible representations can be accessed using GAP's [[GAP:IrreducibleRepresentations\|IrreducibleRepresentations]] function:

			<pre>gap> IrreducibleRepresentations(DihedralGroup(8));
			[ Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
			Pcgs([ f1, f2, f3 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
			Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ] ],
			Pcgs([ f1, f2, f3 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ] ],
			Pcgs([ f1, f2, f3 ]) ->
			[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ],
			[ [ -1, 0 ], [ 0, -1 ] ] ] ]</pre>

Vipul: /* Representations */

2012-04-29T17:21:00Z

Representations

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	{{compare and contrast irreps\|order = 8}}		{{compare and contrast irreps\|order = 8}}

	==~~Representations~~==		==Irreducible representations==

	===Summary information===		===Summary information===

Vipul: /* Family contexts */

2012-04-18T02:42:20Z

Family contexts

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	\| [[dihedral group]] \|\| degree <math>n = 4</math>, order <math>2n = 8</math> \|\| [[linear representation theory of dihedral groups]]		\| [[dihedral group]] \|\| degree <math>n = 4</math>, order <math>2n = 8</math> \|\| [[linear representation theory of dihedral groups]]
	\|-		\|-
	\| [[unitriangular matrix group of degree three over a finite field]] \|\| Case [[field:F2]] \|\| [[linear representation theory of unitriangular matrix group of degree three over a finite field]]		\| [[unitriangular matrix group of degree three]] over a [[finite field]] \|\| Case [[field:F2]] \|\| [[linear representation theory of unitriangular matrix group of degree three over a finite field]]
	\|}		\|}