Linear representation theory of cyclic group:Z3 - Revision history

Vipul at 23:09, 9 May 2011

2011-05-09T23:09:02Z

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 ==Representations==
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 ===Trivial representation===

Vipul at 23:08, 9 May 2011

2011-05-09T23:08:33Z

← Older revision		Revision as of 23:08, 9 May 2011
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	! Item !! Value		! Item !! Value
	\|-		\|-
	\| [[Degrees of irreducible representations]] over a splitting field \|\| 1, 1, 1<br>[[maximum degree of irreducible representation\|maximum]]: 1, [[lcm of degrees of irreducible representations\|lcm]]: 1, [[number of irreducible representations equals number of conjugacy classes\|number]]: 3, [[sum of squares of degrees of irreducible representations\|sum of squares]]: 3		\| [[Degrees of irreducible representations]] over a splitting field \|\| 1, 1, 1<br>[[maximum degree of irreducible representation\|maximum]]: 1, [[lcm of degrees of irreducible representations\|lcm]]: 1, [[number of irreducible representations equals number of conjugacy classes\|number]]: 3, [[sum of squares of degrees of irreducible representations equals group order\|sum of squares]]: 3
	\|-		\|-
	\| Smallest ring of realization of all representations (characteristic zero) \|\| <math>\mathbb{Z}[e^{2\pi i/3}]</math>, also expressible as <math>\mathbb{Z}[x]/(x^2 + x + 1)</math>. Degree two integral extension of <math>\mathbb{Z}</math>.		\| Smallest ring of realization of all representations (characteristic zero) \|\| <math>\mathbb{Z}[e^{2\pi i/3}]</math>, also expressible as <math>\mathbb{Z}[x]/(x^2 + x + 1)</math>. Degree two integral extension of <math>\mathbb{Z}</math>.
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	\|}		\|}

			==Family contexts==

			{\| class="sortable" border="1"
			! Family !! Parameter value !! General discussion of linear representation theory of family
			\|-
			\| [[finite cyclic group]] \|\| 3 \|\| [[linear representation theory of finite cyclic groups]]
			\|-
			\| [[alternating group]] \|\| 3 \|\| [[linear representation theory of alternating groups]]
			\|}
	===Trivial representation===		===Trivial representation===

Vipul: /* Summary */

2011-05-09T23:06:55Z

Summary

← Older revision		Revision as of 23:06, 9 May 2011
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	! Item !! Value		! Item !! Value
	\|-		\|-
	\| [[Degrees of irreducible representations]] over a splitting field \|\| 1, 1, 1		\| [[Degrees of irreducible representations]] over a splitting field \|\| 1, 1, 1<br>[[maximum degree of irreducible representation\|maximum]]: 1, [[lcm of degrees of irreducible representations\|lcm]]: 1, [[number of irreducible representations equals number of conjugacy classes\|number]]: 3, [[sum of squares of degrees of irreducible representations\|sum of squares]]: 3
	\|-		\|-
	\| [[~~Maximum degree of irreducible representation~~]~~] over a splitting field \|\|~~ 1		\| Smallest ring of realization of all representations (characteristic zero) \|\| <math>\mathbb{Z}[e^{2\pi i/3}]</math>, also expressible as <math>\mathbb{Z}[x]/(x^2 + x + 1)</math>. Degree two integral extension of <math>\mathbb{Z}</math>.
	\|-		\|-
	\| ~~[[lcm of degrees of irreducible representations]] over a splitting~~ field ~~\|\| 1~~		\| Smallest field of realization of all representations (characteristic zero) \|\| <math>\mathbb{Q}(e^{2\pi i/3})</math>, also expressible as <math>\mathbb{Q}[x]/(x^2 + x + 1)</math>. Degree two cyclotomic extension of <math>\mathbb{Q}</math>.
	\|-
	~~\| Smallest ring~~ of realization of all representations (characteristic zero) \|\| <math>\mathbb{Z}[e^{2\pi i/3}]</math>
	\|-
	~~\| Smallest field of realization of all representations (characteristic zero) \|\|~~ <math>\mathbb{Q}(e^{2\~~pi i/3~~})</math>
	\|-		\|-
	\| Criterion for a field to be a splitting field \|\| Any field of characteristic not 3 that contains a primitive cube root of unity, i.e., the polynomial <math>x^2 + x + 1</math> splits.		\| Criterion for a field to be a splitting field \|\| Any field of characteristic not 3 that contains a primitive cube root of unity, i.e., the polynomial <math>x^2 + x + 1</math> splits.
	\|-		\|-
	\| [[Degrees of irreducible representations]] over a non-splitting field \|\| 1, 2		\| [[Degrees of irreducible representations]] over a non-splitting field. Includes the case of [[field of real numbers]] <math>\R</math>, also [[field of rational numbers]] <math>\mathbb{Q}</math> \|\| 1, 2<br>[[maximum degree of irreducible representation\|maximum]]: 2, [[lcm of degrees of irreducible representations\|lcm]]: 2, [[number of irreducible representations over reals equals number of equivalence classes under real conjugacy\|number]]: 2
	\|-
	\| [[~~Maximum~~ of degrees of irreducible representations]] over ~~a non-splitting field \|~~\| 2
	\|-		\|-
	\| [[~~lcm of degrees of irreducible representations~~]] ~~over a non-splitting~~ field ~~\|\| 2~~		\| Smallest size splitting field \|\| [[field:F4]], i.e., field with four elements.
	\|}		\|}

Vipul: /* Character table over a non-splitting field */

2011-05-09T16:43:47Z

Character table over a non-splitting field

← Older revision		Revision as of 16:43, 9 May 2011
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	Over a finite field, the character values are interpreted as integers modulo the field characteristic; over an infinite field, they are interpreted as rational numbers and hence field elements.		Over a finite field, the character values are interpreted as integers modulo the field characteristic; over an infinite field, they are interpreted as rational numbers and hence field elements.

	If doing character theory over the real numbers, we know that the [[number of irreducible representations over reals equals number of real ~~conjugacy classes]] and for the rational numbers, we know that the [[number of irreducible representations over rationals equals number of rational~~ conjugacy classes]]. The above is the character table both over the rationals and over the reals.		If doing character theory over the real numbers, we know that the [[number of irreducible representations over reals equals number of real conjugacy classes]]. The above is the character table both over the rationals and over the reals.

Vipul: /* Character table */

2011-04-13T03:16:25Z

Character table

← Older revision		Revision as of 03:16, 13 April 2011
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	Let <math>\omega</math> be a primitive cube root of unity. The character table over a splitting field is as follows:		Let <math>\omega</math> be a primitive cube root of unity. The character table over a splitting field is as follows:

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	!Representation/Conjugacy class !! <math>e</math> (identity element) !! <math>x</math> (generator) !! <math>x^2</math> (generator)		!Representation/Conjugacy class !! <math>e</math> (identity element) !! <math>x</math> (generator) !! <math>x^2</math> (generator)
	\|-		\|-

Vipul: /* Two-dimensional representation that makes sense over all fields and is irreducible over fields not containing a primitive cube root of unity */

2011-04-13T03:15:09Z

Two-dimensional representation that makes sense over all fields and is irreducible over fields not containing a primitive cube root of unity

← Older revision		Revision as of 03:15, 13 April 2011
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	The two one-dimensional nontrivial representations both send the identity element to <math>(1)</math> and the two non-identity elements to the two primitive cube roots of unity. The representations differ in terms of which element of the group is matched with which primitive cube root of unity.		The two one-dimensional nontrivial representations both send the identity element to <math>(1)</math> and the two non-identity elements to the two primitive cube roots of unity. The representations differ in terms of which element of the group is matched with which primitive cube root of unity.

	===Two-dimensional representation ~~that makes sense over all fields and is~~ irreducible ~~over fields not containing a primitive cube root of unity~~===		===Two-dimensional representation: irreducible in the non-splitting case===

	If we denote the group as <math>\{ e,x,x^2\}</math> where <math>e</matH> is the identity element, we can construct the following representation with integer matrices:		If we denote the group as <math>\{ e,x,x^2\}</math> where <math>e</matH> is the identity element, we can construct the following representation with integer matrices:

Vipul: /* Two-dimensional irreducible representation */

2011-04-13T03:13:58Z

Two-dimensional irreducible representation

← Older revision		Revision as of 03:13, 13 April 2011
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	The two one-dimensional nontrivial representations both send the identity element to <math>(1)</math> and the two non-identity elements to the two primitive cube roots of unity. The representations differ in terms of which element of the group is matched with which primitive cube root of unity.		The two one-dimensional nontrivial representations both send the identity element to <math>(1)</math> and the two non-identity elements to the two primitive cube roots of unity. The representations differ in terms of which element of the group is matched with which primitive cube root of unity.

	===Two-dimensional irreducible ~~representation~~===		===Two-dimensional representation that makes sense over all fields and is irreducible over fields not containing a primitive cube root of unity===

	If we denote the group as <math>\{ e,x,x^2\}</math> where <math>e</matH> is the identity element, we can construct the following representation with integer matrices:		If we denote the group as <math>\{ e,x,x^2\}</math> where <math>e</matH> is the identity element, we can construct the following representation with integer matrices:

Vipul: /* Representations */

2011-04-13T03:10:42Z

Representations

← Older revision		Revision as of 03:10, 13 April 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Name of representation type !! Number of representations of this type !! Values not allowed for field characteristic !! Criterion for field !! What happens over a splitting field? !! Kernel !! [[Degree of a linear representation\|Degree]] !! [[Schur index]]		! Name of representation type !! Number of representations of this type !! Values not allowed for field characteristic !! Criterion for field !! What happens over a splitting field? !! Kernel !! [[Degree of a linear representation\|Degree]] !! [[Schur index]] !! What happens by reducing the <math>\mathbb{Z}</math>-representation over bad characteristics?
	\|-		\|-
	\| trivial \|\| 1 \|\| -- \|\| any \|\| remains the same \|\| whole group \|\| 1 \|\| 1		\| trivial \|\| 1 \|\| -- \|\| any \|\| remains the same \|\| whole group \|\| 1 \|\| 1 \|\| --
	\|-		\|-
	\| one-dimensional nontrivial \|\| 2 \|\| 3 \|\| contains a primitive cube root of unity \|\| remains the same \|\| trivial subgroup, i.e., it is faithful \|\| 1 \|\| 1		\| one-dimensional nontrivial \|\| 2 \|\| 3 \|\| contains a primitive cube root of unity \|\| remains the same \|\| trivial subgroup, i.e., it is faithful \|\| 1 \|\| 1 \|\| irrelevant question
	\|-		\|-
	\| two-dimensional irreducible \|\| 1 \|\| 3 \|\| does not contain a primitive cube root of unity \|\| splits into the two one-dimensional nontrivial representations \|\| trivial subgroup, i.e., it is faithful \|\| 2 \|\| 1		\| two-dimensional irreducible \|\| 1 \|\| -- \|\| does not contain a primitive cube root of unity \|\| splits into the two one-dimensional nontrivial representations \|\| trivial subgroup, i.e., it is faithful \|\| 2 \|\| 1 \|\| Taking the <math>\mathbb{Z}</math>-representation and mapping to [[field:F3]] gives an indecomposable two-dimensional representation that is not irreducible.
	\|}		\|}

Vipul: /* Two-dimensional irreducible representation */

2011-04-13T03:08:35Z

Two-dimensional irreducible representation

← Older revision		Revision as of 03:08, 13 April 2011
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	===Two-dimensional irreducible representation===		===Two-dimensional irreducible representation===

	There is a two-dimensional representation that makes sense over any field of characteristic not equal to three, but it is ''irreducible'' only in the case that the field does not have a primitive cube root of unity. Otherwise, it can be decomposed as the direct sum of the two one-dimensional nontrivial representations mentioned above.		If we denote the group as <math>\{ e,x,x^2\}</math> where <math>e</matH> is the identity element, we can construct the following representation with integer matrices:

	If we denote the group as <math>\{ e,x,x^2\}</math> where <math>e</matH> is the identity element, the representation ~~can be described as follows~~ with integer matrices ~~(hence it makes sense over any field)~~:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
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	\|}		\|}

	In the case ~~that~~ the field is the [[field of real numbers]], or more generally, any subfield of the reals containing <math>\mathbb{Q}[\sqrt{3}]</math>, we can provide an alternative description of this representation as rotations by multiples of <math>2\pi/3</math>. ''Note that this alternative description, though it gives an equivalent representation, does not work'' over fields that lack a square root of <math>3</math>.		We can interpret this representation over any field <math>F</math>, and three cases arise as to the nature of <math>F</math>:

			{\| class="sortable" border="1"
			! Case for the field <math>F</math> !! What happens in this case
			\|-
			\| <math>F</math> does not have characteristic <math>3</math> does not contain a primitive cube root of unity \|\| The representation is irreducible. However, it is not absolutely irreducible, it decomposes as a direct sum in a quadratic extension containing a primitive cube root of unity (See next point).
			\|-
			\| <math>F</math> does not have characteristic <math>3</math> and contains a primitive cube root of unity \|\| The representation decomposes as a direct sum of the two one-dimensional nontrivial representations with kernel [[V4 in A4]]
			\|-
			\| Characteristic of <math>F</math> is <math>3</math> \|\| The representation is indecomposable but not irreducible, because there is an invariant one-dimensional subspace.
			\|}

			Our case of current interest is the first one, i.e., where <math>F</math> does not have characteristic <math>3</math> and does not contain a primitive cube root of unity.

			In the case <math>F</math> is the field is the [[field of real numbers]], or more generally, any subfield of the reals containing <math>\mathbb{Q}[\sqrt{3}]</math>, we can provide an alternative description of this representation as rotations by multiples of <math>2\pi/3</math>. ''Note that this alternative description, though it gives an equivalent representation, does not work'' over fields that lack a square root of <math>3</math>.

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Character table over a non-splitting field */

2011-04-13T02:29:40Z

Character table over a non-splitting field

← Older revision		Revision as of 02:29, 13 April 2011
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	\|-		\|-
	\| trivial representation \|\| 1 \|\| 1		\| trivial representation \|\| 1 \|\| 1
	\|		\|-
	\| nontrivial two-dimensional representation \|\| 2 \|\| -1		\| nontrivial two-dimensional representation \|\| 2 \|\| -1
	\|}		\|}