Determination of character table of alternating group:A5 - Revision history

Vipul: /* Filling in the four remaining entries (for the three-dimensional representations) */ fix some intermediate errors in algebra

2019-07-06T03:47:27Z

Filling in the four remaining entries (for the three-dimensional representations): fix some intermediate errors in algebra

← Older revision		Revision as of 03:47, 6 July 2019
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	This simplifies to:		This simplifies to:

	<math>j + k = -1</math>		<math>j + k = 1</math>

	and:		and:
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	Plugging into a single equation, we get:		Plugging into a single equation, we get:

	<math>j^2 + (-1 - j)^2 = 3</math>		<math>j^2 + (1 - j)^2 = 3</math>

	This becomes:		This becomes:

	<math>2j^2 + 2j - 2 = 0</math>		<math>2j^2 - 2j - 2 = 0</math>

	We solve to get <math>j = (1 \pm \sqrt{5})/2</math>. Note that since the same constraints apply to the other character as well, the two solutions represent the two possible characters. Plugging in, we get the full character table:		We solve to get <math>j = (1 \pm \sqrt{5})/2</math>. Note that since the same constraints apply to the other character as well, the two solutions represent the two possible characters. Plugging in, we get the full character table:

2017-03-20T21:19:35Z

← Older revision		Revision as of 21:19, 20 March 2017
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	{{quotation\|'''This page is incomplete'''. At present, it only includes the determination of the degrees of irreducible representations, not the entire character table.}}		{{quotation\|'''This page is incomplete'''. At present, it only includes the determination of the degrees of irreducible representations, not the entire character table.}}

	This page describes various ways to determine the character table of [[alternating group:A5]]. The idea is to determine as much as possible ~~wit has~~ little knowledge of the representation theory of these groups as we can manage, therefore making the discussion suitable for people who know only basic facts about the groups and basic facts of linear representation theory. For a detailed discussion of the character theory, see [[linear representation theory of alternating group:A5]].		This page describes various ways to determine the character table of [[alternating group:A5]]. The idea is to determine as much as possible with as little knowledge of the representation theory of these groups as we can manage, therefore making the discussion suitable for people who know only basic facts about the groups and basic facts of linear representation theory. For a detailed discussion of the character theory, see [[linear representation theory of alternating group:A5]].

2017-03-15T00:41:49Z

Filling in the four remaining entries (for the three-dimensional representations)

← Older revision		Revision as of 00:41, 15 March 2017
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	\| other irreducible constituent of restriction of exterior square of standard \|\| 3 \|\| -1 \|\| 0 \|\| <math>(-\sqrt{5} + 1)/2</math> \|\| <math>(\sqrt{5} + 1)/2</math>		\| other irreducible constituent of restriction of exterior square of standard \|\| 3 \|\| -1 \|\| 0 \|\| <math>(-\sqrt{5} + 1)/2</math> \|\| <math>(\sqrt{5} + 1)/2</math>
	\|}		\|}

	~~This simplif~~

2017-03-15T00:20:21Z

Filling in the column of order three

@@ Line 234: / Line 234: @@
 <math>a^2 + b^2 = 0</math>
-This forces <math>a = b = 0</math>>
+This forces <math>a = b = 0</math>.
 We thus get:
@@ Line 251: / Line 251: @@
 | second irreducible three-dimensional representation || 3 || -1 || 0 || ? || ?
 |}

2017-03-15T00:12:28Z

Filling in the column of order three

2017-03-15T00:12:16Z

Putting in the remaining representations

@@ Line 109: / Line 109: @@
 | second irreducible three-dimensional representation || 3 || ? || ? || ? || ?
 |}
 Let's fill in the second column (corresponding to the conjugacy class <math>(1,2)(3,4)</math>). The element has order 2, so its image under any representation has eigenvalues <math>\pm 1</math>. Therefore, the character value is a sum of values that are <math>\pm 1</math>, with the number of summands being the degree. Therefore:
@@ Line 155: / Line 157: @@
 | second irreducible three-dimensional representation || 3 || -1 || ? || ? || ?
 |}
 Next, let us try to fill in the irreducible five-dimensional representation. We know that it is the only five-dimensional representation, so it equals its complex conjugate. Therefore, it has a real-valued character. We also know that the two conjugacy classes <math>(1,2,3,4,5)</math> and <math>(1,2,3,5,4)</math> are in the same automorphism class. Since the representation is the only one of its degree, it is automorphism-invariant, hence the character must be equal on these two classes. We therefore get:
@@ Line 202: / Line 206: @@
 |-
 | second irreducible three-dimensional representation || 3 || -1 || ? || ? || ?
 |}

2017-03-15T00:07:43Z

Putting in the remaining representations

@@ Line 166: / Line 166: @@
 |-
 | irreducible five-dimensional representation || 5 || 1 || <math>p</math> || <math>q</math> || <math>q</math>
 |-
 | first irreducible three-dimensional representation || 3 || -1 || ? || ? || ?

2017-03-14T23:58:45Z

Putting in the remaining representations

@@ Line 150: / Line 150: @@
 |-
 | irreducible five-dimensional representation || 5 || 1 || ? || ? || ?
 |-
 | first irreducible three-dimensional representation || 3 || -1 || ? || ? || ?

2017-03-14T23:51:23Z

Putting in the remaining representations

@@ Line 108: / Line 108: @@
 |-
 | second irreducible three-dimensional representation || 3 || ? || ? || ? || ?
 |}

2017-03-14T23:42:59Z

Computing the irreducible five-dimensional representation

← Older revision		Revision as of 23:42, 14 March 2017
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	\|}		\|}

	=== ~~Computing~~ the ~~irreducible five-dimensional representation~~ ===		=== Putting in the remaining representations ===

	We need to fill in entries:		We need to fill in entries:
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	\|-		\|-
	\| irreducible five-dimensional representation \|\| 5 \|\| ? \|\| ? \|\| ? \|\| ?		\| irreducible five-dimensional representation \|\| 5 \|\| ? \|\| ? \|\| ? \|\| ?
			\|-
			\| first irreducible three-dimensional representation \|\| 3 \|\| ? \|\| ? \|\| ? \|\| ?
			\|-
			\| second irreducible three-dimensional representation \|\| 3 \|\| ? \|\| ? \|\| ? \|\| ?
	\|}		\|}