Irreducible Characters & Representations of a Cube
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Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three representations whose characters we denote by $chi^{faces}$, $chi^{edges}$, $chi^{vertices}$.
(1) How would you express each of these characters as linear combinations of the irreducible characters (and representations) of $A_4$? (Give a visual explaination as well)
(2) $S_4$ acts by conjugation on the normal subgroup $A_4$. How does this action operate on the isomorphism classes of irreducible representations of $A_4$?
group-theory geometry representation-theory characters
asked Nov 24 at 21:52
JB071098
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The current answers do not contain enough detail.
A visual and written explanation with LOTS of details that do not use modules or tensor products. Only elementary group theory (i.e., Sylow Theorems, semidirect products, etc.)
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Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three representations whose characters we denote by $chi^{faces}$, $chi^{edges}$, $chi^{vertices}$.
(1) How would you express each of these characters as linear combinations of the irreducible characters (and representations) of $A_4$? (Give a visual explaination as well)
(2) $S_4$ acts by conjugation on the normal subgroup $A_4$. How does this action operate on the isomorphism classes of irreducible representations of $A_4$?
group-theory geometry representation-theory characters
asked Nov 24 at 21:52
JB071098
351112
This question has an open bounty worth +50
reputation from JB071098 ending in 5 days.
The current answers do not contain enough detail.
A visual and written explanation with LOTS of details that do not use modules or tensor products. Only elementary group theory (i.e., Sylow Theorems, semidirect products, etc.)
1
Compute the number of fixed elements, and use the character table of $A_4$.
– user10354138
Nov 26 at 16:24
add a comment |
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1
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up vote
1
down vote
favorite
Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three representations whose characters we denote by $chi^{faces}$, $chi^{edges}$, $chi^{vertices}$.
(1) How would you express each of these characters as linear combinations of the irreducible characters (and representations) of $A_4$? (Give a visual explaination as well)
(2) $S_4$ acts by conjugation on the normal subgroup $A_4$. How does this action operate on the isomorphism classes of irreducible representations of $A_4$?
group-theory geometry representation-theory characters
asked Nov 24 at 21:52
JB071098
351112
Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three representations whose characters we denote by $chi^{faces}$, $chi^{edges}$, $chi^{vertices}$.
(1) How would you express each of these characters as linear combinations of the irreducible characters (and representations) of $A_4$? (Give a visual explaination as well)
(2) $S_4$ acts by conjugation on the normal subgroup $A_4$. How does this action operate on the isomorphism classes of irreducible representations of $A_4$?
group-theory geometry representation-theory characters
group-theory geometry representation-theory characters
asked Nov 24 at 21:52
JB071098
351112
asked Nov 24 at 21:52
JB071098
351112
edited yesterday
asked Nov 24 at 21:52
JB071098
351112
asked Nov 24 at 21:52
JB071098
351112
asked Nov 24 at 21:52
JB071098
351112
351112
This question has an open bounty worth +50
reputation from JB071098 ending in 5 days.
The current answers do not contain enough detail.
A visual and written explanation with LOTS of details that do not use modules or tensor products. Only elementary group theory (i.e., Sylow Theorems, semidirect products, etc.)
This question has an open bounty worth +50
reputation from JB071098 ending in 5 days.
The current answers do not contain enough detail.
A visual and written explanation with LOTS of details that do not use modules or tensor products. Only elementary group theory (i.e., Sylow Theorems, semidirect products, etc.)
1
Compute the number of fixed elements, and use the character table of $A_4$.
– user10354138
Nov 26 at 16:24
add a comment |
1
Compute the number of fixed elements, and use the character table of $A_4$.
– user10354138
Nov 26 at 16:24
1
1
Compute the number of fixed elements, and use the character table of $A_4$.
– user10354138
Nov 26 at 16:24
Compute the number of fixed elements, and use the character table of $A_4$.
– user10354138
Nov 26 at 16:24
add a comment |
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Compute the number of fixed elements, and use the character table of $A_4$.
– user10354138
Nov 26 at 16:24