Irreducible Characters & Representations of a Cube
$begingroup$
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 '18 at 21:52
JB071098JB071098
363212
$endgroup$
add a comment |
$begingroup$
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 '18 at 21:52
JB071098JB071098
363212
$endgroup$
1
$begingroup$
Compute the number of fixed elements, and use the character table of $A_4$.
$endgroup$
– user10354138
Nov 26 '18 at 16:24
add a comment |
$begingroup$
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 '18 at 21:52
JB071098JB071098
363212
$endgroup$
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 '18 at 21:52
JB071098JB071098
363212
asked Nov 24 '18 at 21:52
JB071098JB071098
363212
edited Nov 30 '18 at 14:01
JB071098
asked Nov 24 '18 at 21:52
JB071098JB071098
363212
asked Nov 24 '18 at 21:52
JB071098JB071098
363212
asked Nov 24 '18 at 21:52
JB071098JB071098
363212
363212
1
$begingroup$
Compute the number of fixed elements, and use the character table of $A_4$.
$endgroup$
– user10354138
Nov 26 '18 at 16:24
add a comment |
1
$begingroup$
Compute the number of fixed elements, and use the character table of $A_4$.
$endgroup$
– user10354138
Nov 26 '18 at 16:24
1
1
$begingroup$
Compute the number of fixed elements, and use the character table of $A_4$.
$endgroup$
– user10354138
Nov 26 '18 at 16:24
$begingroup$
Compute the number of fixed elements, and use the character table of $A_4$.
$endgroup$
– user10354138
Nov 26 '18 at 16:24
add a comment |
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$begingroup$
Compute the number of fixed elements, and use the character table of $A_4$.
$endgroup$
– user10354138
Nov 26 '18 at 16:24