No onto map between group algebras $FS_5$ onto $M_6(F)$.
$begingroup$
I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $mathbb{Z}_2$ and $M_6(F)$ denotes the matrix algebra of $6times 6$ matrices over the field $F$.
I have no idea how to prove it exactly. I am thinking which matrix doesn’t comes in range if particular map is defined. The dimension of domain algebra also bigger one. I already have link of the problem Artin-Wedderburn decomposition of $mathbb{F}_2[S_5]/J$. But I do not know representation theory. Please give me a suggestion that does not use representation theory. Thanks.
ring-theory finite-groups algebras group-rings ring-homomorphism
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
$endgroup$
|
show 7 more comments
$begingroup$
I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $mathbb{Z}_2$ and $M_6(F)$ denotes the matrix algebra of $6times 6$ matrices over the field $F$.
I have no idea how to prove it exactly. I am thinking which matrix doesn’t comes in range if particular map is defined. The dimension of domain algebra also bigger one. I already have link of the problem Artin-Wedderburn decomposition of $mathbb{F}_2[S_5]/J$. But I do not know representation theory. Please give me a suggestion that does not use representation theory. Thanks.
ring-theory finite-groups algebras group-rings ring-homomorphism
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
$endgroup$
$begingroup$
Can you recall what $FS_5$ is?
$endgroup$
– mathcounterexamples.net
Jan 1 at 5:22
1
$begingroup$
I think the group algebra on the symmetric group $S_5$ over the field $F$.
$endgroup$
– mouthetics
Jan 1 at 5:48
$begingroup$
@mathcounterexamples.net Yes i think i already told about group algebra...
$endgroup$
– neelkanth
Jan 1 at 5:53
3
$begingroup$
If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations?
$endgroup$
– Lord Shark the Unknown
Jan 1 at 6:37
$begingroup$
@LordSharktheUnknown I am not having knowledge of Representation theory...
$endgroup$
– neelkanth
Jan 1 at 6:38
|
show 7 more comments
$begingroup$
I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $mathbb{Z}_2$ and $M_6(F)$ denotes the matrix algebra of $6times 6$ matrices over the field $F$.
I have no idea how to prove it exactly. I am thinking which matrix doesn’t comes in range if particular map is defined. The dimension of domain algebra also bigger one. I already have link of the problem Artin-Wedderburn decomposition of $mathbb{F}_2[S_5]/J$. But I do not know representation theory. Please give me a suggestion that does not use representation theory. Thanks.
ring-theory finite-groups algebras group-rings ring-homomorphism
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
$endgroup$
I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $mathbb{Z}_2$ and $M_6(F)$ denotes the matrix algebra of $6times 6$ matrices over the field $F$.
I have no idea how to prove it exactly. I am thinking which matrix doesn’t comes in range if particular map is defined. The dimension of domain algebra also bigger one. I already have link of the problem Artin-Wedderburn decomposition of $mathbb{F}_2[S_5]/J$. But I do not know representation theory. Please give me a suggestion that does not use representation theory. Thanks.
ring-theory finite-groups algebras group-rings ring-homomorphism
ring-theory finite-groups algebras group-rings ring-homomorphism
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
edited Jan 3 at 14:32
user593746
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
asked Jan 1 at 3:54
neelkanthneelkanth
2,0742928
2,0742928
$begingroup$
Can you recall what $FS_5$ is?
$endgroup$
– mathcounterexamples.net
Jan 1 at 5:22
1
$begingroup$
I think the group algebra on the symmetric group $S_5$ over the field $F$.
$endgroup$
– mouthetics
Jan 1 at 5:48
$begingroup$
@mathcounterexamples.net Yes i think i already told about group algebra...
$endgroup$
– neelkanth
Jan 1 at 5:53
3
$begingroup$
If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations?
$endgroup$
– Lord Shark the Unknown
Jan 1 at 6:37
$begingroup$
@LordSharktheUnknown I am not having knowledge of Representation theory...
$endgroup$
– neelkanth
Jan 1 at 6:38
|
show 7 more comments
$begingroup$
Can you recall what $FS_5$ is?
$endgroup$
– mathcounterexamples.net
Jan 1 at 5:22
1
$begingroup$
I think the group algebra on the symmetric group $S_5$ over the field $F$.
$endgroup$
– mouthetics
Jan 1 at 5:48
$begingroup$
@mathcounterexamples.net Yes i think i already told about group algebra...
$endgroup$
– neelkanth
Jan 1 at 5:53
3
$begingroup$
If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations?
$endgroup$
– Lord Shark the Unknown
Jan 1 at 6:37
$begingroup$
@LordSharktheUnknown I am not having knowledge of Representation theory...
$endgroup$
– neelkanth
Jan 1 at 6:38
$begingroup$
Can you recall what $FS_5$ is?
$endgroup$
– mathcounterexamples.net
Jan 1 at 5:22
$begingroup$
Can you recall what $FS_5$ is?
$endgroup$
– mathcounterexamples.net
Jan 1 at 5:22
1
1
$begingroup$
I think the group algebra on the symmetric group $S_5$ over the field $F$.
$endgroup$
– mouthetics
Jan 1 at 5:48
$begingroup$
I think the group algebra on the symmetric group $S_5$ over the field $F$.
$endgroup$
– mouthetics
Jan 1 at 5:48
$begingroup$
@mathcounterexamples.net Yes i think i already told about group algebra...
$endgroup$
– neelkanth
Jan 1 at 5:53
$begingroup$
@mathcounterexamples.net Yes i think i already told about group algebra...
$endgroup$
– neelkanth
Jan 1 at 5:53
3
3
$begingroup$
If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations?
$endgroup$
– Lord Shark the Unknown
Jan 1 at 6:37
$begingroup$
If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations?
$endgroup$
– Lord Shark the Unknown
Jan 1 at 6:37
$begingroup$
@LordSharktheUnknown I am not having knowledge of Representation theory...
$endgroup$
– neelkanth
Jan 1 at 6:38
$begingroup$
@LordSharktheUnknown I am not having knowledge of Representation theory...
$endgroup$
– neelkanth
Jan 1 at 6:38
|
show 7 more comments
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$begingroup$
Can you recall what $FS_5$ is?
$endgroup$
– mathcounterexamples.net
Jan 1 at 5:22
1
$begingroup$
I think the group algebra on the symmetric group $S_5$ over the field $F$.
$endgroup$
– mouthetics
Jan 1 at 5:48
$begingroup$
@mathcounterexamples.net Yes i think i already told about group algebra...
$endgroup$
– neelkanth
Jan 1 at 5:53
3
$begingroup$
If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations?
$endgroup$
– Lord Shark the Unknown
Jan 1 at 6:37
$begingroup$
@LordSharktheUnknown I am not having knowledge of Representation theory...
$endgroup$
– neelkanth
Jan 1 at 6:38