Group rings decomposition
$begingroup$
Let $G$ be a group with $|G| = 8$.
By the Artin-Wedderburn Theorem, $mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a decomposition?
Now, I know the dimensions on each side must be equal, so are the only possibilities $mathbb{C}^{(8)}$, $M_2({mathbb{C}}) oplus M_2({mathbb{C}}) $, $M_2({mathbb{C}}) oplus mathbb{C}^{(4)}?$
I'm just not sure if anything else apart from $mathbb{C}$ can appear as a division ring?
Say if we looked at $mathbb{R}G$, then we could have things like $mathbb{H} oplus mathbb{H}$, $M_{2}(mathbb{C})$ and so on. Where only $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ itself can appear as division rings.
abstract-algebra noncommutative-algebra group-rings
asked Jan 20 at 18:32
the manthe man
773715
$endgroup$
add a comment |
$begingroup$
Let $G$ be a group with $|G| = 8$.
By the Artin-Wedderburn Theorem, $mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a decomposition?
Now, I know the dimensions on each side must be equal, so are the only possibilities $mathbb{C}^{(8)}$, $M_2({mathbb{C}}) oplus M_2({mathbb{C}}) $, $M_2({mathbb{C}}) oplus mathbb{C}^{(4)}?$
I'm just not sure if anything else apart from $mathbb{C}$ can appear as a division ring?
Say if we looked at $mathbb{R}G$, then we could have things like $mathbb{H} oplus mathbb{H}$, $M_{2}(mathbb{C})$ and so on. Where only $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ itself can appear as division rings.
abstract-algebra noncommutative-algebra group-rings
asked Jan 20 at 18:32
the manthe man
773715
$endgroup$
$begingroup$
Not just division rings. Division algebras over your base field, so in the complex case there is only one possibility. Also, not all those decpmpositions are possible, just two of them.
$endgroup$
– Tobias Kildetoft
Jan 20 at 18:41
$begingroup$
Does a decomposition have to include $mathbb{C}$?
$endgroup$
– the man
Jan 20 at 18:43
$begingroup$
Yes, since those copies correspond to 1-dimensional representations.
$endgroup$
– Tobias Kildetoft
Jan 20 at 19:03
$begingroup$
Ok, thank you. Is this the case with any field $K$ such that the characteristic of $K$ doesn't divide the order of the group?
$endgroup$
– the man
Jan 20 at 19:05
add a comment |
$begingroup$
Let $G$ be a group with $|G| = 8$.
By the Artin-Wedderburn Theorem, $mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a decomposition?
Now, I know the dimensions on each side must be equal, so are the only possibilities $mathbb{C}^{(8)}$, $M_2({mathbb{C}}) oplus M_2({mathbb{C}}) $, $M_2({mathbb{C}}) oplus mathbb{C}^{(4)}?$
I'm just not sure if anything else apart from $mathbb{C}$ can appear as a division ring?
Say if we looked at $mathbb{R}G$, then we could have things like $mathbb{H} oplus mathbb{H}$, $M_{2}(mathbb{C})$ and so on. Where only $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ itself can appear as division rings.
abstract-algebra noncommutative-algebra group-rings
asked Jan 20 at 18:32
the manthe man
773715
$endgroup$
Let $G$ be a group with $|G| = 8$.
By the Artin-Wedderburn Theorem, $mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a decomposition?
Now, I know the dimensions on each side must be equal, so are the only possibilities $mathbb{C}^{(8)}$, $M_2({mathbb{C}}) oplus M_2({mathbb{C}}) $, $M_2({mathbb{C}}) oplus mathbb{C}^{(4)}?$
I'm just not sure if anything else apart from $mathbb{C}$ can appear as a division ring?
Say if we looked at $mathbb{R}G$, then we could have things like $mathbb{H} oplus mathbb{H}$, $M_{2}(mathbb{C})$ and so on. Where only $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ itself can appear as division rings.
abstract-algebra noncommutative-algebra group-rings
abstract-algebra noncommutative-algebra group-rings
asked Jan 20 at 18:32
the manthe man
773715
asked Jan 20 at 18:32
the manthe man
773715
asked Jan 20 at 18:32
the manthe man
773715
asked Jan 20 at 18:32
the manthe man
773715
asked Jan 20 at 18:32
the manthe man
773715
773715
$begingroup$
Not just division rings. Division algebras over your base field, so in the complex case there is only one possibility. Also, not all those decpmpositions are possible, just two of them.
$endgroup$
– Tobias Kildetoft
Jan 20 at 18:41
$begingroup$
Does a decomposition have to include $mathbb{C}$?
$endgroup$
– the man
Jan 20 at 18:43
$begingroup$
Yes, since those copies correspond to 1-dimensional representations.
$endgroup$
– Tobias Kildetoft
Jan 20 at 19:03
$begingroup$
Ok, thank you. Is this the case with any field $K$ such that the characteristic of $K$ doesn't divide the order of the group?
$endgroup$
– the man
Jan 20 at 19:05
add a comment |
$begingroup$
Not just division rings. Division algebras over your base field, so in the complex case there is only one possibility. Also, not all those decpmpositions are possible, just two of them.
$endgroup$
– Tobias Kildetoft
Jan 20 at 18:41
$begingroup$
Does a decomposition have to include $mathbb{C}$?
$endgroup$
– the man
Jan 20 at 18:43
$begingroup$
Yes, since those copies correspond to 1-dimensional representations.
$endgroup$
– Tobias Kildetoft
Jan 20 at 19:03
$begingroup$
Ok, thank you. Is this the case with any field $K$ such that the characteristic of $K$ doesn't divide the order of the group?
$endgroup$
– the man
Jan 20 at 19:05
$begingroup$
Not just division rings. Division algebras over your base field, so in the complex case there is only one possibility. Also, not all those decpmpositions are possible, just two of them.
$endgroup$
– Tobias Kildetoft
Jan 20 at 18:41
$begingroup$
Not just division rings. Division algebras over your base field, so in the complex case there is only one possibility. Also, not all those decpmpositions are possible, just two of them.
$endgroup$
– Tobias Kildetoft
Jan 20 at 18:41
$begingroup$
Does a decomposition have to include $mathbb{C}$?
$endgroup$
– the man
Jan 20 at 18:43
$begingroup$
Does a decomposition have to include $mathbb{C}$?
$endgroup$
– the man
Jan 20 at 18:43
$begingroup$
Yes, since those copies correspond to 1-dimensional representations.
$endgroup$
– Tobias Kildetoft
Jan 20 at 19:03
$begingroup$
Yes, since those copies correspond to 1-dimensional representations.
$endgroup$
– Tobias Kildetoft
Jan 20 at 19:03
$begingroup$
Ok, thank you. Is this the case with any field $K$ such that the characteristic of $K$ doesn't divide the order of the group?
$endgroup$
– the man
Jan 20 at 19:05
$begingroup$
Ok, thank you. Is this the case with any field $K$ such that the characteristic of $K$ doesn't divide the order of the group?
$endgroup$
– the man
Jan 20 at 19:05
add a comment |
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$begingroup$
Not just division rings. Division algebras over your base field, so in the complex case there is only one possibility. Also, not all those decpmpositions are possible, just two of them.
$endgroup$
– Tobias Kildetoft
Jan 20 at 18:41
$begingroup$
Does a decomposition have to include $mathbb{C}$?
$endgroup$
– the man
Jan 20 at 18:43
$begingroup$
Yes, since those copies correspond to 1-dimensional representations.
$endgroup$
– Tobias Kildetoft
Jan 20 at 19:03
$begingroup$
Ok, thank you. Is this the case with any field $K$ such that the characteristic of $K$ doesn't divide the order of the group?
$endgroup$
– the man
Jan 20 at 19:05