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Which partitions realise group algebras of finite groups?
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Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $mathbbC$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the semisimple algebra $A_p:=M_a_1(K) times cdots times M_a_m(K)$.
Two questions:
Given a partition $p$, which finite groups $G$ have their group algebra over $K$ being isomorphic to $A_p$?
Given a natural number $n$, how many partitions with sum $n$ are there such that $A_p$ is isomorphic to a group algebra?
How does the sequence of numbers of such partitions begin depending on $K$? Probably the answer is very complicated, but can something be said about the rough behavior of the sequence?
I dont know how complicated those questions are, so partial answers are also welcome.
co.combinatorics gr.group-theory rt.representation-theory finite-groups
edited 12 mins ago
YCor
26.4k380127
asked 1 hour ago
Mare
3,13121029
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up vote
2
down vote
favorite
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $mathbbC$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the semisimple algebra $A_p:=M_a_1(K) times cdots times M_a_m(K)$.
Two questions:
Given a partition $p$, which finite groups $G$ have their group algebra over $K$ being isomorphic to $A_p$?
Given a natural number $n$, how many partitions with sum $n$ are there such that $A_p$ is isomorphic to a group algebra?
How does the sequence of numbers of such partitions begin depending on $K$? Probably the answer is very complicated, but can something be said about the rough behavior of the sequence?
I dont know how complicated those questions are, so partial answers are also welcome.
co.combinatorics gr.group-theory rt.representation-theory finite-groups
edited 12 mins ago
YCor
26.4k380127
asked 1 hour ago
Mare
3,13121029
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $mathbbC$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the semisimple algebra $A_p:=M_a_1(K) times cdots times M_a_m(K)$.
Two questions:
Given a partition $p$, which finite groups $G$ have their group algebra over $K$ being isomorphic to $A_p$?
Given a natural number $n$, how many partitions with sum $n$ are there such that $A_p$ is isomorphic to a group algebra?
How does the sequence of numbers of such partitions begin depending on $K$? Probably the answer is very complicated, but can something be said about the rough behavior of the sequence?
I dont know how complicated those questions are, so partial answers are also welcome.
co.combinatorics gr.group-theory rt.representation-theory finite-groups
edited 12 mins ago
YCor
26.4k380127
asked 1 hour ago
Mare
3,13121029
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $mathbbC$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the semisimple algebra $A_p:=M_a_1(K) times cdots times M_a_m(K)$.
Two questions:
Given a partition $p$, which finite groups $G$ have their group algebra over $K$ being isomorphic to $A_p$?
Given a natural number $n$, how many partitions with sum $n$ are there such that $A_p$ is isomorphic to a group algebra?
How does the sequence of numbers of such partitions begin depending on $K$? Probably the answer is very complicated, but can something be said about the rough behavior of the sequence?
I dont know how complicated those questions are, so partial answers are also welcome.
co.combinatorics gr.group-theory rt.representation-theory finite-groups
co.combinatorics gr.group-theory rt.representation-theory finite-groups
edited 12 mins ago
YCor
26.4k380127
asked 1 hour ago
Mare
3,13121029
edited 12 mins ago
YCor
26.4k380127
asked 1 hour ago
Mare
3,13121029
edited 12 mins ago
YCor
26.4k380127
edited 12 mins ago
YCor
26.4k380127
edited 12 mins ago
YCor
26.4k380127
26.4k380127
asked 1 hour ago
Mare
3,13121029
asked 1 hour ago
Mare
3,13121029
asked 1 hour ago
Mare
3,13121029
3,13121029
add a comment |Â
add a comment |Â
1 Answer
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For $K=mathbb C$, if $mathbb C Gcong prod_i M_a_i(mathbb C)$,
then the sequence
$d(G)=[a_1,ldots,a_m]$ is the sequence of character
degrees of the irreducible representations of $G$ over $mathbb C$.
That is, it is the sequence of numbers in the first column
of the character table for $G$.
There is a lot that is known and a lot that is
unknown about these sequences. For example:
$|G|=a_1^2+cdots+a_m^2$.- $m$ is the number of conjugacy classes of $G$.
$a_i$ divides $a_1^2+cdots+a_m^2$ for each $i$.
$|G/G'|$ is the number of $a_i$ equal to $1$. In particular,
at least one $a_i$ is $1$, and the number of $1$'s is a divisor of $a_1^2+cdots+a_m^2$.
There exists nonisomorphic groups $G$ and $H$ with
$d(G)=d(H)$. It is even possible
to choose one to be solvable and the other to be nonsolvable.- If $sum_i=1^m a_i <16m/5$
and $p$ is any prime that divides $sum_i=1^m a_i^2$, then $mgeq 2sqrtp-1$.
The McKay Conjecture can be phrased as a statement about how the sequence $d(G)=[a_1,ldots,a_m]$ is related to the corresponding sequences for normalizers of Sylows. For a prime $p$, let $i_p'(G)$ be the number of $a_i$ such that $p$ does not divide $a_i$. The McKay conjecture is that $i_p'(G)=i_p'(N)$ if $N$ is a normalizer of a Sylow $p$-subgroup of $G$.
None of this answers your questions. I'm only saying that the questions are hard.
answered 1 hour ago
Keith Kearnes
6,14422945
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
For $K=mathbb C$, if $mathbb C Gcong prod_i M_a_i(mathbb C)$,
then the sequence
$d(G)=[a_1,ldots,a_m]$ is the sequence of character
degrees of the irreducible representations of $G$ over $mathbb C$.
That is, it is the sequence of numbers in the first column
of the character table for $G$.
There is a lot that is known and a lot that is
unknown about these sequences. For example:
$|G|=a_1^2+cdots+a_m^2$.- $m$ is the number of conjugacy classes of $G$.
$a_i$ divides $a_1^2+cdots+a_m^2$ for each $i$.
$|G/G'|$ is the number of $a_i$ equal to $1$. In particular,
at least one $a_i$ is $1$, and the number of $1$'s is a divisor of $a_1^2+cdots+a_m^2$.
There exists nonisomorphic groups $G$ and $H$ with
$d(G)=d(H)$. It is even possible
to choose one to be solvable and the other to be nonsolvable.- If $sum_i=1^m a_i <16m/5$
and $p$ is any prime that divides $sum_i=1^m a_i^2$, then $mgeq 2sqrtp-1$.
The McKay Conjecture can be phrased as a statement about how the sequence $d(G)=[a_1,ldots,a_m]$ is related to the corresponding sequences for normalizers of Sylows. For a prime $p$, let $i_p'(G)$ be the number of $a_i$ such that $p$ does not divide $a_i$. The McKay conjecture is that $i_p'(G)=i_p'(N)$ if $N$ is a normalizer of a Sylow $p$-subgroup of $G$.
None of this answers your questions. I'm only saying that the questions are hard.
answered 1 hour ago
Keith Kearnes
6,14422945
add a comment |Â
up vote
5
down vote
For $K=mathbb C$, if $mathbb C Gcong prod_i M_a_i(mathbb C)$,
then the sequence
$d(G)=[a_1,ldots,a_m]$ is the sequence of character
degrees of the irreducible representations of $G$ over $mathbb C$.
That is, it is the sequence of numbers in the first column
of the character table for $G$.
There is a lot that is known and a lot that is
unknown about these sequences. For example:
$|G|=a_1^2+cdots+a_m^2$.- $m$ is the number of conjugacy classes of $G$.
$a_i$ divides $a_1^2+cdots+a_m^2$ for each $i$.
$|G/G'|$ is the number of $a_i$ equal to $1$. In particular,
at least one $a_i$ is $1$, and the number of $1$'s is a divisor of $a_1^2+cdots+a_m^2$.
There exists nonisomorphic groups $G$ and $H$ with
$d(G)=d(H)$. It is even possible
to choose one to be solvable and the other to be nonsolvable.- If $sum_i=1^m a_i <16m/5$
and $p$ is any prime that divides $sum_i=1^m a_i^2$, then $mgeq 2sqrtp-1$.
The McKay Conjecture can be phrased as a statement about how the sequence $d(G)=[a_1,ldots,a_m]$ is related to the corresponding sequences for normalizers of Sylows. For a prime $p$, let $i_p'(G)$ be the number of $a_i$ such that $p$ does not divide $a_i$. The McKay conjecture is that $i_p'(G)=i_p'(N)$ if $N$ is a normalizer of a Sylow $p$-subgroup of $G$.
None of this answers your questions. I'm only saying that the questions are hard.
answered 1 hour ago
Keith Kearnes
6,14422945
add a comment |Â
up vote
5
down vote
up vote
5
down vote
For $K=mathbb C$, if $mathbb C Gcong prod_i M_a_i(mathbb C)$,
then the sequence
$d(G)=[a_1,ldots,a_m]$ is the sequence of character
degrees of the irreducible representations of $G$ over $mathbb C$.
That is, it is the sequence of numbers in the first column
of the character table for $G$.
There is a lot that is known and a lot that is
unknown about these sequences. For example:
$|G|=a_1^2+cdots+a_m^2$.- $m$ is the number of conjugacy classes of $G$.
$a_i$ divides $a_1^2+cdots+a_m^2$ for each $i$.
$|G/G'|$ is the number of $a_i$ equal to $1$. In particular,
at least one $a_i$ is $1$, and the number of $1$'s is a divisor of $a_1^2+cdots+a_m^2$.
There exists nonisomorphic groups $G$ and $H$ with
$d(G)=d(H)$. It is even possible
to choose one to be solvable and the other to be nonsolvable.- If $sum_i=1^m a_i <16m/5$
and $p$ is any prime that divides $sum_i=1^m a_i^2$, then $mgeq 2sqrtp-1$.
The McKay Conjecture can be phrased as a statement about how the sequence $d(G)=[a_1,ldots,a_m]$ is related to the corresponding sequences for normalizers of Sylows. For a prime $p$, let $i_p'(G)$ be the number of $a_i$ such that $p$ does not divide $a_i$. The McKay conjecture is that $i_p'(G)=i_p'(N)$ if $N$ is a normalizer of a Sylow $p$-subgroup of $G$.
None of this answers your questions. I'm only saying that the questions are hard.
answered 1 hour ago
Keith Kearnes
6,14422945
For $K=mathbb C$, if $mathbb C Gcong prod_i M_a_i(mathbb C)$,
then the sequence
$d(G)=[a_1,ldots,a_m]$ is the sequence of character
degrees of the irreducible representations of $G$ over $mathbb C$.
That is, it is the sequence of numbers in the first column
of the character table for $G$.
There is a lot that is known and a lot that is
unknown about these sequences. For example:
$|G|=a_1^2+cdots+a_m^2$.- $m$ is the number of conjugacy classes of $G$.
$a_i$ divides $a_1^2+cdots+a_m^2$ for each $i$.
$|G/G'|$ is the number of $a_i$ equal to $1$. In particular,
at least one $a_i$ is $1$, and the number of $1$'s is a divisor of $a_1^2+cdots+a_m^2$.
There exists nonisomorphic groups $G$ and $H$ with
$d(G)=d(H)$. It is even possible
to choose one to be solvable and the other to be nonsolvable.- If $sum_i=1^m a_i <16m/5$
and $p$ is any prime that divides $sum_i=1^m a_i^2$, then $mgeq 2sqrtp-1$.
The McKay Conjecture can be phrased as a statement about how the sequence $d(G)=[a_1,ldots,a_m]$ is related to the corresponding sequences for normalizers of Sylows. For a prime $p$, let $i_p'(G)$ be the number of $a_i$ such that $p$ does not divide $a_i$. The McKay conjecture is that $i_p'(G)=i_p'(N)$ if $N$ is a normalizer of a Sylow $p$-subgroup of $G$.
None of this answers your questions. I'm only saying that the questions are hard.
answered 1 hour ago
Keith Kearnes
6,14422945
edited 40 mins ago
answered 1 hour ago
Keith Kearnes
6,14422945
answered 1 hour ago
Keith Kearnes
6,14422945
answered 1 hour ago
Keith Kearnes
6,14422945
6,14422945
add a comment |Â
add a comment |Â
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