23
votes
How to find normal subgroups from a character table?
This is quite well-known and can be found in books on representation theory. Here is an explanation, which is far from being original.
First fact : $N$ is a normal subgroup of a finite group $G$ if ...
- 1,246
13
votes
Dirichlet characters and quadratic fields
This is probably not a complete answer to your question but here it is anyways.
Usually, one calls a Dirichlet character $\chi$ quadratic, if it has order 2 in the dual of $(\mathbb{Z}/m\mathbb{Z})^\...
- 2,168
11
votes
Intuition behind the choice of the trace-function in character theory
Let's recall that, concretely, the data of an $n$-dimensional representation of a finite group $G$ is a collection $\rho(g)$ of $n \times n$ matrices, one for each element $g \in G$. Classifying such ...
- 409k
10
votes
Accepted
Absolute values of complex irreducible characters of finite groups
Answers:
No. A way of seeing this is that all integers of cyclotomic fields occur as character values. Among them are numbers of the form $2-2\cos(\pi/n)$ for any positive integer $n$, and those ...
- 130k
9
votes
Accepted
If $\phi$ is a character of $G$ such that $\langle \phi,\phi \rangle=4$, then there exists a character $\chi$ of $G$ such that $\phi=2\chi$
The claim is false. It is possible that $\phi=\chi_1+\chi_2+\chi_3+\chi_4$ for some four distinct irreducible characters of $G$.
The smallest groups with four distinct characters are the abelian ...
9
votes
Accepted
Do characters distinguish real representations of a finite group?
See the following answer of Derek Holt to a related question: link. A reference for what you need is Theorem 29.7 in Curtis and Reiner.
Let $G$ be a finite group and let $K \subseteq L$ be a field ...
- 11.7k
8
votes
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In what sense are the linear characters among the irreducible characters
The linear characters are precisely the characters (in the trace sense) of $1$-dimensional representations, which are automatically irreducible.
- 409k
8
votes
Accepted
Irreducibility of a character if and only if inner product equals 1
You are not allowed to have rational numbers in the linear combination of the characters. The $m_i$'s are natural numbers. Theorem 10 in your notes says that
...
- 43k
8
votes
Accepted
why the column sums of character table are integers?
Let $K$ be the Galois field extension of $\mathbb{Q}$ generated by the entries of the character table. Then the action of $\operatorname{Gal}(K/\mathbb{Q})$ on character values permutes the rows of ...
- 27.5k
8
votes
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character degrees of finite simple groups: an elementary question
To use 2.3, suppose $G$ is simple with irreducible character $\chi$ of degree $2$. If $\rho\colon G\to GL_2(\mathbb{C})$ affords $\chi$, then $\ker\rho\unlhd G$ must be trivial, since $\rho$ is not ...
- 12.1k
8
votes
Accepted
Why are characters of algebraic groups interesting?
The character group $X(G)$ of a connected, reductive group $G$ is as interesting as the character group of its radical $A = \mathscr R(G)$. Here $A$ is the unique maximal torus lying in the center of ...
- 33k
8
votes
What is a non-alternating simple group with big order, but relatively few conjugacy classes?
I'm a big fan of the group $\operatorname{PSL}(2,7)$, also known as $\operatorname{GL}(3,2)$, which is finite, simple, not isomorphic to an alternating group, and has precisely $6$ conjugacy classes.
- 61.4k
8
votes
Accepted
If F < E are fields, how is it possible for a representation X, to be irreducible as an F-representation, but reducible as an E-representation?
I think this is based in a misunderstanding of how we can convert between $E$ and $F$ vector spaces. For concreteness, let's work with $\mathbb{C}$ and $\mathbb{R}$, but you'll see that the same idea ...
- 35.6k
7
votes
Accepted
Is it true that $\langle \psi, \omega\rangle$ is the dimension of $Hom(V,W)$?
If $\newcommand{\Hom}{\text{Hom}}\Hom(V,W)$ denotes the vector space
homomorphisms from $V$ to $W$ then $\Hom(V,W)$
is a $G$-module with character $\overline\psi\omega$ and $\Hom_G(V,W)=\Hom(V,W)^G$. ...
- 157k
7
votes
Accepted
Irreducible characters of SU(3)
Yes, in fact there is! The formula for the character of the irreducible representation of $SU(3)$ with highest weight $(p,q)$ is
\begin{align}
\chi^{p,q}(\theta, \phi)
=
e^{i \theta (p+2q)}\sum\...
- 1,083
7
votes
Accepted
Relation between the order of an element of a group and their character in a simple group
Let $\rho$ be a representation affording $\chi$. Then the eigenvalues of $\rho(g)$ are $\pm 1$. Suppose $m$ of them are $1$ and $n$ are $-1$. So $\chi(g) = m-n$ and $\chi(1) = m+n$.
If $\chi(g) \equiv ...
- 86.8k
7
votes
Accepted
How to construct the $C_{4v}$ character table given its Cayley table
OK, so this author has some really wonky notation, but it's all fine in the end.
The first comment is that the pyramid is a red herring. All symmetries of the pyramid leave the peak fixed, so actually ...
- 10.9k
7
votes
Accepted
Perfect groups whose character degrees square divide its order
If you extend to perfect groups, this is pretty easy. Let $G$ be your favourite simple group, and let $A$ be a very large abelian group. A semidirect product $X=A\rtimes G$ has character degrees ...
- 10.9k
6
votes
Accepted
Can one always construct a square-root of a homomorphism to $\mathbb{C}^*$?
Suppose that $G=C_2$ is cyclic of order 2 with generator $x$. Define $\chi$ by $\chi(1)=1$, $\chi(x)=-1$. A reasonable choice for the square root would be $\sqrt{1}=1$, $\sqrt{-1}=i$, but then
$$ \...
- 53.2k
6
votes
Accepted
a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$
Fix $n$ and define
$$f(d)=\sum_{o(\chi)=d}\chi(n).$$
Let's show $f$ is multiplicative. First off, let $g$ be a generator for $(\mathbb{Z}/p\mathbb{Z})^\times$ and write $n=g^k$, then let $\psi$ be a ...
- 150k
6
votes
Accepted
sum of roots of unity is algebraic integer
Let us assume that all the $\alpha_j$ are $m$-th roots of unity.
So if we fix some primitive $m$-th root of unity $\zeta$ then
$\beta=\alpha_1+\cdots+\alpha_k=\zeta^{a_1}+\cdots+\zeta^{a_n}$ for some ...
- 157k
6
votes
Accepted
characters of finite groups which vanish except identity
This character $\chi$ is a multiple of the character of the regular representation: $\chi=a\chi_{reg}$. A priori $a\in\Bbb Q$, but the regular representation contains one copy of the trivial ...
- 157k
6
votes
Accepted
The condition between $\chi(1)$ and $[G:H]$ which gives us a normal subgroup.
Any reference here is to the book of Marty Isaacs, Character Theory of Finite Groups. Now for (b) you must be a bit more precise: if $H$ is abelian, and $\chi$ is an irreducible character of $G$, then ...
- 46.9k
6
votes
Accepted
Monster coefficients
In GAP, you could simply iterate in a triple loop over $\phi,\psi,\rho$, calculate the $g$-values and find maximum and sum values:
...
- 17.4k
6
votes
Accepted
How does knowing that $\sqrt{7}\notin\mathbb{Q}(e^{2\pi i/7})$ help construct this character table?
Okay, I finally worked it out. I'll give a complete solution to the exercise:
First of all we have the trivial character, which we'll denote $\chi_1$. We have $\left<\chi_1,\alpha\right>=\left&...
- 16k
6
votes
Find all irreducible characters of a matrix group on finite field $\mathbb F_5$
(Partial answer / hint, starting with character degrees) I am assuming that you are taking about representations over $\mathbb{C}$. As indicated $G=HN$, $N \lhd G$, $H \cap N=1$ with $H \cong C_4$ and ...
- 46.9k
6
votes
Validating a Character Table for a Given Finite Group
In general, it is not easy to decide if a given matrix is a character table. For instance, the matrix
$$\begin{pmatrix}
1&1&1&1\\
1&1&-1&1\\
2&2&0&-1\\
6&-1&...
- 3,231
6
votes
Accepted
Application of character theory to structure of groups
If you do not wish to use character theory or calculations in a group ring for that matter, try this one: for a finite group $G$, let $g \in G$ be a commutator (that is, $g=[x,y]=x^{-1}y^{-1}xy$ for ...
- 46.9k
6
votes
If F < E are fields, how is it possible for a representation X, to be irreducible as an F-representation, but reducible as an E-representation?
Let's apply your logic to the example, since clearly something must go wrong.
Here $F = \mathbb{R}$, $E = \mathbb{C}$, $X: C_3 \rightarrow GL_2(\mathbb{R})$ given by $g \mapsto X(g) = \begin{pmatrix} ...
- 862
6
votes
Possible degrees of group characters
Yes, for instance the symmetric group $S_{n+1}$ has an irreducible representation of degree $n$. Precisely, take the canonical permutation representation $V$ of $S_{n+1}$; it has degree $n+1$, and we ...
- 24.8k
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