All Questions
41 questions
37
votes
0
answers
1k
views
Is this generalized character always a character?
Let $G$ be a finite group, and $p$ be a prime. Then there is a generalized character $\Psi$ of $G$ which takes value $0$ on all elements of order divisible by $p$, and has $\Psi(y)$ equal to the ...
- 44.4k
9
votes
0
answers
254
views
An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
- 2,306
6
votes
1
answer
288
views
Which finite simple groups are rational-relative-real?
A finite group $G$ is called rational if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$.
Analogously, I'll call $G$ real ...
- 54.6k
2
votes
0
answers
220
views
Characters of alternating groups
I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...
- 370
1
vote
0
answers
179
views
Character extension about $Q_8$
Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise:
(Exercise 5.9)
Let $G$ be a finite group and $N\unlhd G$, suppose ...
- 195
6
votes
0
answers
320
views
(CFSG-free) Finite simple groups whose character degrees square divide its order
Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...
12
votes
2
answers
926
views
Finite groups with integral character table
The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
0
votes
1
answer
302
views
Lower bound of the largest irreducible character degree of alternating group $A_n$
$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
- 1
20
votes
0
answers
451
views
Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 24.2k
3
votes
0
answers
228
views
Subquotients of representations and character
Let $G$ be a group, $K$ an algebraically closed field of characteristic zero and $\rho_1,\rho_2:G\to \mathrm{GL}_n(K)$ be two semi-simple representations. What I would like to be able to determine is ...
- 4,418
21
votes
0
answers
473
views
Is there a "direct" proof of the Galois symmetry on centre of group algebra?
Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$
This induces a linear ...
- 1,949
3
votes
2
answers
222
views
Equality of subsets of abelian groups
Let $G$ be a finite abelian group, $X$ and $Y$ be two non-empty subsets of $G$ of equal size. Suppose that for each irreducible character $\chi$ of $G$ we have $\sum_{x\in X}\chi(x)=\sum_{y\in Y}\chi(...
- 105
4
votes
0
answers
227
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
20
votes
2
answers
948
views
The finite groups with a zero entry in each column of its character table (except the first one)
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
17
votes
2
answers
860
views
The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group
Throughout $G$ is a finite, non-abelian group.
$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$
Let $\Irr(G)$ be the set of ...
- 25.8k
9
votes
2
answers
525
views
Characterization of the family of simple groups PSL(2,q) by tensor multiplicity
Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...
4
votes
0
answers
260
views
Indexed character tables for wreath products in Sage and GAP
I am trying to obtain character table for the Hyperoctahedral group $\mathcal{H}_n$ in Sage using GAP.
This group arises as the wreath product $\mathcal{C}_2 \wr \mathcal{S}_n$, so of course I can ...
- 41
22
votes
3
answers
1k
views
On permanents and determinants of finite groups
$\DeclareMathOperator\perm{perm}$Let $G$ be a finite group. Define the determinant $\det(G)$ of $G$ as the determinant of the character table of $G$ over $\mathbb{C}$ and define the permanent $\perm(G)...
- 26.5k
3
votes
0
answers
400
views
Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
1
vote
0
answers
68
views
Relationship between non-zero values of characters and normality in finite groups
Note: Let $G$ be a finite group and $H \leq G$. Then it is clear that if $H \unlhd G$, then $\chi(h) \neq 0$ for irreducible constituents $\chi$ of the permutation character $(1_H)^G$ and $h\in H$. ...
- 366
6
votes
1
answer
298
views
Is a finite group given by its character table if its Sylow subgroups are so?
As pointed out by Mikko Korhonen in this answer, Özdem Çelik proved (in 1976 here) that a finite group whose Sylow subgroups are cyclic (called a Z-group) is determined by its character table.
...
4
votes
1
answer
518
views
Finite groups with the same character table *including* class types, and square-free order
There are non-isomorphic finite groups with the same (complex) character table, as $D_4$ and $Q_8$.
$$\scriptsize\begin{array}{c|c}
\text{class}&1&2A&2B&2C&4 \newline
\text{...
11
votes
1
answer
289
views
Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters
Let $G$ be a transitive permutation group of degree $d$ having a cyclic regular subgroup $K = \langle k \rangle \cong C_d$. Let $\pi(g) = |\mathrm{Fix}(g)|$ be the permutation character of $G$ and let
...
- 11.2k
9
votes
1
answer
286
views
Divisors of the regular character of a finite group
Recall that the regular character $\rho=\hspace{-.2cm}\sum\limits_{\chi\in\operatorname{Irr}(G)}\hspace{-.2cm}\chi(1)\chi$ of a finite group $G$ takes values
$$
\rho(g)=
\left\{\begin{array}{cl}
...
- 1,090
1
vote
0
answers
103
views
Character degrees of a finite group?
Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
- 841
7
votes
1
answer
647
views
On the structure of a finite group of order $144$
Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.
QUESTION: Prove $G\cong A_4\times A_4$.
By using Magma, we know ...
- 577
20
votes
3
answers
940
views
What did Frobenius prove about $M_{12}$?
I am interested in this paper which I can't read because it's in German:
Frobenius, G., Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02.
A free ...
- 11.2k
3
votes
1
answer
353
views
Computing characters of $\alpha$-projective representations
Given a finite group $G$, a finite cyclic group $A$ (viewed as a subgroup of $\mathbb{C}^{\times}$, i.e. generated by a $|A|$-th root of unity), and a 2-cocycle $\alpha\in Z^{2}(G,A)$. Recall that an $...
- 547
6
votes
1
answer
225
views
A sum over characters of $S_{2n}$ and zonal spherical functions of $(S_{2n},H_n)$
The hyperoctahedral group $H_n$ can be seen as the centralizer of the permutation $(12)(34)\cdots (2n-1\,2n)$ in $S_{2n}$. It has $2^nn!$ elements.
The quantities $$ \omega_\lambda(\pi)=\frac{1}{2^...
- 2,552
8
votes
3
answers
1k
views
Beyond Brauer's theorem
Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...
- 17.6k
2
votes
1
answer
220
views
The lower bound of a group with characters of special degrees
Is there any lower bound for the order of a group with an irreducible character of degree $p$, where $p$ is a prime.
Is there any similar result for $p^2$ or $p^3$ instead of $p$?
Thanks for your ...
- 1,168
6
votes
1
answer
208
views
Sum identities with immanants
For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} \chi(...
- 93
1
vote
1
answer
270
views
On the character degrees of a finite group with special structure
Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a ...
- 1,168
2
votes
0
answers
219
views
The tallest possible lattice?
Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
- 298
27
votes
3
answers
2k
views
How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
- 2,450
4
votes
3
answers
565
views
Finding a character of height zero
My character theory is rather weak, so excuse me if this is a triviality.
I have read on the encyclopedia of maths that for any group $G$, every block of $G$ contains an irreducible character of ...
- 298
43
votes
2
answers
4k
views
How much of character theory can be done without Schur's lemma or the Artin-Wedderburn theorem?
This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear.
...
- 114k
9
votes
2
answers
1k
views
Character table entries and sums of roots of unity
It is well-known that the entries of the character table of a finite group are sums of roots of unity.
Question: Is the converse true? Explicitly, given $z\in \mathbb{Z}[\mu_\infty]$, can I find a ...
- 1,261
52
votes
0
answers
1k
views
Class function counting solutions of equation in finite group: when is it a virtual character?
Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
- 6,919
4
votes
0
answers
348
views
Interplay between two definitions of the transfer homomorphism.
The transfer homomorphism can be defined in a couple of ways. For the purposes of this question, assume that all groups are finite.
Defintion 1. Let $1\leqslant K'\leqslant L \leqslant K\leqslant G$ ...
- 1,173
5
votes
5
answers
1k
views
Irreducible representations of the unitriangular group
Hi,
I wonder how much is known about the irreducible representations of the nxn unitriangular group over a finite field with q elements.
I know that all characterdegrees are a power of q and all ...
- 891