The characters tag has no usage guidance.

**1**

vote

**0**answers

40 views

### Parabolic characters of subgroups $\Gamma \subset \operatorname{SL}_2(\textbf{Z})$ generated by parabolic and elliptic elements [on hold]

In the paper Generalized Modular Forms from Knopp and Mason, one can read in page $6$:
Remark. It is not too hard to prove that a subgroup $\Gamma$ of finite index in $\operatorname{SL}_2(\textbf{Z})$...

**3**

votes

**2**answers

415 views

### Relation of these two Dirichlet $L$-functions

Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions.
Is there a realtion between these two Dirichlet $L$-functions:...

**2**

votes

**0**answers

88 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 $...

**1**

vote

**1**answer

139 views

### Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula
$$O_\gamma(f) = \int_G f(x^{-1}\...

**2**

votes

**0**answers

210 views

### On a sequence of L-functions having same zeros in critical strip and GRH

I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...

**0**

votes

**0**answers

109 views

### Abel summation formula versus Perron's formula to bound a partial sum

Taking $\chi$ a primitive character, with Abel summation it is easy to show that for $\epsilon >0$, there is a constant $M$ such that for all $x$ we have :
$$|\sum_{n<x} \frac{\chi(n)}{n^{\...

**1**

vote

**1**answer

70 views

### Character kernels in the lattice of subgroups of a finite abelian group

I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian group....

**4**

votes

**2**answers

155 views

### Evolution of partial sum of a sequence of induced Dirichlet characters

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.
Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...

**5**

votes

**0**answers

84 views

### Gaussian Hypergeometric Functions and Legendre Character

I was hoping somebody might be able to point me to a good reference on Gaussian hypergeometric functions defined over a finite field. the reason I'm interested is that I've encountered sums of the ...

**3**

votes

**1**answer

189 views

### On partial sum of non-primitive Dirichlet characters

Consider a Dirichlet character, $\chi(n)$, and the partial sum :
$$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$
There are many works to bound this sum when $\chi$ is a primitive character, but ...

**5**

votes

**1**answer

110 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^...

**12**

votes

**1**answer

290 views

### A sum over characters of the symmetric group

Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$.
Let $\...

**8**

votes

**2**answers

254 views

### Irreducible reps and characters of $G \rtimes A$

Is there a theorem which classifies irreducible representations of semi-direct product of finite groups $G \rtimes A$, where $A$ is a finite abelian group and hence write down the character table for $...

**5**

votes

**1**answer

154 views

### Connection between cyclic group and exponential function

I have been thinking about this for a while, but now got to the point where I got stuck. I don't know if it might be considered as a research level question, but I would be very happy if somebody knew ...

**1**

vote

**1**answer

128 views

### construct a Hecke character in MAGMA with given infinity type

I need to do some numerical computation on special values of a Hecke L-function $L(s,\chi)$. To do this, I want to construct a Hecke character in MAGMA, given that I know its infinity type.
In other ...

**7**

votes

**3**answers

560 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 ...

**1**

vote

**0**answers

69 views

### associativity of the extension of finie groups [closed]

Following my previous question I have two questions:
1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then $2....

**5**

votes

**0**answers

75 views

### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...

**7**

votes

**0**answers

123 views

### Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...

**1**

vote

**1**answer

140 views

### Groups with many vanishing elements

It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is
What can be said about a finite group $G$ for which $\...

**2**

votes

**0**answers

68 views

### When does a finite group have a lower-dimensional representation than one of its quotients?

The minimal dimension of a nontrivial representation of a central extension of a group $G$ can be smaller than the minimal dimension of a nontrivial representation of $G$. What about non-central ...

**2**

votes

**1**answer

171 views

### Sum of irreducible complex character degrees for alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups?
The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...

**2**

votes

**1**answer

117 views

### Irreducible characters of finite abelian groups

Let $G$ be finite abelian group and $K$ a field such that $char(K)$ does not divide the order $r$ of $G$. For each divisor $d$ of $r$ let $\omega_d$ be a primitive $d$-root of unity and $a_d:=\frac{\...

**2**

votes

**1**answer

185 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 ...

**6**

votes

**1**answer

105 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(...

**3**

votes

**1**answer

200 views

### Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?
For a primitive Dirichlet character $\chi$ we have:
...

**1**

vote

**2**answers

101 views

### Non-degenerate characters of the unitriangular group $U$

I made a previous post which was unclear and mistaken in fundamental aspects, so that it was actually more worthy making this new post than actually editing the previous one.
I'm studying the ...

**1**

vote

**2**answers

281 views

### Rational Conjugacy Classes of Finite Groups

Suppose $G$ is a finite group and $A$ is the set of all character values of $G$. By character values, I mean entries of the character table of $G$. Let $\Gamma = \operatorname{Gal}({\mathbb{Q}(A)}/{\...

**5**

votes

**1**answer

227 views

### Dirichlet Characters as Eigenvectors

This was asked in Math Stackexchange here but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at ...

**1**

vote

**1**answer

181 views

### Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...

**2**

votes

**1**answer

205 views

### Extension of a formula for the quadratic Gauss sums

I am interested in the sums
$$g(a,k)=\sum_{n=0}^{p-1}e^{2\pi i a n^k/p}$$
where $p\equiv1\mod k$ is a prime and $a$ is coprime with $p$.
When $k=2$, it is a classical fact that $g(a,2)=\chi(a)g(1,2)$ ...

**2**

votes

**1**answer

216 views

### Explicit bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$

I'm looking for an explicit bound for $f(x) = \sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$, where $\chi$ is a Hecke character for a number field $K$ of degree $n$, on the ideals $I_\...

**1**

vote

**1**answer

222 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 ...

**2**

votes

**1**answer

232 views

### In which finite groups is there a non-central g such that, for all irreducible characters, Chi(i)(g) <> zero?

What is the character of Pi(G), the tensor product of all inequivalent irreducible representations of G?

**2**

votes

**0**answers

205 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 ...

**5**

votes

**1**answer

178 views

### To whom is the internal characterization of $Q$-groups due?

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:
$G$ is a $Q$-group if ...

**22**

votes

**3**answers

1k 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 ...

**4**

votes

**3**answers

327 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 ...

**3**

votes

**2**answers

177 views

### differences between character distributions of supercuspidal representations and others

Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to $\...

**10**

votes

**1**answer

375 views

### Asymptotic Weyl Character Formula

Let $G$ be a complex semi-simple group along with a chosen pair of opposite Borel subgroups (so we get all the root-theoretic data we need). Let $\lambda$ be a dominant weight, and let $V(\lambda)$ be ...

**0**

votes

**1**answer

281 views

### Missing formula! [closed]

I am doing a project on group association schemes, in particular looking at the structure constant
$$p_{KL}^M = \#\{(x, y, xy) : x \in K, y\in L, xy \in M\}$$
where $K, L$ and $M$ are conjugacy ...

**1**

vote

**1**answer

172 views

### Hecke Character vs Grossencharakter

I would like to know if there is any difference between
(1) an algebraic Hecke character
(2) a Hecke character
(3) a Grössencharakter
All of the above in the setting of ellitpic curves with complex ...

**8**

votes

**1**answer

879 views

### On Applications of Murnaghan Nakayama Rule

This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ...

**37**

votes

**2**answers

2k 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.
...

**3**

votes

**2**answers

473 views

### On finite groups with same complex-valued character table

What are the necessary and sufficient conditions for two finite groups $G$ and $H$
to have same complex-valued character table?
Is there any criterion for which one could know about the character ...

**7**

votes

**3**answers

441 views

### Characters of p-groups

Berkovich mentioned the following result of Mann in his book on p-groups:
The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1.
Do you know any reference for ...

**8**

votes

**2**answers

497 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 ...

**8**

votes

**1**answer

675 views

### Permutation character of the symmetric group on subsets of certain size

The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...

**34**

votes

**0**answers

611 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,...

**4**

votes

**0**answers

223 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 ...