# Questions tagged [characters]

The characters tag has no usage guidance.

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### A $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...

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143 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}
...

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118 views

### Conditions for a solvable group to have a non-trivial center

I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for ...

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92 views

### A generalization of the character group

Let $G$ be a group.
We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$
where $Tor(\mathbb{T})$ is the group of torsion elements of the unit ...

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**1**answer

198 views

### Reference request: an elementary result on characters of finite abelian groups

The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:
Let $A$ be a finite abelian group of order $...

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414 views

### Closed formulas for the character of the symmetric group

I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:
$$\chi_{n}(\sigma) = 1$$
$$\chi_{11...1}(\sigma) ...

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146 views

### Are there partially algebraic Hecke characters?

$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$
Let $F$ be a number field.
Let $\chi\colon \mathbb{A}_F^\...

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156 views

### Independence of characters with respect to polynomials

I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\...

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310 views

### $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?

Let $G$ be a finite group of order $240$.
If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are
$
[1,1,1,1,~3,3,3,3,3,3,3,3, ~...

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165 views

### Is $G$ non-solvable?

Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that
(1) there is a character $\chi\in\mathrm{...

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69 views

### A class function defined using Frobenius-Schur indicators

Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...

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87 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 $\theta$ ...

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109 views

### Does $G$ have a normal abelian Sylow $2$-subgroup?

Let $G$ be a finite group. Let $|G|=2^\alpha n$ where $(2,n)=1$ and $\alpha$ is a positive integer. Suppose that $\def\cd{\operatorname{cd}} n=\max \cd(G)$, and $n^2>\frac{1}{2}|G|$, where $\cd(G)$...

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### Modular transformation of affine characters of non-simply connected groups$.$

Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078)
$$
\chi_\mu\to\sum_{\nu\...

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502 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 ...

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**1**answer

267 views

### Is a $G$-invariant character $\theta$ of $H$ extendible to $G$?

Let $G/H\cong PSL(2,11)$, and $\theta$ be an irreducible $\mathbb{C}$-character of $H$. Suppose $\theta$ is invariant in $G$ and $\theta(1)=9$.
Question: Is $\theta$ extendible to $G$?

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### The degrees of ordinary characters of $PSp(2n,q)$ and $P\Omega O(2n+1,q)$

The finite simple groups $PSp(2n,q)$ and $P\Omega O(2n+1,q)$ have a same order, where $n\geqslant3$ and $q$ is odd.
What are the degrees of the ordinary characters of these two groups?
Thanks!!!

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247 views

### Is there a converse to the Brauer–Nesbitt theorem?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in ...

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### Characters of the kernel of the norm map of an extension of local fields

Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm ...

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416 views

### Character theoretic proof of the Littlewood–Richardson rule?

The Littlewood–Richardson coefficients are the multiplicities
$$
c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu))
$$
and the Littlewood–Richardson rule says that ...

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367 views

### About the existence of characters on $B(X)$

Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M_n(\mathbb{C})$ ...

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99 views

### Characters on $PGL(2)$

I am interested in computations with characters on $PGL(2, F)$ and not in $GL(2, F)$, and have some issues concerning definitions.
The notion of conductor is standard for characters $\chi$ of a $p$-...

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**3**answers

647 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 ...

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216 views

### How to estimate a mixed character sum $\sum_{h \in \mathbb{Z}/q \mathbb{Z}} \chi(f(h)) e(Ch/q)$?

Let $q = p^t$ where $p$ is prime. I am interested in estimating the complete exponential sum, which looks like
$$
\sum_{0 \leq h < q} \chi( (h-a_1)(h-a_2)(h-a_3))
\ \bar{\chi}( (h-b_1)(h-b_2)(h-...

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### Every quasicharacter of an open subgroup extends to a quasicharacter on the whole group

Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous ...

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151 views

### For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$?

Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ ...

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### Expressing quartic Dirichlet characters modulo primes $p\equiv 1\bmod 4$ with Legendre symbols

Looking through some old notes of mine from two years ago I found some crude notes writing what amounted to the statement that for any prime $p\equiv 1\bmod 4$ one could express for any odd integer $p\...

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187 views

### Conductor of characters

Consider characters of $PGL(2)$. The conductor of a local component $\pi_p$ of an automorphic representation is defined as the smallest index such that there is nontrivial fixed vector by the ...

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### If $\chi\in\operatorname{Irr}(L_J)$ and $^\ast R^{L_J}_T(\chi)\neq 0$, does $(\mu,R^{L_I}_{L_J}(\chi))=0$ when $^\ast R^{L_I}_T(\mu)=0$?

Suppose $G$ is a finite group with a split $BN$-pair, satisfying the commutator relations, and such that representatives in $N$ of elements of the Weyl group can be chosen in a nice way: if $W\ni w=s_{...

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580 views

### Tannakian Formalism for the Quaternions and Dihedral Group

It is a basic fact in representation theory of finite groups over complex numbers that the character tables of $Q_8$ and $D_8$ are identical. I believe, this implies that the corresponding categories ...

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### Characters of a quadratic extension and convergence

Let $F$ be a non-archimedean local field, $\chi$ a quasi-character of $F^\star$ and $\psi$ a positive character of $E^\star$. I would like to understand why the usual Rankin-Selberg zeta integrals ...

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### Character table of a group determines the set of commutators of the group

We write $[x,y]$ for the commutator $x^{-1}y^{-1}xy$ of $x$ and $y$ in a group $G$.
(A) Let $g \in G$ and fix $x \in G$. Show that $g$ is conjugate to $[x,y]$ for some $y \in G$ iff
$$\sum_{\chi \...

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482 views

### Conductor as volume of the integers ring

I am working on Tate's thesis, and I have some problems with computations, yet the result seems to be a good natural motivation for introducing the arithmetic conductor of a character.
Let $F$ be a ...

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161 views

### When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...

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### Length *vs* table of characters

Let $G$ be a finite group. Its length $\ell(G)$ is the length $n$ of any Jordan-Hölder filtration $1=G_n\subset\cdots\subset G_0=G$ ($G_i$ is normal in $G_{i-1}$ and $G_{i-1}/G_i$ is simple).
Can $\...

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274 views

### Spectrum of induced subgraphs of Paley graph

Let $G_q$ be a Paley graph on $q$ vertices, where $q=1 \text{ (mod 4)}$, i.e., the vertices of $G_q$ are the elements of the finite field $\mathbb{F}_q$, and there is an edge between vertices $a,b \in ...

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### embedding local characters to global (Hecke) characters

Let $F$ be a number field, and $S$ be a non-empty finite set of places of $F$. Suppose that for each $v\notin S$, we have a character $\chi_v$ of $F_v^\times$ (for my purpose, we can require that $\...

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172 views

### Legendre symbols as homomorphisms in number fields, and quadratic reciprocity [closed]

$\newcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}$
Suppose we have a finite set of rational primes $B=\{p_1,\ldots,p_k\}$, and $V=\{ x\in\mathbb{Q}^*~|~x\text{ contains only primes in B} \}$. So ...

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### Constructing an explicit extension of a continuous character on a closed subgroup of a certain locally compact abelian group

Let $ G $ be a locally compact abelian group and $ \omega: G \times G \to \mathbb{T} $ a continuous multiplier on $ G $, i.e.,
\begin{align}
\forall r,s,t \in G: \qquad
\omega(s,t) ~ \omega(r,s + t) &...

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### Large spin expansion of affine $\mathfrak{su}(2)_k$ characters

There is a problem I am trying to solve for some time now which in a few words boils down to computing the coset characters for
$$
\frac{\mathfrak{su}(2)_k\oplus\mathfrak{su}(2)_\ell}{\mathfrak{su}(2)...

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

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 $\operatorname{SL}_2(\textbf{Z})$...

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568 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:...

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141 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 $...

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**1**answer

218 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}\...

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240 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 ...

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149 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....

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208 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 ...

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101 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 ...

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**1**answer

387 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 ...

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**1**answer

144 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^...