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### How to find the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$? [migrated]

Let $\mathfrak g$ be a Kac-Moody algebra.
Then
$$
\mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha}
$$
and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the ...

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130 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:
...

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77 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 ...

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207 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 = ...

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165 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 ...

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**1**answer

165 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 ...

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**1**answer

175 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 $g(a,2)=\chi(a)g(1,2)$ ...

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194 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 ...

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204 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 ...

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218 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?

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199 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 ...

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157 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 ...

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985 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 ...

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260 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 ...

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**2**answers

157 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 ...

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322 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 ...

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269 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 ...

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113 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 ...

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**1**answer

625 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 ...

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### 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.
...

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389 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 ...

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349 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 ...

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399 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 ...

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503 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 ...

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**1**answer

418 views

### When does the modulus of a sum of an integer and an algebraic integer equal an integer?

Let say Z is a sum of n-roots of unity and thus an algebraic integer, and D is an rational integer.
If |z+D| is an integer, what can we conclude regarding Z? can we say |Z| is integer?
Another ...

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432 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 ...

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199 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 ...

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### Degrees of irreducible characters of groups of order 48 [closed]

Hello
i wonder if there is a simple argument to show that no group of order 48 can have an irreducible character of degree larger than 4?
Thanks, Karim

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### Induced character for non-injective homomorphisms

Any group homomorphism $\phi\colon H\to G$ gives rise to an induction/restriction adjunction between $G$-representations and $H$-representations:
$$ \hom_G(\phi_! M, N) \cong \hom_H(M, \phi^* N) $$
...

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### Finite, abelian, yet “fugitive” orthogonal subgroups

Update July 29, 2013.
I have still not found a good textbook for this topic, if you point one to me I will be grateful :) I have accepted BS's answer anyway, since their explanation was useful to me ...

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975 views

### Two Definitions of “Character” of topological groups

When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
Let $G$ be a topological group. A character of $G$ is a ...

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### When is a class function on a group G (finite abelian) into the rational numbers Q an element of the rational representation ring of G?

Given a class function $f: G \to \mathbb Q$, where $G$ is a finite abelian group, is there an easy way to decide whether $f$ is an element of the rational representation ring $R_{\mathbb Q}(G)$, i.e. ...

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454 views

### Weil bound for characters sums. (reference-request )

Do you know on any good reference on Weil bound for charcter sums over algebraic curves.
I prefer reference which assume few previous knowlage.

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370 views

### Compute formal character of semisimple Lie algebras.

Let $\mathfrak{g}$ be a semisimple Lie algebra and $V_{\lambda}$ be the irreducible $\mathfrak{g}$-module with highest weight $\lambda$. Are there some softwares which can compute the formal character ...

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### Introduction to L-series and Dirichlet characters?

I'm looking for an introductory text on Dirichlet characters and the L-series of a field K, specifically for quartic extensions of $\mathbb{Q}$. I have Davenport's Multiplicative Number Theory, ...

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407 views

### Some special characters of finite groups

Let $G$ be a finite group, for each irreducible character $\chi$, we define ${\bf Z}(\chi)$ to be the set of all $x\in G$ such that $|\chi(x)|=\chi(e)$ when $e$ is the identity of the group.
For every ...

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368 views

### A question about the existence of a specific extension of a character.

general situation:
Let $ N \leq G $ be a subgroup,and let $ \chi \in Irr(G) $ be an irreducible character of G such that $\chi_N $ is not irreducible( i dont think that this is really needed) and let ...

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### Groups of order $n$ with a character whose degree is at least $0.8\sqrt{n}$ (say)

[edited in response to some corrections by Geoff Robinson and F. Ladisch]
Throughout, all my groups are finite, and all my representations are over the complex numbers.
If $G$ is a group and $\chi$ ...

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**5**answers

717 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 ...

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531 views

### Is 2-sylow subgroup of a rational group also a rational group?

As we know, a finite group $G$ is a rational group if $\chi (g)\in\mathbb{Q}$, where $\chi$ is every irreducible charahter and $g\in G$. I have an interesting question that is "Is 2-Sylow subgroup of ...

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408 views

### Primitive Characters

Let $f(n)$ be an arithmetic function with period $q$ such that $f(n)=0$ whenever $(n,q)>1$. Call $d$ a quasiperiod of $f$ if $f(m)=f(n)$ whenever $m\equiv n \bmod d$ and $(mn,q)=1$.
Suppose $d_1$ ...

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201 views

### On the decomposition of two representations of the Iwahori-Hecke algebra of type A_n

Consider the Iwahori Hecke algebra $H_q(n)$ of the symmetric group $S(n)$, the $\Bbb{Z}[q^{1/2},q^{-1/2}]$ algebra with generators $T_i$, $1 \leq i \leq n-1$, subject to the braid relations and the ...

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### Character group of Frobenius kernels

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequently, the character ...

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### The natural inclusion of an infinite abelian group $G$ into $\widehat{\widehat{G}}$

I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into
...

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### characters on a finite group with `extremal' behaviour

The following question is a bit technical, and I haven't got to grips with it enough to be able to present it as a well-focused question. However, my hope is that the collective group-theoretic ...

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489 views

### On the determinant of an odd, continuous Galois representation.

In his paper, Duke paper, Serre consider continuous, odd Galois representation
$\rho: G_{\mathbb{Q}}\longrightarrow GL_{n}(\overline{\mathbb{F}}_{p})$ where $p$ is a rational prime.
Roughly, (I don't ...

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289 views

### Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$

Hi,
I am faced with a finite-dimensional representation $V$ of $\mathfrak{gl}_{2n}$, whose character I know. I know how to use this character to determine the irreducibles for $\mathfrak{gl}_{2n}$ ...

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### Proving interesting theorems about S_n using its character table.

Hi,
i wonder if there are interesting proofs about $S_n$ (group theoretic or not) using its character table.
Using the Murnaghan-Nakayama rule you can for example prove that for $n>4$ $A_n$ is the ...

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1k views

### Are there “real” vs. “quaternionic” conjugacy classes in finite groups?

The complex irreps of a finite group come in three types: self-dual by a
symmetric form, self-dual by a symplectic form, and not self-dual at all.
In the first two cases, the character is real-valued, ...

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**1**answer

716 views

### Fundamental characters of level 1,2, etc.

I'm trying to understand the concept (definition) of the fundamental characters of level 1,2, etc. as Serre defined those in his Inventiones'72 paper and how they are related to Serre's conjecture. ...