# Questions tagged [characters]

For questions about characters (traces of representations of a group on a vector space).

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### Existence of subgroups of index $2$

I would like to seek some clarification to the solution to the following problem from Representations and Characters of Groups (Gordon, Liebeck): Let $\rho$ be a representation of the finite group $G$...
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### Basis change of $Z(\mathbb C [G])$

Let $G$ be a finite group with . Consider the two bases of $Z(\mathbb C [G])$: One is the elements of the form $e_{g} =\Sigma_{x\in c_g} x$, where $c_g$ is a conjugacy class, and another is elements ...
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### Algebraic characters versus analytic characters for algebraic Tori over non-archimedean field

Given an split torus $T$ over a field $K$ complete with respect to a non-archimedean absolute value, the (algebraic) character group can be defined as $$X(T):=\text{Hom}_{K-gr}(T,\mathbb{G}_m)$$ ...
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### Confirmation of Relation between Degree of Induced Character and Degree of Initial Character

I have been reading about induced characters, and I haven't found explicitly written anywhere, the relation between degree of the induced character of a character and the dimension of the character ...
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### Absolute value of sum of additive characters of $\mathbb{F}_p$

Consider the absolute value of the following exponential sum: $\left|\sum_{x \in \mathbb{F}_p} \sum_{y \in \mathbb{F}_p}e^{\frac{2\pi i}{p}(ux+vy-wxy)}\right|$ for given $u,v,w\in\mathbb{F}_p$ with ...
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### A special type of Gauss sum

In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof. Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
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### How can I easily find character table of Sergeev group (finite)?

I am looking for the character table of the Sergeev group S_d for small d (say, 'd' up to 10 or up to whatever is possible). The Sergeev group $S_d$ is defined as follows: Let $\mathfrak{S}_d$ be ...
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### Prove that: Z(G) = { g∈G ∣ |χ(g)| = χ(1) for all χ ∈ Irr(G)} (Representation Theory)

I'm working on representation theory and I came across a statement that Z(G) = { g ∈ G ∣ |χ(g)| = χ(1) for all χ ∈ Irr(G)}. I'm trying to see why this is the case. (1) For the backwards direction, ...
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### Character of regular representation.

Let $G$ be a finite group and $k$ a finite dimensional field. I am asked to 'find the character' of the regular representation of $G$. And determine what happens in the modular case when ...
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### Are finite abelian groups isomorphic to k-character groups for arbitrary alg.closed k?

Let $G$ be a finite abelian group, $k$ an algebraically closed field with group of units $k^{\times}$, and let $Ch_{k}(G)$ be the set of $k$-characters of $G$ whose elements are multiplicative ...
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### Character of representations of SO(3) derived from the irreducible representations of SU(2)

I have been given that there is a 2:1 surjective homomorphism from $SU(2)$ to $SO(3)$ with kernel $\{-I,+I\}$ and that $V_n$ are irreducible representations of $SU(2)$ where $V_n$ is the space of ...
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### Prove that the value of $\chi_{(n−1,1)}$ on a permutation $\sigma\in S_n$ is one less than the number of fixed points in $\sigma$.

Fix $n$ such that $n > 2$. Let $\chi_{(n−1,1)}$ denote the character of the Specht module $S_{(n−1,1)}$ How do I prove that the value of $\chi_{(n−1,1)}$ on a permutation $\sigma\in S_n$ is one ...
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### Character sum involving conjugacy classes

I am tasked with showing $$\sum_{ x \in C_1, y \in C_2, z \in C_3} \chi(xyz) = \frac{|C_1| |C_2| |C_3| \chi(C_1)\chi(C_2)\chi(C_3)}{\chi(1)^2}$$ where $C_1,C_2,C_3$ are conjugacy classes in a ...
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### An Exercise from Representation of Finite Groups by Serre (section 10)

I was trying to solve a statement from Serre's Linear Representation of Finite Groups (taken from the hint of Exercise 10.6(b) in the book) : Suppose that $H$ is normal in $G$ and that $G/H$ is ...