# Questions tagged [characters]

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

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### Solution to system of nonlinear equations

I am trying to comple the character table of a finite group with seven conjugacy classes $c_1, \cdots, c_7$: Character table If I use the orthogonality of the column vectors of table I get a system of ...
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### Help with proof of $\langle\chi,\psi\rangle=\delta_{\chi,\psi}$ with $\chi$ and $\psi$ characters of a group.

In The Symmetric Group, we are presented with the following theorem Let $\chi$ and $\psi$ be irreducible characters of a group $G$. Then $$\langle\chi,\psi\rangle=\delta_{\chi,\psi}\tag{1}$$ In the ...
1 vote
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### p-adic Norms and Character Values [closed]

Let $G$ be a group with a $\mathbb{Q}_p$-representation $\mathfrak{X}: G \to \mbox{GL}_n(\mathbb{Q}_p)$ affording the character $\chi$. We have the $p$-adic norm $\| \cdot \|$ defined on an extension ...
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### Trouble with fourth type of irreducible character of GL2(F) Serge Lang algebra

so it's been 1 month since I started reading Serge Lang Algebra, I'm now pretty advanced in my reading of the book but I am stuck at p 721, here it is : So here Lang is computing and classifying all ...
1 vote
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### Explicit Computation of Characters of Finite Field $\mathbb{F}_d$

I have a question about the computation of characters of $\mathbb{F}_{2^n}$ in arXiv:quant-ph/0410155. My question might be trivial but I'm not very familiar with details in the theory of finite ...
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### A question about hypothesis of Clifford's Theorem in Isaacs Character Theory book

I've been studying the book of Isaacs of Character Theory of Finite Groups. Clifford's Theorem (Theorem 6.2 of the book) states the following. Theorem: Let $H\lhd G$ and let $\chi\in\text{Irr}(G)$. ...
1 vote
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### Indecomposable/Extreme Group Characters and Direct Sum Decompositions

I am currently working through the details of theorem 2.10 in this paper. There are a few things which don't make sense to me, and I'm hoping someone could help me work through the details of the ...
1 vote
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### Complex conjugation and outer automorphisms of finite groups

Context: A finite group has all characters real valued if and only if every element $g \in G$ is in the same conjugacy class as $g^{-1}$. This property has some special name but I can't remember ...
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### Character of restricted representation.

My lecture notes state that the restriction of an irreducible character is in general not irreducible and gives the following example: "Restricting any non-linear irreducible character to the ...
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### Generalization of the Great Orthogonality Theorem?

Consider the sum $$\Pi_\tau = \frac{1}{|G|} \sum_{g \in G} \chi_\tau^*(g) \, (\mu^\star \otimes \nu)(g),$$ where $G$ is a finite group, $\mu$, $\nu$, and $\tau$ are irreps of $G$, and $\chi_\tau$ is ...
1 vote
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### Character of a representation and dimensione of $G$-invariant space

Let $G$ be a finite group. Let $V$ be a finite dimensional vectorial space on an algebraically closed field and let $\rho: G \to Gl(V)$ an irreducible linear representation of $G$ in $V$. Since $V$ is ...
1 vote
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### If $H$ has a normal complement then it has the character restriction property

I am trying to show that if a subgroup $H$ of a group $G$ has a normal complement then $H$ has the character restriction property. That is every irreducible character of $H$ is a restriction of some ...
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### Why is the determinant of a character well-defined?

In Karpilovsky's Group Representations (vol. 1) it can be read in chapter 27, that if $\chi$ is a character of $G$, and $\rho \colon G \to \mathrm{GL}_{n}(\mathbb{C})$ is a representation which ...
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### Generalization of convolution of characters to irreps

Let $G$ be a finite group. Let $\pi_i, \pi_j, \pi_k$ be irreps of $G$. Then consider the sum $$f(x):= \frac{1}{|G|} \sum_{g \in G} \chi_i(xg^{-1}) \chi_j(g) \pi_k(g)$$ If $\pi_k$ is the ...
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### Is there some sort of irrep orthogonality for finite groups?

Disclaimer: As pointed out by several people the original orthogonality condition I was asking about seems to be incorrect so I'm just going to change the question to the orthogonality condition ...
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### Isomorphic group of $D_8/C_2$

In my exercise I have to find all one dimensional representations of $D_8/C_2$ and their characters and they give us the hint 'which group is this isomorphic to?'. I have tried to maybe find a ...
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### Confused about the domain of characters

In Stein's Fourier Analysis Ch.7 he defines characters as follows. Let $G$ be a finite Abelian Group with operation $*$, and $S^1$ be the unit circle in $\mathbb{C}$. $e: G \to S^1$ is a character on ...
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### Character table guarantees certain property of groups

How to show there exist three conjugacy classes in a certain group $G$ such that the products $xyz, x\in C_1, y\in C_2, z\in C_3$ divide equally over all elements of $G$ by looking at the character ...
1 vote
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### Let $\chi(g)$ is a nonnegative real number. Show that if $\chi$ is irreducible, then $\chi$ is the trivial character

I am working on a problem from my abstract algebra class: Let $\chi$ be a character of a group $G$ with the property that $\chi(g)$ is a nonnegative real number for all $g \in G$. Use the row ...
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### Irreducible representation over finite field GAP

I have a finite group $G$ with $p\not \mid |G|$ and I want to compute an explicit modular representation of $G$ given a character $\chi$. I can compute a representation by ...
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