# Tag Info

## Hot answers tagged characters

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### If $\chi$ is a complex-valued character of a representation of a finite group, is it always true that $\overline{\chi(g)}=\chi(g^{-1})$?

Another way to see that is as follows: Let $G$ be a finite group, $ρ:G\rightarrow \text{GL}_n(\mathbb{C})$ a representation and $χ$ the corresponding character. We know that for any $g\in G$, all the ...
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### Validating a Character Table for a Given Finite Group

Adding to the excellent answer of @BrauerSuzuki I wanted to comment on the point: For $H≤G$, one gets a permutation character $π$ from the (transitive) action of $G$ on the set of (left) cosets by (...
• 51
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### Does any valid character table correspond to a group?

In his famous report for the ICM, Richard Brauer proposed a number of questions on representations of finite groups. His Problem 6 asks to give necessary and sufficient criteria on a complex matrix to ...
• 3,597
4 votes
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### Confused about the domain of characters

Usually, when people ask about potential errors in books, either the book is wrong or they’re wrong. In this case, you’re both a bit wrong. The book is wrong in that it introduces the circle group ...
• 230k
4 votes
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### Character table guarantees certain property of groups

This happens if and only if $$\sum_{\chi\in\mathrm{Irr}(G)}\frac{\chi(x)\chi(y)\chi(z)\chi(g)}{\chi(1)^2}$$ is constant (i.e. does not depend on $g$). See Corollary 4.14 in [Navarro, Characer theory ...
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### If $g$ is commutator then so is $g^m$ for $(m,o(g))=1$

A character-free proof of this result of K. Honda (1953) has recently been found by Hendrik Lenstra. It has been published in Operations Research Letters, Powers of Commutators, Volume 51, Issue 1, ...
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4 votes
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### Why does character of identity element always positive integer?

The character of the identity element is the trace of the identity matrix, i.e. the dimension (the "degree") of the representation.
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4 votes
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### Definition of a character of a linear algebraic group

Yes, please don't feel bad about this confusion. It is for semi-historical reasons as KReiser rightly pointed out above, and can be quite confusing. Namely, let us fix a field $K$ and an (affine) ...
• 53.2k
4 votes
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### On the proof of: If $0<\frac{|χ(g)|}{χ(1)}<1$ then $\frac{χ(g)}{χ(1)}\notin\overline{\mathbb{Z}}$

This has something to do with Galois theory. Since the minimal polynomial is irreducible, its roots are permuted transitively by the Galois group of the splitting field. But the Galois group sends ...
• 3,597
4 votes
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### Why is the determinant of a character well-defined?

For any two choices $\rho$ and $\rho'$ giving the same character $\chi$, there exists $P\in GL_n(\mathbb{C})$ such that $\rho'(g)=P\rho(g)P^{-1}$ for all $g\in G$: in other words, $\rho$ and $\rho'$ ...
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• 638
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### A question about hypothesis of Clifford's Theorem in Isaacs Character Theory book

The irreducibility of $\chi$ is necessary to assert that $\chi$ is a constituent of $\vartheta^G$. By Frobenius reciprocity we have that $0\not=[\chi_H,\vartheta]=[\chi,\vartheta^G]$. Then $\chi$ is a ...
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2 votes
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### The set of characters

Consider $C$ be an ${}^*$-algebra (not $C^*$ yet) generated by $C_\infty$ and $\{e^{it}\}.$ $C = C_\infty \oplus P(e^{it}, e^{-it})$ where $P$ is polynomial algebra. Direct sum is viewed in category ...
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