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Questions tagged [characters]

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

The character of a representation $\rho:G\to\mathrm{GL}(V)$ is the function $\chi:G\to \mathbb F$ given by $\chi(g)=\mathrm{trace}(\rho(g))$ (where $V$ is a finite-dimensional vector space over the field $\mathbb F$).

The term is also use for homomorphisms $G\to \mathbb F^\times$, which can be seen as a special case of the above definition when the representation is one-dimensional.

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Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some cases, there are so many kinds of orbits of G on…
Jack Schmidt
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If $g$ is commutator then so is $g^m$ for $(m,o(g))=1$

There are certain theorems in finite group theory whose proofs involve character theory and for which there are still no character-free proofs. Among such is Frobenius theorem on transitive permutation groups. (Another was Burnside's $pq$ themrem;…
Beginner
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How to show a representation is irreducible?

I have a professor who says that I should be able to show a representation is irreducible simply by looking at its trace (with other possible conditions), but after researching this for a while, I have still not been able to see this. I found a…
TheMobiusLoops
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exercise in Isaacs's book on Character Theory

I'm stuck on an exercise in Isaacs's book "Character Theory of Finite Groups" - it relates to something I'm looking at as part of ongoing research, but I guess it belongs here rather than on MathOverflow, since it's an exercise and hence ought to be…
user16299
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Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/F_1\right)$. If $F_2$ is a Frobenius group…
Alexander Gruber
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Do all algebraic integers in some $\mathbb{Z}[\zeta_n]$ occur among the character tables of finite groups?

The values of irreducible characters of a finite groups are always sums of roots of unity; do all sums of roots of unity (i.e. algebraic integers in the maximal abelian extension of $\mathbb{Q}$) actually appear among the character tables of finite…
Will
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Formula for number of solutions to $x^4+y^4=1$, from Ireland and Rosen #8.18.

There is a sequence of three exercise in Ireland and Rosen's Introduction to Modern Number Theory, Chapter 8, page 106. I can do the first two, but can't finish the third. I can include the proofs to the first two if they are wanted. Suppose that…
Noomi Holloway
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Monomial characters of direct product

Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of irreducible complex characters of $G$. A character $\chi\in\text{Irr}(G)$ is monomial if there exists a subgroup $H\leq G$ and a linear character $\lambda\in\text{Lin}(H)$ such that…
DRossi
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Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character theory). I would like to know what are some of the…
Mohan
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Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to finish the following question (I am ok with general…
Otis
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The characteristic of the field does not divide the dimension of an irreducible representation

Let $k$ be an algebraically closed field of characteristic $p$, and $G$ be a finite group whose order is not divisible by $p$. I would like to prove the following: if $V$ is an irreducible representation of $kG$, then $\dim V \neq 0$ in $k$, i.e.…
Joppy
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Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces $X_\lambda$ that are the smooth complete model of…
Alex
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Show that each character of $G$ which is zero for all $g \ne 1$ is an integral multiple of the character $r_G$ of the regular representation

This is a question from J.P.Serre's book 'Linear representation of finite groups',section 2.4 The question: Let $G$ be a finite group. Show that each character of $G$ which is zero for all $g \ne 1$ is an integral multiple of the character $r_G$ of…
Hajime_Saito
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Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of $\lambda$ and $\ell(\lambda):=\lambda^t_1$ the length of…
Jesko Hüttenhain
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What is an irreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b \in S_n$ [NOTE: this is not 100% accurate - see…
Stefan Smith
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