For questions about characters (homomorphisms from a group into the multiplicative group of a field).

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6
votes
3answers
550 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
7
votes
2answers
332 views

Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
8
votes
2answers
416 views

What is an irrreducible 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 ...
4
votes
2answers
384 views

How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
5
votes
1answer
686 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
0
votes
1answer
382 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
20
votes
0answers
565 views

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 ...
8
votes
1answer
781 views

What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
4
votes
1answer
167 views

Order of group elements from a character table

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I'm trying to answer a question which goes the ...
3
votes
2answers
567 views

Why does the tensor product of an irreducible representation with the sign representation yield another irreducible representation?

I was writing this question, and I came up with an answer, so I thought I would answer it myself: In considering representations of $S_n$, among others, we have the "sign representation", that is the ...
3
votes
1answer
370 views

Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary ...
8
votes
1answer
125 views

Characters of the symmetric group corresponding to partitions into two parts

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
5
votes
1answer
265 views

Generalizing Artin's theorem on independence of characters

Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$. Can this theorem be ...
4
votes
0answers
440 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
1
vote
0answers
56 views

Character representation of right regular representation as sum of irreducible characters

Let $G$ be a finite group acting on itself by the action $g \ast x := xg^{-1}$. Then this corresponds to an representation $\rho : G \to GL(L^2(G))$, where $L^2(G)$ denotes the space of function on ...
1
vote
1answer
28 views

Fourier inversion formula on finite abelian groups

The following exercise is exercise 2.2.3 from these lecture notes by Daniel Bump. Let $\mathcal F : L^2(G) \to L^2(G^{\ast})$ be the Fourier transform, defined by $\mathcal{F}f = \hat f$, where ...
1
vote
1answer
137 views

Weyl character formula and finding the trace.

Let $v$ be a positive integer. I have a representation $\rho_v$ of $USp(4) = \{g\in M_2(\mathbb{H})\,|\,g^{T}\bar{g}\}$, where $\mathbb{H}$ is Hamiltons quaternions. The representation $\rho_v$ has ...
5
votes
1answer
89 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
2
votes
1answer
109 views

$\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ .

Let $G = H \times K$. Let $\phi \in \operatorname{Irr}(H)$ and $\theta \in \operatorname{Irr}(K)$ be faithful. Show that $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$. Problem 4.3 of ...
1
vote
1answer
63 views

Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $ H $ be a subgroup with index $ m $ in the finite group $ G $. Let $ F $ be an algebraic closed field of characteristic $ 0 $. ...
1
vote
1answer
81 views

Complete the character table of group of order $21$

You are given the incomplete character table of a group $G$ with order $21$ which has $5$ conjugacy classes, $C_1,\dots,C_5$, which have sizes $1,7,7,3,3$. $$ \begin{array}{|c|c|c|c|c|} \hline & ...
1
vote
0answers
54 views

Summation of Legendre symbol

Let $\chi_{2,q}$ be the real Dirichlet character modulo a prime $q>2$, which is not the principal one (the so-called Legendre symbol). Is it true that $$ \sum_{n=0}^{+\infty} ...
0
votes
2answers
41 views

Standard representation of $S_5$

I am trying to determine the standard representation of $S_5$. I understand that it will be a map from group elements to $\mathbb{C}^4$. The character table is as follows. I understand that the ...