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

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2
votes
0answers
16 views

What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...
4
votes
1answer
37 views

If $H<G$ is abelian, and $\chi(1)=[G:H]$ for irreducible $\chi$, then $H$ contains a nontrivial normal subgroup?

Suppose $\chi$ is an irreducible character of a finite group $G$, and $H$ is a nontrivial abelian subgroup such that $\chi(1)=[G:H]$. Why does $H$ contain a nontrivial normal subgroup? I understand ...
0
votes
0answers
55 views
+100

Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
3
votes
1answer
38 views

Generalizing Dirichlet characters

Suppose I want to consider Dirichlet characters $$\chi: \mathbb{F}_p(\zeta_r)^{*} \longrightarrow \mathbb{C}$$ Can I prove something similar to the Polya Vinogradov inequality for these characters? ...
1
vote
0answers
20 views

Linear character and p-rationality

Let $G$ a group and $\chi$ a linear character, p$\ne$2 a prime number with $|G|$=$p^a$$m$ and $(p,m)=1$. Show that $\chi$ is p-rational if and only if p doesn't divide the order of $\chi$ (as element ...
1
vote
1answer
69 views

Primitive, quadratic Dirichlet character to odd prime power modulus

Let $p$ be an odd prime number and let $\alpha \geq 1$ be an integer. Let $\chi$ be a real, non-principal, primitive Dirichlet character mod $p^{\alpha}$. How does one show that $\alpha = 1$? If we ...
2
votes
0answers
28 views

How to do this integral? $\int_{-i\infty}^{i\infty}F_{\psi}(z) dz$?

Let $\psi$ be a character with conductor $f_\psi$. Define $$F_\psi(z)=\begin{cases}\sum_{n=1}^{\infty}\psi(n)e^{2\pi i nz}&\text{ if }& \text{Im}(z)>0\\ -\sum_{n=1}^{\infty}\psi(-n)e^{-2 \...
0
votes
0answers
21 views

Irreducible representation of $1$-transposition groups

I would like to know the theory of irreducible representation of $1$-transposition groups. Could anyone provide me a pointer from where I can proceed?
1
vote
0answers
29 views

Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
0
votes
1answer
13 views

For a compact abelian group, what can I conclude if range of a character is finite?

Suppose $G$ is a compact abelian group, and suppose $<g^n>$ is dense in $G$ where $g$ is a particular element of $G$ and $<g^n>$ is the subgroup generated by $g$. Let $\chi$ be a character ...
2
votes
0answers
38 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
12
votes
5answers
456 views

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 ...
1
vote
1answer
28 views

Faithful monomial representation induced from faithful character

Let $\rho: G \rightarrow GL_n(\mathbb{C})$ be a faithful irreducible representation such that $\rho = Ind_N^G \phi$ for some 1-dimensional representation $\phi$ and normal subgroup $N$. Does $\phi$ ...
1
vote
1answer
41 views

Irreps of products between dihedral group and any finite group

Let $D_n$ be the dihedral group with order $2 n$. The total number of irreducible representations for $D_n$ is as follows. When $n$ is even, the total number is $\frac{n-2}{2} + 4 = \frac{n}{2} + 3$. ...
2
votes
0answers
32 views

Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
1
vote
1answer
20 views

Definition of $\hat{G_1}\times \hat{G_2}$ where $G_1,G_2$ are abelian groups and $\hat{G}$ is the dual of $G$

The question is in the title. I want to know what happens to $(\chi_G,\chi_H)\in\hat{G}\times \hat{H}$. Are they just passively sitting there as a pair or they give something when applied on $(g,h)$? ...
0
votes
0answers
13 views

Inner product of Induced permutation representation and an irrep $\langle {\chi \uparrow^{S_n}_{D_n}}_{\mathbf{ 1}_{D_n}} , \chi_\rho \rangle_{S_n}$

I am trying to compute the inner product of the characters of the induced permutation representation from the trivial representation of a dihedral group $D_n$ of order $2 n$ to $S_n$ and an irrep $\...
2
votes
2answers
22 views

Characters of $d$-dimensional representations have size at most $d$.

Let $\chi$ be the character of a $d$-dimensional complex representation $\rho$ of a finite group $G$. Then $\vert \chi (g)\vert \le d$ for all $g$ and if we have equality (for all $g$) then $\rho(g)=\...
0
votes
1answer
40 views

How do we evaluate this Dirichlet L-series

In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character ...
1
vote
3answers
29 views

When does a Character Table have Non-real Entries, and how do I Compute them?

When can one conclude that a character table has non-real entries? In particular, by constructing the character table for $\mathbb{Z}/3\mathbb{Z}$ or $A_4$ how does one determine that some of the ...
2
votes
1answer
58 views

Characters on rings of residue classes modulo polynomials over finite fields

First recall the following orthogonality relation on $\mathbb{Z}/n\mathbb{Z}$. Fix $n \in \mathbb{Z}$, $n \neq 0$. For $r \in \mathbb{Q}$, let $e(r) := e^{2 \pi i r}$. Let $x \in \mathbb{Z}$. Then ...
1
vote
0answers
20 views

Character of an Induced Representation

In Fulton's book, page 34, it is stated that, To compute the character of $V = Ind\space W$, note that $g \in G$ maps $\sigma W $ to $g \sigma W$, so the trace is calculated from those cosets $\...
1
vote
1answer
25 views

Bi-character for finite, commutative monoids?

If I have a finite commutative monoid $M$ (which is not a group), is it possible to get a bi-character on this? By bi-character, I mean a map $\beta:M\times M\rightarrow \mathbb{C}^*$ such that, $\...
1
vote
2answers
36 views

Identifying conjugacy classes in GAP

I would like to know which column of the character table of $SL(2,\mathbb{Z}/q\mathbb{Z})$ that GAP produces with the command Display(CharacterTable(SL(2,q))); ...
0
votes
0answers
46 views

Computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$

How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$? Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 ...
0
votes
1answer
54 views

Something about Galois character

Let $p$ be odd prime. $\Sigma=(p, \infty)$, $\mathbf{Q}_{\Sigma}$ is maximal Galois extension of $\mathbf{Q}$ unramified outside $\Sigma$. Let $G=\mathrm{Gal}(\mathbf{Q}_{\Sigma}/\mathbf{Q})$. $\...
0
votes
1answer
30 views

In some basis for a vector space $V$, its matrix is diagonal.

Let $\phi: G \to \text{Aut}(V)$ be an irreducible representation of a finite group $G$, where in some basis for $V$, all matrices $\phi(g)$ have real entries. For this basis, is it true that $\phi(g)$...
2
votes
1answer
53 views

Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
2
votes
1answer
57 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, \operatorname{sgn}...
3
votes
1answer
39 views

Question on character theory.

Let $\phi :G \to \text{Aut}(V_0)$ be the regular representation of $G$. Define a representation $\Theta$ of $G \times G$ on $V_0$ by $\Theta(g_1,g_2)e_g=e_{g_1gg^{-1}_2}$, i.e it maps the element $(...
0
votes
0answers
15 views

Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
0
votes
0answers
25 views

Find character of exterior power.

Consider the symmetric group $S_4$ and let $\phi: S_4 \to \text{Aut}(V)$ be a representation of $S_4$ of degree $3$ whose character $\chi$ is given by $$1^4 \mapsto 3$$$$2^2 \mapsto -1$$$$(3,1) \...
0
votes
0answers
19 views

Calculating the eigenvalues of a representation with an unknown order

If I know that $\chi$ is a character of degree $2$ on $G$ and suppose $\chi(g) = 1$. Prove that $\chi(g^2)= -1$. Now, I know that $\rho(g)$ is a 2x2 matrix, so therefore there are 2 eigenvalues (...
2
votes
1answer
33 views

How to find normal subgroups from a character table?

I know that normal subgroups are the union of some conjugacy classes Conjugacy classes are represented by the the columns in a matrix How could we use character values in the table to determine ...
0
votes
1answer
28 views

Dimension of a direct sum of characters (example with $S_3$)

Here is the character table of $S_3$: I was wondering how one can determine the dimension of for example the sign character $sgn$. Could we get it from the character table? Also, if we define $A$ ...
1
vote
1answer
33 views

Character of a representation on $S_3$ and irreducible representations

Here is the character table of S3: Consider $V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ with basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ Let $\pi$ be the representation of $...
0
votes
0answers
22 views

Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$ I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible representations ...
1
vote
2answers
83 views

Character of an $\mathbb{R}G$-module constructed from a $\mathbb{CG}$-module

I have been reading Representations and Characters of Groups by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module. ...
0
votes
0answers
29 views

Is G isomorphic to a Subgroup of $GL(2,\mathbb C)$

I'm stuck on a question on a past exam paper that asks if a group $G$ is isomorphic to a subgroup $GL(2,\mathbb C)$. We are given the character table for $G$ which I've attached below. It's the last ...
0
votes
0answers
24 views

Why is conductor of a Dirichlet character the product of conductors of other Dirichlet characters?

Let $n=\prod_{i=1}^np_i^{e_i}$ with $p_i$ different prime numbers and $e_i$ positive integers. Given a Dirichlet character $\chi$ modulo $n$ we can define the characters $\chi_i$ (modulo $p_i^{e_i}$) ...
9
votes
0answers
82 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
1
vote
1answer
21 views

Simplify $\langle \operatorname{Ind}^G_1 1, \operatorname{Ind}^G_H\phi\rangle_G$

Let $G$ be a finite group and $H$ a subgroup. Let $\phi$ be an irreducible character of $H$ and $\mathbb 1$ the trivial character of the trivial subgroup $1$. Let $\langle,\rangle_G$ be the usual ...
1
vote
1answer
37 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation $\...
1
vote
1answer
39 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
0
votes
1answer
31 views

characters in semi-direct product.

The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 ...
1
vote
1answer
24 views

Question on Fourier-Stieltjes transform (in Rudin, p. $15$)

I have a question on an inequality written on the bottom of this page. Let $G$ be a locally compact group and let $\gamma : G \to S^1 \subset \Bbb C$ be a character of $G$. If $\mu$ is a complex ...
1
vote
0answers
18 views

Identity for exponential character sums

I was confused about the following identity I ran into. I would appreciate it if somebody could clear this up for me. Suppose that we have an exponential sum $$g(a)=\sum_{t=0}^{p-1} \exp( 2\pi i at^k/...
0
votes
0answers
9 views

Deformation of Gamma function integral contour

Terence Tao has described the gamma function as the inner product of a multiplicative and an additive character with respect to the Haar measure on $\Bbb R^+$. The gamma function is defined as follows:...
2
votes
2answers
27 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...
0
votes
1answer
37 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...