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

learn more… | top users | synonyms

0
votes
0answers
36 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
34 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
29 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 ...
2
votes
1answer
49 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, ...
3
votes
1answer
34 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
12 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
18 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
23 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
25 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 $A$ ...
0
votes
1answer
22 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
21 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 ...
1
vote
2answers
80 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. ...
1
vote
0answers
19 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, ...
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
20 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
76 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
19 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
30 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
36 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 ...
-1
votes
0answers
26 views

Character of subgroup of index 2

Let $\chi$ be an irreducible character of a finite group $G$; if $H\leq G$ is a subgroup of index 2, is $Res_H^G\chi$ irreducible? How do conjugacy classes change from $G$ to $H$?
0
votes
1answer
27 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
21 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 ...
0
votes
0answers
16 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 ...
0
votes
0answers
8 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 ...
2
votes
2answers
24 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
35 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 ...
4
votes
0answers
17 views

Computing Multiplicative Character Values over Finite Fields [duplicate]

Let $\mathbb F_q$ be the finite field of order $q$, where $q\equiv 1\pmod 4$ is some prime power. Let $\chi_4\colon\mathbb F_q^\times\to\mathbb C^\times$ be a multiplicative character of exact order 4 ...
3
votes
1answer
42 views

Using Irreducible Group Characters to Count nth Roots of Group Elements

Given $n\in\mathbb{N}$, define $\tau_n(g)=|\lbrace h\in G: h^n=g\rbrace|$. Let $\chi_i,1\leq i\leq r$ be the distinct complex irreducible characters of a finite group $G$, and let ...
8
votes
2answers
456 views

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 ...
2
votes
1answer
79 views

Sum of sum of elements in conjugacy class is a multiple of them if and only if $G=G'$

I have another question on character/group theory. This one seems to be a bit harder. Let $Cl(g_1),...Cl(g_r)$ be the conjugacy classes of a finite group, $G$ and let $C_i \in \mathbb{C}(G)$ (the ...
2
votes
1answer
59 views

A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

Let $G$ be a finite group and $H$ a subgroup. Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ ...
1
vote
0answers
36 views

Is character of a group representation the same as trace?

If so, why cannot the Klein group's character be zero? The group element of Klein group matrices can be traceless, right?
2
votes
1answer
46 views

Faithful character of degree less than p gives abelian p-Sylow groups.

I am trying to prove: Suppose $p$ divides $|G|$ and let $\chi \in Irr(G)$ if $\chi$ is faithful and its degree is less than $p$ then any $p$-Sylow subgroup of $G$ is abelian. I have tried to ...
2
votes
1answer
13 views

Question on normal subgroup and sum of characters.

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Prove that $$\mid G:N \mid = \sum \chi(1)^2$$ where the sum ranges over all irreducible characters of $G$ such that $N \subseteq \ker ...
0
votes
1answer
55 views

Character of $Sym^2(V)$ and decomposition into irreducible representations

Let $G=S_3$ be the symmetric group on three elements, whose character table is given as follows: Let $V$ be the unique irreducible representation of dimension $2$ Question 1: Compute the character ...
2
votes
1answer
28 views

Character regular representation

Consider the regular representation of a finite group $G$ and let $X_{reg}$ be its character. Let $(\pi, V)$ be any finite dimensional representation of $G$ with character $X$. Show that ...
0
votes
1answer
44 views

Character Table, Row and Column orthogonality, Conjugacy Classes

Let $G$ be a finite group with conjugacy classes $C_1, C_2, ..., C_k$ and let $g_i \in C_i$ be an element for each $i=1, ..., k$ Part 1: State the theorems on row and column orthogonality in the ...
1
vote
1answer
34 views

Wedderburn component of $\mathbb C[G]$ corresponding to a contragredient character

Let $G$ be a finite group and let $\phi$ be an irreducible character of $G$ over $\mathbb C$. How does the Wedderburn component of $\mathbb C[G]$ corresponding to the contragredient character of ...
2
votes
1answer
35 views

Let ($\pi, V$) be a representation of $G$ with character $X$: if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations

Let $(\pi, V)$ be a representation of $G$ with character $X$. Prove that if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations I was under the impression that the inner ...
1
vote
0answers
19 views

Kloosterman Sums and K3 Surfaces

In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface ...
0
votes
0answers
32 views

Is this statement true? (characterize elements of dual group)

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
1
vote
0answers
29 views

Is there a non-trivial character on any locally compact Abelian group?

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$? indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.
0
votes
0answers
30 views

Subspace generated by irreducible characters

It is mentioned in the answer for this question: Perhaps we should also note that the subspace generated by irreducible characters is the same as that generated by their conjugates. Can you ...
0
votes
0answers
23 views

Proof of existence of a non-trivial character on a local field of positive characteristic

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
3
votes
1answer
38 views

A question concerning an exercise from Tao Vu

This is the exercise 4.1.5 from Tao Vu Additive Combinatorics. $Z$ is a finite additive group with a fixed symmetric non-degenerate bilinear form $\cdot$ Define $e: \mathbb{R}/\mathbb{Z} \to ...
1
vote
0answers
24 views

Hints in proving the irreducibility of the permutation representation of $S_n$

Can anyone give me any hints of how to proceed: I have scoured this forum and exhausted all online notes. I am trying to prove that there is no room for an irreducible permutation representation, ...
0
votes
0answers
42 views

discontinuous group homomorphism from $(\mathbb{R},+)$ to $S^1$,unit circle

It is well known that if $\gamma:(\mathbb{R},+)\rightarrow S^1$ is continuous homomorphism,then $\exists y\in\mathbb{R}$,such that $\gamma(x)=e^{ixy}$. Show that there is a discontinuous homomorphism ...
0
votes
1answer
50 views

Hypothesis: $\sum\limits_{n \in A} ccl_p(n) = p-1 $ [closed]

Let G be a the finite set of integers $1, \dots , p-1 $ One can see, if $|G| = p-1 $ with $p$ prime, then $\forall a \in G: \exists n \in \mathbb{N}$ with $ a^n = 1$ mod $p$ and $ 1 \le n < p $ I ...