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

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Factoring the conductor of a Dirichlet character and factoring generalized Gauss sums

Let $m=m_1m_2\cdots m_r$ with $m_i$ positive integers with $\gcd(m_i,m_j)=1$ for $i\neq j$. Given a Dirichlet character $\chi$ modulo $m$ we can define the characters $\chi_i$ (modulo $m_i$) by ...
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1answer
16 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 ...
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29 views
+100

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 ...
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1answer
37 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. ...
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1answer
25 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 ...
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1answer
33 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 ...
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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 $H$?
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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 ...
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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 ...
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15 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 ...
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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 ...
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2answers
23 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 ...
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1answer
32 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 ...
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16 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 ...
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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 ...
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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?
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1answer
58 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 = \{ ...
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1answer
75 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 ...
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1answer
44 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 ...
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1answer
43 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 ...
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1answer
26 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 ...
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1answer
30 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 ...
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1answer
32 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 ...
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1answer
32 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 ...
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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 ...
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31 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 $\mathbb{K}^+$ ...
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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 ...
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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.
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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 ...
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22 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 $\mathbb{K}^+$ ...
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0answers
23 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, ...
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41 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 ...
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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 ...
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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 ...
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1answer
26 views

Existence of homomorphism from $GO(2n,\mathbb{C})$ to $\mathbb{C}^*$ that doesn't lose too much information

L.S., Denote $GO(2n,\mathbb{C})$ the group of matrices that leave an inner product invariant up to a scalar. We have for all $A \in GO(2n,\mathbb{C}), v,w$ vectors, that: $\langle A(v),A(w)\rangle = ...
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1answer
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Can one always construct a square-root of a homomorphism to $\mathbb{C}^*$?

L.S., Let $G$ a group and $\chi: G \rightarrow \mathbb{C}^*$ a morphism of groups. If we set $\delta: G \rightarrow \mathbb{C}$ by $\delta(g) = \sqrt{\chi(g)}$, where we choose one branch of ...
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The character group of $G$ for an abelian group $G$.

Problem Statement: Prove that the one-dimensional characters of a group $G$ form a group under multiplication of functions, i.e. where the group operation is: ...
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“linear independence” of characters? Artin's theorem.

Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$. One fact ...
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31 views

About character table and normal subgroups

I have read that: All normal subgroups of a group $G$ can be recognized from its character table. The kernel of a character $\chi$ is the set of elements $g$ in $G$ for which $\chi(g) = \chi(1)$; ...
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15 views

Computing character for the partition $(2, 2, 1, 1)$ using Murnaghan–Nakayama rule

I am trying to understand an example from Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. The group ...
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22 views

Understanding the Murnaghan–Nakayama rule

I am trying to understand the Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. Here is the ...
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Twisted Kloosterman Sums

A twisted Kloosterman sum is a character sum of the form $$S(\chi, \psi, \eta)=\sum_{t\in (\mathbb{F}_q)^{\times}} \chi(t) \psi(t) \eta(t^{-1}).$$ Here, $\chi$ is a multiplitive character of ...
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2answers
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Character Sums and Characteristic Functions

Let $\mathbb{F}_q$ be the finite field with $q$ elements with $q$ odd. Consider the subgroup $H$ of $(\mathbb{F}_q)^{\times}$ consisting of squares: $H=\{x^2\,:\,x\in(\mathbb{F}_q)^{\times}\}$. We can ...
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A character on a subgroup could be written as a difference with some character on the whole group, help on argumentation

Let $G$ be a finite group, and $U \le G$ of odd order such that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t$ and assume that $U^g \ne U$ implies $U^g \cap U = 1$. Then I can not ...
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1answer
36 views

On the decomposition of the group ring $\mathbb Q[G]$ over the rationals if $G$ is finite and cyclic

Let $G$ be a cyclic finite group of order $n$. I tried to determine the structure of the group ring $\mathbb Q[G]$ over the rationals $\mathbb Q$, what I got for even $n$ is $$ \mathbb Q[G] = A ...
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37 views

Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove ...
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23 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
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4answers
40 views

Class function and character of $S_3$ representation

I particularly need help with question 2. The $\textit{character table}$ for $S_3$ is given as follows:$$\begin{array}{c|c|c|} & \text{1} & \text{(12)} & \text{(123)} \\ \hline ...
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1answer
58 views

(Infinite) Non-abelian group with only linear characters

If $G$ is an abelian group, then every irreducible character has dimension one (i.e. is linear), for finite group we also have a converse. Do we have a converse for infinite groups? Or: Does there ...
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1answer
38 views

A direct proof that for finite $G$ we have “$G$ abelian iff all irreducible characters are linear”

A finite group is abelian iff all its irreducible characters have dimension one (hence are linear). A common proof uses that the number of irreducible representations equals the number of conjugacy ...