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

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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
21 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 $$X=\{...
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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 $\mathbb{K}^+$ ...
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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
30 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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31 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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0answers
26 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
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, $\...
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43 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
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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 \mathbb{...
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1answer
27 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
45 views

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 $\mathbb{...
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59 views

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: $$(\chi\cdot\chi')(g)=\chi(g)\chi'(g)$$...
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32 views

“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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34 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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16 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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24 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 screenshot....
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25 views

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 $(\mathbb{...
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2answers
22 views

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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35 views

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
40 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 this?...
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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
47 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 \text{$...
3
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1answer
61 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 ...
3
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1answer
44 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 ...
2
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1answer
29 views

Relating a Character sum to a Gauss sum

Let $q$ be a prime power. Consider the mapping $f:(\mathbb{F}_q)^{\times} \to (\mathbb{F}_q)^{\times},$ where $x\mapsto x^2$. I was interested in sums of the form $$\sum_{t\in \mbox{Im}(f)} \psi_a(t)\...
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0answers
28 views

Constructing character table of subgroup from character table of whole group

If $\psi : G \to \mathbb C$ is a character and $U \le G$, then $\psi_{|U} : U \to \mathbb C$ is a character, but I guess this might not be irreducible with respect to $U$. But is it possible to ...
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0answers
40 views

How to construct a character table? (E.g Klein 4 group)

Could someone explain to me how you make a character table? Say I wanted to give the character table for the Klein $4$ group, $K$. $K$ is isomorphic to the product $\mathbb{Z}/2\mathbb{Z} \times \...
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1answer
49 views

Conjugacy classes, irreducible representations, character table of $D_{10}$ (order 20)

$D_{10}=\langle r,s \rangle$ is the dihedral group of order 20 . I have been struggling a bit with this question, particularly c, regarding values in the character table (a). Find the conjugacy ...
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1answer
33 views

Why do we have these values of the generalized character when evaluated with the scalar product?

Let $U \le G$ be a subgroup of odd order of the finite group $G$. Suppose $t \notin U$ is an involution with $u^t \in uU'$ for all $u \in U$, where $U'$ denotes the commutator subgroup of $U$. Set $T :...
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1answer
36 views

Extending a linear character of $U$ to $TU$, where $T$ is generated by an involution normalising $U$

Let $U \le G$ be a subgroup of the finite group $G$ of odd order. And suppose $t \notin U$ is an involution normalising $U$, i.e. $U^t = U$ and $t^2 = 1$. Assume $t$ centralizes $U / U'$, i.e. $u^tU' =...
2
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1answer
48 views

Does same character table imply isomorphic abelianizations?

We know two finite groups with the same character table might not be isomorphic (e.g. $D_4$ and $Q_8$), but the sizes of their abelianizations are equal (in fact equal to the number of linear ...
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0answers
23 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
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0answers
35 views

All the characters on $S^1$

Would anyone please give me a hint to find all the characters on $S^1$? By characters I mean complex valued functions on $ S ^ 1 $ that their values have norm $1 $ and they satisfy: $f (x+y)=f (x) ...
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0answers
29 views

Inverse of character table

The character table of a group is always invertible, because the rows are orthogonal. Is there a general formula to compute the inverse of the character table?
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2answers
39 views

Degree one irreducible representations

In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be $S_3$...
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17 views

Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
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1answer
47 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
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38 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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1answer
21 views

For a semisimple algebra and two $M$- and $W$-homogeneous parts for $M \ncong W$, why we have $M(A)W(A) = 0$.

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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0answers
55 views

Product of standard and sign representation of $S_5$

I am able to work out the sign representation of $S_5$ and standard representation of $S_5$. How do I compute the product of standard and sign representation of $S_5$? What kind of product do I need ...
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2answers
58 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 ...
2
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1answer
59 views

Character table for $G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle$

Let $G$ be the group of order $20$ defined in terms of generators and relations: $$G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle.$$ Can anyone help me to derive the character table? ...
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0answers
52 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
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0answers
44 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
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2answers
43 views

Exactly two irreducible characters of dimension 1

I've been working through Artin's Algebra on my own time, and I'm stuck on one of the questions, namely 10.5.3: Suppose that a group G has exactly two irreducible characters of dimension 1, and ...
2
votes
1answer
100 views

why the column sums of character table are integers?

There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number. It is mentioned in Martin Isaacs Character ...
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48 views

Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...