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Questions tagged [characters]

For questions about characters (traces of representations of a group on a vector space).

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evaluating a (character) sum

What is the value of the following for fixed $b, P$, and $Q$: $$\frac{1}{2^m}\sum_{a\in \mathbb{F}_2^m} i^{\big(2(b+c)+(P+Q)a\big)^Ta},$$ where $b,c\in \mathbb{F}_2^m$, $P,Q$ are symmetric $m\times ...
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Question on irreducible character of regular representation of the symmetric group

Consider the symmetric group $S_n$ acting on $A=\{1,..,n\}$, for any nonnegative integer $k\leq n/2$, denote $A_k$ to be the collection of all $k$-element subsets of $A$. Let $\chi_k$ be the character ...
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Characters table of non abelian group of order 21

I am trying to understand the Characters table of non abelian group G of order 21. In the attached picture, I understand how we get the 3 first irreducible representations of dimension 1. I guess we ...
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Product of characters that is not irreducible

I am looking for an example of product of irreducible characters that is not irreducible. I tried the character table of $S_4$ and $D_4$ and thought about $\Bbb Z/6\Bbb Z$ but I couldn't find such ...
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1answer
104 views

What is a non-alternating simple group with big order, but relatively few conjugacy classes?

I'm not sure if this question is legal. I'm writing my BsC thesis on character theoretical calculations and I have already calculated a lot of character tables (a ...
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Characters of the fundamental representations of $SU(3)$

Let us denote $3$ and $\bar{3}$ the fundamental representations of $SU(3)$. According to my lecture notes, the characters read as follows: $\chi_{[3]} = e^{\omega_1} + e^{\omega_1 - \alpha_1} + e^{\...
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Isomorphic representations have equal characters

I have read the following sentence in Vinberg "linear representations of groups" pg.58: "Since isomorphic representations are given in compatible basis by identical matrices, we see that isomorphic ...
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2answers
27 views

Product of a class function with a conjugate of an irreducible character

Let $\chi$ be an irreducible complex character of a finite group $G$ and define $f:G \to \mathbb{C}$ by $f(g)=|\{h \in G:h^2=g\}|$. From a question which I am trying to solve it appears that $f(g)\...
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41 views

Irreducible complex characters of a finite group

Let $G$ be a finite group. If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$. We write $\nu(\chi):= ...
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19 views

Hermitian inner product of a character and its complex conjugate

We let $G$ be a finite group. If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$, and define $\chi^{(2)...
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Question on irreducible complex characters

We let $G$ be a finite group. If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$. We write $\nu(\chi)...
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Inner product of a character

We let $G$ be a finite group. If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$, and define $\chi^{(2)...
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1answer
33 views

Character of representation of $D_5$ on $\mathbb{R}^5$.

How does one stablish the character of a representation of $D_5$ on $\mathbb{R}^5$? The problem lies in the fact that I am unsure as to what the rotation and reflection matrices will look like in ...
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1answer
37 views

Complex character of a finite group $G$

We let $G$ be a finite group. If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$, and define $\chi^{(2)...
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1answer
28 views

Tensor product between a representation and its dual

If $\chi$ is the character of an irreducible representation of a finite group $G$ such that $\chi(1) > 1$, then I want to prove $\chi \chi^{*}$ is never irreducible. My idea was to show $\sum_{g \...
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Show that the characters of the representations $\phi_{n}$ of $SU(2)$ constitute a complete orthogonal set.

The question is given below: And the other questions mentioned are (I know the solutions of all of them): Sorry for the bad formulation of the my question at the first time I have ...
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Frobenius-Schur indicator and 1-dimensional representations

If I denote $FS(V)$ the Frobenius-Schur indicator of the representation $V$ with character $\chi$, I know $FS(V) \in \{0,1,-1\}$. Now, if I suppose that $V$ is 1-dimensional representation of a finite ...
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1answer
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Why the character of any representation is a central function?

Why the character of any representation is a central function? Could anyone explain this for me please?
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1answer
91 views

Relation between a finite group and the Galois group for the field extension generated by the character table entries

Let $E$ be the extension of $\mathbb{Q}$ generated by the character table entries of a finite group $G$. Let $F$ be the Galois closure of $E$. Examples: - $G=C_n$, $E=F=\mathbb{Q}(\zeta_n)$ and $...
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2answers
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Is the homomorphism $\mathbb{Q}G\to \prod M_{\chi_i(1)}(\mathbb{Q})$ given by $x \mapsto (\rho_i(x))_i$ an isomorphism?

If we have the group algebra $\mathbb{Q}G$ and ${\chi_1,...,\chi_n}$ the irreducible characters of $G$ afforded by the representation $\rho_1,...,\rho_n$, is it true that the map: $\mathbb{Q}G\to \...
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1answer
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Additive characters of $\mathbb{C}_p$

Consider $x \in \mathbb{C}_p$ with $|x|<1$ then for $a \in \mathbb{Z}_p$ we have the characters $$ a \mapsto (1+x)^a $$ where $(1+x)^a= \exp(a\log_p(1+x))$ My question is : it's possible to ...
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1answer
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Calculate the characters of the left and right regular representationsof an arbitrary finite group.

Calculate the characters of the left and right regular representations of an arbitrary finite group. The answer of the question is given below: But I do not know why the character of the left and ...
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Convolution of trig polynomials over a Group

I want to prove that $T(G)=T(G)*T(G)$ where G is an infinite compact abelian Hausforff Topological group. I'm trying to start this but really im confused with the convolution. Say $f,g \in T(G)$ I ...
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1answer
36 views

What is the span of the symmetric group $S_{5}$?

My professor while writing the character table of $A_{5}$, uses that the span of $S_{5}$ is $\{e_{1} - e_{2}, e_{2} - e_{3}, e_{3} - e_{4}, e_{4} - e_{5} \}$ but I do not know why this is the span of $...
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1answer
25 views

Solution Verification: Irreducible representations of $C_n$ present in $V \otimes V$.

Let $C_n$ be the cyclic group order n. Let V be the faithful two dimensional representation (over complex field) denoted by: \begin{equation} \rho(g^j)= \begin{pmatrix} \omega^j & 0 \\ 0 &...
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1answer
38 views

Character of An Element in the Center of a Group Representation

Let V be an n-dimensional irriducible complex representation of a finite group G, let C be it's center. Show that $|\chi (s)| = n $ when $s \in C$. Where $\chi$ is the character function. My ...
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Prove that the character induced from $\langle x \rangle$ to $D_n$ is irreducible

If we take the dihedral group of order $2n$ for odd $n$, $D_n=\langle x, y | \, x^n=y^2=1, yxy=x^{-1} \rangle$, then we have $\frac{n-1}{2}$ complex characters, that are induced from $\langle x \...
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Prove that $\left|\sum\limits_{m=N+1}^M\frac{\chi(m)}{m}\right|< \frac{2}{N+1}\sqrt{k}\log k$.

This problem is from Apostol's Analytic Number Theory, Chapter $9$. The problem: Let $\chi$ be a primitive Dirichlet character mod $k$. Prove that if $N< M$ we have $$ \bigg|\sum_{m=N+1}^M\...
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1answer
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Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.

This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...
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Group characters are eigenfunctions of the Laplacian with eigenvalue proportional to the quadratic Casimir

For the finite dimensional irreducible representations of $SU(2)$ we have that the group characters $\chi_n(g)$ for the $n^{th}$ representation are eigenfunctions of the Laplacian over the group ...
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2answers
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6-Dimensional Irrep of $S_5$

So I am computing the character table for $S_5$, and the only thing I have yet to understand is how we know the character values for the row relating to the 6-dimensional irrep. The irreps I have are: ...
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Character of representation of map from a finite set $M$ to $\mathbb{C}$

Let $G$ a finite group which act on a finite set $M$. Let $C(M) : = Map(M, \mathbb{C}:= \{f: M \rightarrow \mathbb{C} \}$ the vector space of complex values functions from $M$. The group $G$ act on $C(...
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1answer
109 views

understanding the irreducible Brauer characters of a defect-1-block

In section XII of his famous paper from 1966, Janko investigated the principal 11-block of his group $J_1$ (and thereby finally proved existence and uniqueness of this group). I would like to learn ...
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1answer
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Embedding $\mathbb C_p$ into $\mathbb C$ and vice versa…?

I'm reading Washington's book on cyclotomic fields, and he mentions that it is sometimes convenient to embed $\mathbb{C}_p$ into $\mathbb C$ and vice versa. In my mind, $\mathbb C_p$ and $\mathbb C$ ...
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1answer
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Proving this Hall algebra is commutative without Matlis duality

For a finite abelian $p$-group $G$ we have that $$ G \simeq \mathbf{Z}/(p)^{\lambda_1} \oplus \dotsb \oplus \mathbf{Z}/(p)^{\lambda_r} $$ for some positive integers $\lambda_1 \geq \dotsb \geq \...
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Cayley and character tables for the lamplighter group

Let us define the lamplighter group $L(G)$ on the group $G$ is defined as a semi direct product, $L(G) := \mathbb{G} \ltimes \sum_{x\in G}\mathbb{Z}_2$, with the direct sum of copies of $\mathbb{Z}_2$ ...
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all one-dimensional representations of a group G (over algebraically closed field) can be obtained us one-dimensional representations of G/G'.

How can I prove that: all one-dimensional representations of a group G (over algebraically closed field) can be obtained us one-dimensional representations of G/G'. Do I have to use this definition ...
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1answer
88 views

About a Corollary of Isaacs' “character theory of finite groups”: does the converse implication hold?

Let $N \lhd G$ (with $G$ finite) and let $\chi \in \mathrm{Irr}(G)$ be such that $\chi_N=\theta \in \mathrm{Irr}(N)$. Then the characters $\beta \cdot \chi$ for $\beta \in \mathrm{Irr}(G/N)$ are ...
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1answer
37 views

Characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 $

Follow up question to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$, how would I find the characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$? The Cayley table for this group: \begin{align*} \...
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1answer
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Characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2$

From the Cayley table: \begin{align*} \begin{array}{c | c c c c } & (0,0) & (0,1) & (1,0) & (1,1)\\ \hline (0,0) & (0,0) & (0,1) & (1,0) & (1,1)\\ (0,1) &...
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Calculate the character degrees of a finite group $G$.

Let $G$ be a finite group and $K$ be a group of order $8$. Suppose that $G/K\cong M_{12}$ where $M_{12}$ is one of the Mathieu group. QUESTION: How to calculate the all complex character degrees ...
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37 views

Show that $\chi(\cdot)$ is a non-trivial character on $(\mathbb{Z}/p\mathbb{Z})^{\times}$.

Let $G = \mathbb{Z}/p\mathbb{Z}$ with $p$ an odd prime. If $p \nmid a$ then multiplication by $a$ on the elements of G is bijective and therfore this is an permutation on G. Define $\chi(a)$ as the ...
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1answer
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How to show that, in this case, all the Sylow $p$-subgroups of $G$ are abelian.

Let $G$ be a finite, simple group of order $n$. Let $p$ be a prime divisor of $|G|$ and suppose that the number of conjugacy classes of $G$ is $> \frac{n}{p^2}$. Then all the Sylow $p$-subgroups of ...
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1answer
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False proof that $\langle\chi,1_G\rangle$ need not be an integer.

I'd like to know where the following calculation has gone wrong. I'm sure it is a silly error. Let $G$ be a finite group acting on the right cosets $G/H$ of $H\le G$. Let $\chi$ be the character of ...
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1answer
28 views

Character group of torsion-free abelian group

Let $G$ be a torsion free abelian group. Consider the character group $\hat G :=Hom_\mathbb Z (G,\mathbb C^\times)$ which is the group of all group homomorphisms from $G$ to $\mathbb C^\times$. When ...
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3answers
102 views

Unitary Central Character by Schur's Lemma

Consider an irreducible smooth representation $\pi$ of the group $G=GL_n(\mathbb{Q}_p)$ with center $Z$. Does there exist a unitary central character for $\pi$? More precisely, is there a (quasi-)...
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1answer
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Relation between the order of an element of a group and their character in a simple group

Let $\chi$ be the representation of a finite group $G$. Let $g \in G$ be an element of order 2. If $G$ is a simple group but not cyclic of order 2, prove that $\chi(g) \equiv \chi(1) \mod 4$. Proof ...
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31 views

How exactly does character inflation work?

Let $A_4$ be the alternating group and let $V = {\{(1), (12)(34), (13)(24), (14)(23)}\}$ be a normal subgroup of $A_4$. Then $A_4/V \simeq C_3$, so $A_4$ has $3$ one dimensional representations, which ...
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3answers
70 views

irreducible representations and character table of $D_6$

Let $$D_6=\langle a,b| a^6=b^2=1, ab=ba^{-1}\rangle$$ $$D_6=\{1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}$$ I would like to compute its character table and its irreducible representations. I will ...
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Why the characters of the Minkowski spacetime translation group involve the Minkowski metric?

The translation group of Minkowski spacetime is just the additive group $\mathbb{R}^4$. Indeed, if $x\in \mathbb{R}^4$ is a point in Minkowski spacetime, the translation $T_v$ acts on $x$ by $$T_vx=x+...