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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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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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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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35 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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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
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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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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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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
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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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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
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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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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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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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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
133 views

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
56 views

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
27 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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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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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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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
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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+...
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Jacobi sums Gaussian Sum. Show $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$

I want to show that $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$, where $\rho^{'},\chi^{'}$ are characters of a finite field $F_{p^s}$ and $\chi,\rho$ are characters a finite field $F_p$. My work: I ...
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Is there any relation between real and complex character functions of irreducible representations of compact lie groups?

Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $\chi_U^\mathbb{R}:G\to\mathbb{R}$ as $\chi_U^\mathbb{R}(g)=\operatorname{Tr}(l_g)$. If $V$ is a complex $...
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Induced characters of $G$ from a normal subgroup $H$

Let $H \lhd G$ and let $\chi$ be a character of $H$. Let $g \in G$ and let $H^g = gHg^{-1}$. Define $\chi^g$ to be the class function on $H^g$ given by $\chi^{g}(x) = \chi(g^{-1}xg)$. Suppose that $\...
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orthogonality relation for characters

Let $\rho : G \to GL(V)$ be a representation on G. Then, its character is defined as $\chi_V(g) := Tr(\rho(g)) $, where $Tr$ denotes the trace function. For an exercise I am trying to solve, I would ...
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Trying to understand a certain form of zeta function

A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate. facts: Let $N_s$ be the number of points on the projective hypersurface $\bar{H}_f(...
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1answer
40 views

The Character/Weight of a Representation of an Algebra

This may well be something of a silly question, but if so, then all the more reason I get it straightened out. I have in the past been working with representations of both groups and algebras, and in ...
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2answers
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Find all irreducible characters of a matrix group on finite field $\mathbb F_5$

Find all irreducible characters of matrix group $G =\left\{ \left( \begin{array}{cc} a & b \\0 & a^{-1}\end{array} \right)|\,\,\, a,b \in\mathbb F_5, a\not=0 \right\}$. The former question is ...
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1answer
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Irreducible $G$ spaces and characters

Now, I've been learning about character theory and I've been building up to showing that the character table is square; the number of irreducible characters is equal to the number of conjugacy classes....
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1answer
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Some easy questions about multiplicative characters and Jacobi sums.

First I want to give you some context. Then I will ask my questions. I think that my questions are easy and fast to answer, so I've decided to put them together in one question here. Context ...
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Why is number of real characters mod $q$ a multiplicative function?

Let $R(q)$ be the number of real characters mod $q$. A character $\chi \mod q$ is called real if $\chi(a)\in\mathbb{R}$ for every $a\in \mathbb{Z}$, which means $\chi(a)\in\{-1,1\}$ for every $a\in\...
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Differentiating a $p$-adic character

Let $L$ be a finite extension of $\mathbb Q_p$ with ring of integers $\mathcal{O}=\mathcal{O}_L$ and let $B_1(L):=\{z \in L \colon \vert z-1 \vert <1 \}$. Let $\widehat{\mathcal{O}}(L)_{\mathbb ...
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$\chi$ varies over characters of $F$ of order dividing $m$, $\chi^{'}$ varies over characters of $F_s$ of order dividing $m$

A month ago I've asked two questions about rationality of the zeta function. The pages that belongs to my question are (linked here) Unfortunately I'm still clueless, but some steps are clear now. ...
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Validity of a result from character theory for finite field.

Corollary $11.29$ of the book Character Theory of Finite Groups by I. Martin Isaacs as given: Corollary $11.29:$ Let $H$ be a normal subgroup of $G$ and $\zeta\in Irr(G), \xi\in Irr(H)$, be a ...
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Do “$K/k$ twisted” representations exist?

Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $V\otimes_k K\cong W\otimes_k K$ as $K$-representations, do we have that $V\cong W$? Being ...
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2answers
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Brauer Character in GAP.

What is command to obtain Brauer Character in GAP? In magma, it is like ...
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2answers
68 views

Unique character of order 2

"The multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$of reduced residue classes modulo an odd prime p is a cyclic group of (even) order p − 1. Thus it has a unique character of order 2." Why ...
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60 views

On a $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...
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2answers
85 views

Proving the trace of a representation is equal to zero

I'm having some trouble in beginner's representation theory and am pretty lost about this problem: Let ($\rho$, $V$) be a representation of $G$, so $\rho$: $G$ $\to$ $GL(V)$ is a group homomorphism. ...
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1answer
56 views

Character Table from Generators of a Group

Let $G=\langle x,y|x^5=y^4=yxy^{-1}x^{-2}=1\rangle$ be a group. How would I construct the full character table of this group with no other given information? Here is what I know regarding characters:...
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1answer
40 views

orthogonal group what does it represent

Let $A$ be an finite abelian group and $B$ be a subgroup of $A$. Then we defined the orthogonal of $B$ : $$B^{\perp} = \{f:(A,+) \to (\mathbb{Q}/\mathbb{Z},+) \mid \forall b \in B ,f(b) = 0 \}$$ I ...