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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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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
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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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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
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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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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
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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
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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
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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 ...
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Charater of induced representation

Suppose we have an induced representation of $\theta: H \to GL(W)$, we define the space $$ V := \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W, $$ $V$ has as a basis $\{e_r \otimes_{\mathbb{C}[H]} w\}_{r \...
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Irreducible Characters & Representations of a Cube

Let $A_4$ act on the four long diagonals (labeled $1,2, 3, 4$) inscribed in a cube (which is $S_4$). Then $A_4$ acts on the faces, the edges, and vertices of the cube. This gives rise to three ...
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Trying to understand why the zeta function is a rational function under certain conditions. Questions about some equations.

Information: I linked the pages below, which relate to my questions. I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th ...
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1answer
90 views

Question about characters ( Section : The Rationality of the Zeta Function associated to $a_0x_0^m+a_1x_1^m+…+a_nx_n^m$ )

I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this ...
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2answers
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A morphism form $G$ to $\mathbb{C}^*$, character what does it represent

I've just begin a course on character theory. Juste to repeat we say : Let $G$ be a finite group. Then a character $\chi$ is a morphism from $G \to \mathbb{C}^*$. We then have some property on ...
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Induction representation of the center is not irreducible

Suppose $\varphi: Z(G) \to GL(V)$ is an irreducible representation of the center of a non-abelian group $G$. I want to show that $Ind^G_{Z(G)} \varphi$ is not irreducible. Any hints? So far, I have ...
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Is there an analogue of the Mertens function for the generalised Riemann conjecture

It is known that the Riemann conjecture is equivalent to $$M(x) = O(x^{\frac12+\epsilon}),$$ where M(x) is the Mertens function. Does there exist an analogue to this equivalence for the generalized ...
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1answer
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Affine $\mathfrak{su}(2)_k$ characters and Jacobi triple product

In this post, the Kac character formula for affine $\mathfrak{su}(2)_k$ $$\chi_{\ell}^{(k)}(\tau,z) = \frac{\Theta_{\ell+1,k+2}(\tau,z)-\Theta_{-\ell-1,k+2}(\tau,z)}{\Theta_{1,2}(\tau,z)-\...
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1answer
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Grading of the character group of a maximal Torus in a reductive group

I am working through the paper "On the Algebraic $K$-Theory of Some Homogeneous Varieties" by Alexey Ananyevskiy and got stuck at the beginning of the second section. The set up is the following: Let ...
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1answer
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Relation between the order of an element of a group and their character

I am struggling with a proof of a two part question: Let $\chi$ be a character of a finite group $G$. a) If $g$ has order 2, then $\chi(g) \in \mathbb{Z}$ and $\chi(g) \equiv \chi(1)$ (mod 2) ...
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2answers
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How to test if a $\mathbb{R} G$-module is irreducible?

Let $V$ be a $\mathbb{C} G$-module with character $\chi$. We know that $V$ is irreducible if and only if the inner product $\left<\chi,\chi\right>=1$. But what if $V$ is a $\mathbb{R} G$-module?...
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1answer
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If $V$ is a $\mathbb{C} G$-module whose character is real then dim$V$ and dim $V^G$ have the same parity.

Let $G$ be a finite group of odd order. Prove that if $V$ is a $\mathbb{C} G$-module whose character is real, then dim$V$ and dim$V^G$ have the same parity. where $V^G:=\{v \in V \mid vg=v \forall g \...
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Square-free numbers congruent to $1$ modulo $p$ (asymptotic formula)

Knowing that $\sum_{n\leq x}\mu^2(n)=\frac{6}{\pi^2}x+O(\sqrt{x})$, prove that: $$\sum_{n\leq x,\,\,n\equiv 1(\text{mod }p)}\mu^2(n)=\frac{6}{\pi^2(p-1)}x+O(\sqrt{x})$$ Using Dirichlet charaters,...
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1answer
60 views

Schur's Lemma and $Z(G)$

Let $Z(G)$ be the centre of G. Let $V$ be an irreducible $G-$space with matrix representation $\rho_v$. Let $z \in Z(G)$, then I'm trying to show that $\rho_v(z)$ is multiplication by a root of ...
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1answer
53 views

Order of an element and its character

Let $G$ be a finite group and let $\chi$ be its character. Suppose $g \in G$ as order $3$ and $\chi(g) \in \mathbb{R}$. Show that, in fact, $\chi(g) \in \mathbb{Z}$ and $\chi(g) \equiv \chi(1)mod3$. ...
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1answer
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$\chi$ is a character and $\chi(g) \in 2\mathbb{Z}$ then $\frac{1}{2}\chi$ is also a character.

Suppose $G$ is a finite group. Let $\chi$ be the character of some $\mathbb{C}G$-module with the property that $\chi(g)$ is an even integer for every $g \in G$. Is it true that $\chi/2$ defined by $\...
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1answer
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Character table and conjugacy class of cyclic subgroups

Let $G$ be a non-abelian finite group. Let $C_1$ and $C_2$ be distinct conjugacy classes in $G$ with following conditions: 1) $C_1$ and $C_2$ contain of elements of same prime order $p$. 2) For ...
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Local factors determine local Weil representations - proof of the Artin representation case

This post can be seen as a continuation of this post I created on MathOverflow. I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
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Why chemists are interested in character tables?

I'm a french PhD student and I'm working on representation theory (of non-compact Lie groups). By the way I try to give a "concrete" sense to character tables of finite groups and I have a lot of ...
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What is a tensor product

I know this question has already been answered many times elsewhere. I now formally the definition of a tensor product, however I do not get the point of it: why is it used ans how to think of it? To ...
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1answer
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construction of a new character by conjugating another character

In the wikipedia page about Clifford theory, they are constructing a new character by conjugating another character. ie, Let N be a normal subgroup of G, If μ is a complex character of N, then for a ...
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What is a primitive character of a Galois groups of a finite cyclic extension of local fields?

Let $K$ be a local field and $F/K$ be a cyclic extension of degree $n$, meaning that $n = [F:K]$ and $\operatorname{Gal}(F/K) \simeq C_n$ is cyclic. In the proof of Lemma 2 of the paper "Euler ...
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The Stickelberger relation in Ireland&Rosen's Number Theory book

Could anyone please take a look at the very last expression at the very bottom of Ireland&Rosen page 214? I am refering to their A Classical Introduction to Modern Number Theory, 2nd Edition. The ...
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Hook-length formula [closed]

Let $\lambda=(\lambda_1,\ldots,\lambda_n)$ be a partition of $d$. Then hook length formula gives us $$dim(\lambda)=\chi_{1^d}^{\lambda}=\frac{d!}{\prod h_{\lambda}(i,j)}$$ where $\chi_{a}^{b}$ denote ...
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2answers
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Interest of some Dirichlet characters

I bumped into papers interested in the following characters $$\chi_m(n) = \left\{ \begin{array}{cl} \left( \frac{m}{n} \right) & \text{if } m \equiv 1 \mod 4 \\ \left( \frac{4m}{n} \right) & \...
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1answer
27 views

Dual Character for Compact Lie Groups

I've been reading Brian Conrad's notes on compact Lie groups and came across the assertion $$\chi_{V^*} = \overline{\chi_V} $$ where $V$ is a finite dimensional complex representation of a compact Lie ...
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1answer
59 views

Complex-valued integrals on Lie groups

I've been learning about some representation theory of compact Lie groups, and one of the $G$-averaging tricks has thrown me for a bit of a loop. Given a finite group $G$, I'm familiar with the trick ...
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0answers
28 views

Partial summation of character $\chi \pmod{p}$?

In Ch. 8 (p. 142) of the Opera de Cribro (Friedlander, Iwaniec) it is written: To this end we apply the sieve to remove from each $p$ in a set $\mathcal{P}$ residue classes which are not squares ...