Questions tagged [characters]
For questions about characters (traces of representations of a group on a vector space).
1,064
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Solution to system of nonlinear equations
I am trying to comple the character table of a finite group with seven conjugacy classes $c_1, \cdots, c_7$:
Character table
If I use the orthogonality of the column vectors of table I get a system of ...
3
votes
1
answer
38
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Why do characters fail to characterize non-abelian LCH-groups?
For any locally compact Hausdorff abelian group (LCA group) $A$, a character $\xi\colon A\mapsto\mathbb{T} $ is definined as a continuous group homomorphism to the unit circle $\mathbb{T}\subseteq\...
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Uniqueness of characters of transitive actions
Let a finite group $G$ act on a set $X$ transitively by permuting its elements. Then, $|X| \big| |G|$ by the orbit-stabilizer theorem. Let this action induce a permutation representation $\rho: G \to ...
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Help with proof of $\langle\chi,\psi\rangle=\delta_{\chi,\psi}$ with $\chi$ and $\psi$ characters of a group.
In The Symmetric Group, we are presented with the following theorem
Let $\chi$ and $\psi$ be irreducible characters of a group $G$. Then $$\langle\chi,\psi\rangle=\delta_{\chi,\psi}\tag{1} $$
In the ...
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p-adic Norms and Character Values [closed]
Let $G$ be a group with a $\mathbb{Q}_p$-representation $\mathfrak{X}: G \to \mbox{GL}_n(\mathbb{Q}_p)$ affording the character $\chi$. We have the $p$-adic norm $\| \cdot \|$ defined on an extension ...
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Character of induced representation - problem
Let $G$ be a group and $H$ a subgroup and let $X_H$ be a representation of $H$. According to the book Algebra by Cohn the character of the representation of $G$ induced by $H$ is given by
$$\tag{1}
\...
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Existence of irrep containing a particular element for a finite group
Let $ G $ be a finite group.
Idea: Can any matrix that "looks like" it belongs in $ G $ be found in the image of some faithful irrep of $ G $?
Formal statement of question:
Let $ G $ be a ...
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35
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Weil-Carlitz-Uchiyama bound for Weil sum of high degree
Let $\mathbb F$ be a finite field of characteristic $p$, and size $q=p^n$.
Let $\chi$ be a multiplicative character of order $m$.
Then, it is well-known that the Weil sum is bounded as follows.
$$W_{\...
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Trouble with fourth type of irreducible character of GL2(F) Serge Lang algebra
so it's been 1 month since I started reading Serge Lang Algebra, I'm now pretty advanced in my reading of the book but I am stuck at p 721, here it is :
So here Lang is computing and classifying all ...
- 61
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Explicit Computation of Characters of Finite Field $\mathbb{F}_d$
I have a question about the computation of characters of $\mathbb{F}_{2^n}$ in arXiv:quant-ph/0410155. My question might be trivial but I'm not very familiar with details in the theory of finite ...
- 11
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A question about hypothesis of Clifford's Theorem in Isaacs Character Theory book
I've been studying the book of Isaacs of Character Theory of Finite Groups.
Clifford's Theorem (Theorem 6.2 of the book) states the following.
Theorem: Let $H\lhd G$ and let $\chi\in\text{Irr}(G)$. ...
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Indecomposable/Extreme Group Characters and Direct Sum Decompositions
I am currently working through the details of theorem 2.10 in this paper. There are a few things which don't make sense to me, and I'm hoping someone could help me work through the details of the ...
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Complex conjugation and outer automorphisms of finite groups
Context: A finite group has all characters real valued if and only if every element $ g \in G $ is in the same conjugacy class as $ g^{-1} $. This property has some special name but I can't remember ...
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2
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Character of restricted representation.
My lecture notes state that the restriction of an irreducible character is in general not irreducible and gives the following example:
"Restricting any non-linear irreducible character to the ...
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Character of induced representation
Let $H$ be a subgroup of a group $G$ and let $\rho^G$ be the representation of $G$ induced by a representation $\rho$ of $H$. My book Algebra Vol. 2 by Cohn states that if the character of $\rho$ is $\...
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Existence of non-degenerate alternating pairing $ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $ and perfect square cardinality
Let $A$ be a finite abelian group, and let
$ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $
be an alternating, non-degenerate bilinear form on $A$.
To prove $A$ have square cardinality, Non-degenerate ...
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Is there a group $G$ with $|G|=8$ and $\chi$ assuming values $-2,-1,0,0,0,0,1,2?$
I want to understand if there is a group $G$ with $|G|=8$ and a character $\chi$ of $G$ assuming values $-2,-1,0,0,0,0,1,2$.
Let $\rho: G \rightarrow GL(V)$. Because $\chi(1_G)=\dim V$, we must have $...
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Generalization of the Great Orthogonality Theorem?
Consider the sum
$$
\Pi_\tau = \frac{1}{|G|} \sum_{g \in G} \chi_\tau^*(g) \, (\mu^\star \otimes \nu)(g),
$$
where $G$ is a finite group, $\mu$, $\nu$, and $\tau$ are irreps of $G$, and $\chi_\tau$ is ...
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1
answer
34
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Character of a representation and dimensione of $G$-invariant space
Let $G$ be a finite group. Let $V$ be a finite dimensional vectorial space on an algebraically closed field and let $\rho: G \to Gl(V)$ an irreducible linear representation of $G$ in $V$.
Since $V$ is ...
- 387
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If $H$ has a normal complement then it has the character restriction property
I am trying to show that if a subgroup $H$ of a group $G$ has a normal complement then $H$ has the character restriction property. That is every irreducible character of $H$ is a restriction of some ...
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Why is the determinant of a character well-defined?
In Karpilovsky's Group Representations (vol. 1) it can be read in chapter 27, that if $\chi$ is a character of $G$, and $\rho \colon G \to \mathrm{GL}_{n}(\mathbb{C})$ is a representation which ...
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Generalization of convolution of characters to irreps
Let $ G $ be a finite group. Let $ \pi_i, \pi_j, \pi_k $ be irreps of $ G $. Then consider the sum
$$
f(x):= \frac{1}{|G|} \sum_{g \in G} \chi_i(xg^{-1}) \chi_j(g) \pi_k(g)
$$
If $ \pi_k $ is the ...
- 7,020
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votes
2
answers
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Is there some sort of irrep orthogonality for finite groups?
Disclaimer: As pointed out by several people the original orthogonality condition I was asking about seems to be incorrect so I'm just going to change the question to the orthogonality condition ...
- 7,020
2
votes
1
answer
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Kernel of an operator constructed from an irrep of a finite group
Let $ \pi: G \to GL_d(\mathbb{C}) $ be a degree $ d $ irrep of a finite group $ G $ with character $ \chi_\pi $. Consider the linear map $ T: \mathfrak{gl}_d(\mathbb{C}) \to \mathfrak{gl}_d(\mathbb{C})...
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2
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What is the number of irreducible characters of $G$ and their dimensions when $|G|=p^3$ is non abelian?
I know $G'=Z(G)$ and $|Z(g)|=p$. $|G/G'|=p^2$ is the number of irreducible characters of dimension $1$. If $\chi $ is the characteristic of the regular representation
$$p^3=|G|=\langle\chi,\chi \...
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When can we divide characters in character table?
For $G$ a finite group and two $\mathbb CG$ modules $V$ and $W$, then (for $\chi_V$ the character ie the trace of the $G$-action) we have that
$$
\chi_{V \otimes W} = \chi_V \cdot \chi_W
$$
is a ...
- 431
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answer
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If $\chi$ is a complex-valued character of a representation of a finite group, is it always true that $\overline{\chi(g)}=\chi(g^{-1})$?
I have learned from Maschke's theorem that there exists an inner product on any $G$-module of a finite group $G$ that is invariant under the action of $G$. That means that we can select an orthonormal ...
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If a hom. $\phi:G\to H$ of diagonalisable linear algebraic groups is injective, then the induced hom. $\phi^*:X^*(H)\to X^*(G)$ is surjective
This is Exercise 3.2.10(2) of Springer's book, "Linear Algebraic Groups (Second Edition)". According to Approach0, it is new to MSE.
The Question:
Let $\phi:G\to H$ be a homomorphism of ...
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Sum of squares of coefficients appearing in $Res_{H}^{G}$
On a representation theory course, we were given the following Proposition:
Let $G$ be a finite group, $H$ a subgroup and $χ_{H,1},\dots,χ_{H,r}$ the irreducible characters of $H$. Let also $χ$ be a ...
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Terminology: Formula for group characters and idempotent
Background.
Let $G$ be a finite group and $K$ an algebraically closed field whose characteristic does not divide the order of $G$. Let $S_1,\ldots S_n$ be representatives of the isomorphism classes of ...
- 1,039
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Same character values iff related by outer automorphism, for perfect groups
Let $ G $ be a finite perfect group. Let $ \chi_1, \chi_2 $ be two different irreducible characters of $ G $. Suppose that the set of values taken by $ \chi_1 $ is the same as the set of values taken ...
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Is the map sending everything to 0 conventionally considered a complex character?
Fix a finite group $G$. Would the function sending each $g \in G$ to $0$ usually be considered a character over $\mathbb{C}$? (character in the sense of representation theory - I'm not sure whether ...
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1
answer
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On the proof of: If $0<\frac{|χ(g)|}{χ(1)}<1$ then $\frac{χ(g)}{χ(1)}\notin\overline{\mathbb{Z}}$
In class we proved the following theorem:
Let $σ:G\rightarrow GL(n,\mathbb{C})$ be a representation of $G$ and let $χ$ be the corresponding character.
Then $\forall g\in G$ we have:
If $0<\frac{|χ(...
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votes
1
answer
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Legendre symbol as a group character
I’ve read that the Legendre symbol $$\left (\frac{\cdot}{p} \right ): x \mapsto \left (\frac{x}{p} \right ) $$
is a character of the group $G=(\mathbb{Z}/p\mathbb{Z})^\times$.
However, that means the ...
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votes
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answer
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Definition of a character of a linear algebraic group
This might sound like a silly question, but I seem to be getting different definitions from different texts.
Let $k$ be some field and $G$ some linear algebraic group over $k$. Now, the standard ...
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Compactness of the "orthogonal" of a maximal isotropic subgroup modulo this subgroup
I am currently facing this question: I have a locally compact abelian group $B$, which is $2$-regular ( $x\mapsto 2x$ is invertible), endowed with a bicharacter $h(x,y)$ so that $x\mapsto h(x,\cdot)$ ...
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A Relation between Dirichlet L functions for Quadratic Character and Gauss sums
This is problem 8 of Chapter 16 in the book A Classical Approach to Modern Number Theory by Ireland and Rosen.
Let $g(\chi)$ be the classical Gauss sum : $\sum_{x = 1}^{p-1} \chi(x) \zeta^{x} $, $\chi$...
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Frobenius reciprocity for reductive groups
Let $G$ be finite group and $H$ its subgroup. We assume that everywhere the base field is $\mathbb{C}$. For any characters $\phi$ of $G$-representation and $\psi$ of $H$-representation Frobenius ...
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Why does character of identity element always positive integer?
I'm a chemist who is interested in mathematics. One of the subject that I'm interested in is group theory and representation theory, because of its application in chemistry.
I was looking at ...
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Show a character of $A_5$ is integer-valued
Let $G$ be the alternating group $A_5$, let $\chi_1$ be the trivial character of $G$, and let $\chi_2$ be the irreducible character of $G$ given by
where $\alpha = (\sqrt{5} + 1)/2$ and $\beta = (-\...
- 2,704
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answer
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Isomorphic group of $D_8/C_2$
In my exercise I have to find all one dimensional representations of $D_8/C_2$ and their characters and they give us the hint 'which group is this isomorphic to?'. I have tried to maybe find a ...
2
votes
1
answer
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Confused about the domain of characters
In Stein's Fourier Analysis Ch.7 he defines characters as follows.
Let $G$ be a finite Abelian Group with operation $*$, and $S^1$ be the unit circle in $\mathbb{C}$. $e: G \to S^1$ is a character on ...
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Character table guarantees certain property of groups
How to show there exist three conjugacy classes in a certain group $G$ such that the products $xyz, x\in C_1, y\in C_2, z\in C_3$ divide equally over all elements of $G$ by looking at the character ...
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Let $\chi(g)$ is a nonnegative real number. Show that if $\chi$ is irreducible, then $\chi$ is the trivial character
I am working on a problem from my abstract algebra class:
Let $\chi$ be a character of a group $G$ with the property that $\chi(g)$ is a nonnegative real number for all $g \in G$. Use the row ...
- 2,704
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votes
0
answers
48
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Irreducible representation over finite field GAP
I have a finite group $G$ with $p\not \mid |G|$ and I want to compute an explicit modular representation of $G$ given a character $\chi$. I can compute a representation by ...
- 545
1
vote
1
answer
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Finding linear relations using character table
My question comes from a section of Representations and Characters of Groups, by James & Liebeck. They discuss the following example in the chapter $15$ on The Number of Irreducible Characters
$...
- 2,704
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1
answer
60
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Irreducible Characters and the Dual Space
I've taken courses previously on representation theory and Harmonic Analysis, and I'm working on some research which has taken me to studying Haar measures and integration over general locally compact ...
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0
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33
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Finding $\chi _{\text{det}(V)} $ for two-dimensional representation
So I am trying to work out what $\chi _{\text{det}(V)} $ is for the two-dimensional matrix representation of $D_8 $ where $D_8$ is the group of symmetries of the square.
I have found the character ...
- 1,798
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1
answer
61
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Intuition for $\int_G\chi_V=\dim(V^G)$?
Let $G$ be a compact topological group with Haar measure $dg$. We can decompose a finite-dimensional complex representation as $V\cong \mathbb C^k\oplus V_1\oplus\dots\oplus V_n$, where $G$ acts ...
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votes
1
answer
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Does any valid character table correspond to a group?
I realize that this question is very open-ended since it's not entirely clear what a "valid" character table is.
I would like to know whether creating a character table that has all of the ...
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