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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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On Frobenius–Schur indicator of real/complex representations

Let $G$ be a finite group with complex irreps $W_i$. Let $V$ be a real irrep of $G$. Denote $\chi_{W_i}$ and $\chi_{V}$ the corresponding characters. Each $V$ has three possibilities: Case 1: $\dim_{\...
khashayar's user avatar
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Set of primitive idempotents of group algebras

Let $G$ be a finite group and $K$ be a field with characteristic zero. Can we construct the set of primitive orthogonal idempotents of $KG$? By this set, I mean the set of idempotents such that $...
khashayar's user avatar
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2 votes
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Irreducible and faithful group action and primitive prime divisor.

I'm trying to show that if a cyclic group $R$ acts Frobeniusly (hence faithfully) and irreducibly on $G=C_p^k$ (that is $C_G(r)=1$ for any $1\ne r\in R$), then $k$ is the smallest integer such that $|...
Simon's user avatar
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Can the sum of a nonlinear irreducible character's values on $Z(\chi)$ be zero? [closed]

I need a lemma for a research problem. Suppose that I sum the values of a nonlinear irreducible character $\chi$ of a finite group over the center of that character $Z(\chi)$. Is it possible for the ...
Hanklin's user avatar
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2 votes
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Brauer characters/ modular characters are not well defined

In chapter $15$ of Isaacs' "Character theory of finite groups", he defines a field $F$ of characteristic $p$, isomorphic to the algebraic closure $\overline{\mathbb{F}_p}$ of the prime field ...
GC.'s user avatar
  • 115
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20 views

Character of a kac-moody module as the sum of characters of Verma modules

I'm trying to prove the next result from Kac's book, Infinite dimensional Lie algebras. Let $V$ be a $\mathfrak{g}(A)$-module with highest weight $\Lambda$. Then $$ \text{ch}(V)=\sum_{\lambda\in B(\...
Rubén Túquerrez's user avatar
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41 views

What are the irreducible representations of $A_3$?

I've got some notes saying the Character table of the alternating group $A_3$ is given as in the attached image. I can't seem to figure out what the representations $\rho_1, \rho_2$ are supposed to be....
Merkel_Bot's user avatar
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24 views

Decomposition of primitive central idempotents in group algebras

Let $W$ be an irreducible $\mathbb{C}$-representation of a finite group $G$ with character $\chi_W$. A primitive central idempotents of the group algebra $CG$ is: $$e=\frac{\dim_{\mathbb{C}}(W)}{|G|}\...
khashayar's user avatar
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3 votes
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Exercise 2.19 in Isaacs's book on character theory

Here is the problem: Let $E=\langle x_1,x_2,x_3,x_4\rangle$ be an elementary abelian group of order 16.Let $P=\langle y\rangle$ be cyclic of order 3.$P$ acts on $E$ by $$x_1^y=x_2,x_2^y=x_1x_2,x_3^y=...
Little GTN's user avatar
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Problem 2.14 from Isaacs's Character Theory of Finite Group

I'm solving this problem from Isaacs's Character Theory of Finite Group: Let $H \subseteq G' \cap Z(G)$ be cyclic of order $n$ and let $m$ be the maximum of the orders of the elements of $G/H$. ...
QiQi's user avatar
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1 answer
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How to determine $\Gamma_{15}$?

For context,yesterday I asked How to determine $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4,\Gamma_5$?. Now I learned of new theorem that made me curious: Theorem: Let $m,n \in \mathbb{Z}$ such that $gcd(m,n)=...
NTc5's user avatar
  • 609
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1 answer
44 views

Non-commutative banach algebra

A theorem states that if $A$ is a commutative Banach algebra, then for all $a\in A$ we have $$\sigma(a)=\{\chi(a) : \chi \text{ is a character }\}$$ My question what if $A$ is not commutative. Does ...
vemapo's user avatar
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0 answers
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Generic bound on quadratic character sum

Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free polynomial over $\mathbb{F}_q[x]$. Then by the Weil bound, we have the generic estimate $|\sum_{x\...
Madarb's user avatar
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5 votes
1 answer
116 views

Dimension of the center of a block

Let $G$ be a finite group and let $F$ be an algebraically closed field of characteristic $p$. I've been studying some modular character theory from Navarro's "Characters and blocks of finite ...
Gauss's user avatar
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Projection of real representations onto the isotypic components

Let $(V, \rho)$ be a representation of a finite group $G$ over field $\mathbb{F}$ and $W_i$ be irreducible representations (irreps) of $G$ over $\mathbb{F}$ with dimension $d_i$ and character $\chi_i$....
khashayar's user avatar
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How can an element of order p define if a $\mathbb{C}G$-module or a representation is reducible or irreducible?

I have a question as homework ,but I don't understand it intiutively at all. Here's the question : Let $G$ be a finite group of order $|G|=pn$,where p is a prime, and U be a subspace of $\mathbb{C}G$....
GGG's user avatar
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2 votes
2 answers
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Let $P \in \text{Syl}_{5}(S_5)$. Show that the normalizer $N := N_{S_5}(P)$ is a monomial group.

Fix the field to $\mathbb{C}$. Let $P$ be a sylow $5$-subgroup of the symmetric group $S_5$. Let $N := N_{S_5}(P)$ be the normalizer of $P$. I want to show that $N$ is a monomial group, that is, that ...
Ben123's user avatar
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If $|C(g)|>|G|/2$, then g$\in Z(G)$?

I'm having trouble solving an exercise on characters of finite groups and I don't know what to do as I don't even understand the idea behind it. The question goes as follows : Let G be a finite group ....
GGG's user avatar
  • 347
-1 votes
1 answer
110 views

Character table for a covering group of $\mathbb{Z}_n \times \mathbb{Z}_m$

I’m considering the group $G = \mathbb{Z}_n \times \mathbb{Z}_m$ and its covering group $$G^* = \langle \alpha, \beta, a|\alpha a = a\alpha, \beta a = a\beta, a^p = 1, \alpha^n = 1, \beta^m = 1, \...
slowspider's user avatar
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7 votes
1 answer
133 views

Let $G$ be a non-abelian finite group whose all non-linear irreducible characters are faithful. Is there a classification of these groups?

Let $G$ be a non-abelian finite group whose all non-linear irreducible characters are faithful. Is there some kind of a classification of these groups? If $G$ is a finite non-abelian simple group, the ...
Riju's user avatar
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3 votes
1 answer
69 views

Kernel of a (complex) character.

Let $\psi$ be a character in $\textbf{Class}(G,\mathbb{C})$. Then $$\psi = \sum\limits_{\chi \in \text{Irr}(G)} n_{\chi}(\psi) \chi$$ where $\text{Irr}(G)$ is a representative set of irreducible ...
Ben123's user avatar
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1 answer
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What is the action on $\mathrm{Sp}_2(q^2)$ which makes $\mathrm{Sp}_2(q^2)\colon 2$ a maximal subgroup of $\mathrm{Sp}_4(q)$ for an even power of $q$?

I know that there's a homomorphic embedding of the finite field $\mathbb{F}_{q^2}$ into $2\times 2$ matrices over the field $\mathbb{F}_q$. But I cannot determine the action of the semi-direct product ...
NewViewsMath's user avatar
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27 views

Automorphism of a character and Frobenius morphism

Let $T$ be a torus, and let $X(T)$ be its characters group (the group of homomorphisms $T \to \mathbb{G}_m$). I am trying to understand the proof of the following result (Proposition 4.2.3) in Digne-...
Conjecture's user avatar
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2 votes
2 answers
130 views

Character values with absolute value $1$ are roots of unity

An algebraic integer with absolute value $1$ may not be root of unity. It is an exercise in Isaac's Charcter theory of finite groups that if a character value $\chi(x)$ has absolute value $1$ (for ...
Maths Rahul's user avatar
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How to find the spanning set of a subspace of particular dimension in terms of basis elements of vector space?

I am reading "Representations and Characters of Groups" by G James and M Liebeck. Here, they attempt to find out the spanning set of a subspace of V $\equiv \mathbb R^{12}$. So, there is a ...
Marisha's user avatar
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1 answer
50 views

find the order of a corresponding group element and prove that the element is not central [closed]

Suppose we're given a single column of a character table for a finite group as $\begin{bmatrix}1 \\ -1\\ 0\\ -1\\ -1\\ (-1+i\sqrt{11})/2\\ (-1-i\sqrt{11})/2\\ 0\\ -1\\ 0\end{bmatrix}$ . 1.Find the ...
user1127's user avatar
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0 votes
1 answer
62 views

how can you tell whether $\chi'-\chi$ is a character

If $\chi$ is an irreducible character and $\chi'$ is any character, how can you tell whether $\chi'-\chi$ is a character without constructing the corresponding representations? In order to verify ...
user1127's user avatar
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1 vote
3 answers
104 views

if the character of a g-representation has real values then it is realizable over the reals

Prove or disprove that if the character of a G-representation (where G is a finite group) has real values (i.e. $\chi_{\phi}(g)\in\mathbb{R}\forall g\in G$) then it is realizable over the reals (i.e. ...
user1127's user avatar
  • 469
2 votes
1 answer
135 views

Does isomorphism between group algebras imply equivalence between character tables?

As is already known, groups $D_8$ and $Q_8$ have equivalent character tables but $\mathbb{Q}D_8 \not= \mathbb{Q}Q_8$. Does anyone know if given an isomorphism between group algebras the character ...
ayphyros's user avatar
  • 323
0 votes
0 answers
24 views

How to decompose $\mathbb{R} G$ module in its irreducible submodules, for $\mathbb{R^{15}}$?

I am reading "Representations and Characters of Group" by Gordon James and Martin Liebeck. I want to apply this technique (14.27) to find irreducible submodules of symmetry group of ...
Marisha's user avatar
4 votes
1 answer
73 views

A polynomial from character theory

Let $G$ be a finite group, and define $a(n)=\#\{g\in G\mid o(g)=n\}$. In problem 5.18 of Isaacs' Character Theory of Finite Groups, the following polynomial is defined: $$F_G(X)=\frac{1}{|G|}\sum_ma(m)...
semisimpleton's user avatar
9 votes
0 answers
108 views

Character table of groups of order 96, coincidence or there is a reason?

It is well known that every column sum in the character table of a finite group is an integer. Nevertheless, it is not easy to find an example where a column sum is negative. The smallest example has ...
Kira G.'s user avatar
  • 149
3 votes
1 answer
192 views

what are the possible dimensions of an irreducible representation of a group of order 21?

What are the possible dimensions of an irreducible representation of a group of order 21? I'm not sure what theorems might be useful for solving this problem. I know that every element of the group ...
user1127's user avatar
  • 469
8 votes
0 answers
147 views

Problem relating $p$-defect zero characters with their values on a field of characteristic $p$

Studying Gabriel Navarro's book "Character Theory and the McKay Conjecture", I've come across the following problem. First, let's fix some notation: $G$ will be a finite group, $R$ will ...
Gauss's user avatar
  • 2,663
1 vote
1 answer
66 views

Central idempotents and their relation to subrepresentations, in finite representation-theory over the group-ring $\mathbb{C}[G]$.

Let $G$ be a finite group. In M. Isaacs "Character theory of finite groups", chap. 2, we cover the fact that $\mathbb{C}[G] = \bigoplus_{i = 1}^{k} M_i(\mathbb{C}[G])$, where $M_i(\mathbb{C}[...
Ben123's user avatar
  • 1,252
0 votes
1 answer
79 views

find the remaining rows of a character table

Assume we're given a group of order 10 with 4 conjugacy classes with representatives $g_i$ for $1\leq i\leq 4$ so that $(|C_G(g_i)|:1\leq \leq 4) = (10,5,5,2)$ where $C_G(g_i) := \{h \in G : g_i h = h ...
user1127's user avatar
  • 469
0 votes
1 answer
62 views

Gauss sum in character theory

Define a non trivial additive character $\psi: \mathbb F_q \rightarrow \mathbb C$ s.t $\psi(x)= \xi^{Tr(x)}$ where $\xi=e^{2\pi i /p}$, where $Tr(x):\mathbb F_q \rightarrow \mathbb F_p$ . First ...
Giuseppe's user avatar
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0 votes
1 answer
67 views

Understanding the proof of Theorem 9.4 in Montgomery & Vaughan's Multiplicative Number Theory

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: I understand everything in the proof except for: How existence of $c$ such ...
Ali's user avatar
  • 281
0 votes
1 answer
72 views

Understanding the proof of Lemma 9.3 in Montgomery & Vaughan's Multiplicative Number Theory

The following is from "Multiplicative number theory I: Classical theory" by Hugh L. Montgomery, Robert C. Vaughan: I understand everything in the proof except for: How $m' \equiv n'$ (mod $...
Ali's user avatar
  • 281
2 votes
0 answers
54 views

An identity about the sum of the reciprocal of the irreducible character of a symmetric group evauated at identity

I just learnt character theory and I am reading the paper https://academic.oup.com/jlms/article-abstract/66/3/623/811347 which is a quite beautiful paper. Section 4.3, on p. 631 contains an inequality ...
Han's user avatar
  • 115
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0 answers
15 views

Orthogonality and completeness of irreducible representations

I am reading a paper https://arxiv.org/abs/1910.07143 I am unable to understand the construction of table III, for the section Orthogonality and completeness of matrix elements in group space. The ...
user519535's user avatar
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0 answers
82 views

Character table of S4

I am trying to understand the character table of $S_4$. I have obtained the trivial, signature and standard representations. The fourth one is the product of signature and standard. Now for the last ...
user519535's user avatar
0 votes
0 answers
72 views

Constructing the character table of Octahedral Group (order 48)

I am trying to construct the character table for the symmetry group of Cube, $O_h$, which as 48 elements. I figured out that there are 10 classes. Now, as a consequence of the great orthogonality ...
user519535's user avatar
4 votes
1 answer
109 views

$\varphi^G$=$\psi^G$ implies that subgroups are conjugate

Let $G$ be a finite group and let $\varphi,\psi$ be linear characters of $H$ and $K$ respectively, where $H,K$ are subgroups of $G$. I'm trying to prove that if the $G$-induced characters of $\varphi,\...
Deif's user avatar
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2 votes
0 answers
36 views

Are all automorphisms of the character table of a quasisimple finite group generated by Galois and outer automorphisms?

This is a follow up question to Automorphisms of a character table Consider the character table of a finite group $ G $. Every outer automorphism of $ G $ permutes the characters of $ G $. If some of ...
Ian Gershon Teixeira's user avatar
0 votes
1 answer
75 views

Additive duality of character of a local fields

I am studying "the local Langlands conjecture for GL(2)" of Bushnell and Henniart. I am having a hard time getting into the mechanics. And I have some problems with one of the first ...
Mario's user avatar
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0 answers
63 views

Property of $(\chi(g))_{g\in G, \chi\in \hat{G}}$ for a finite abelian group $G$.

Let $k$ be a field, $G$ be a finite abelian group such that $\operatorname{ch}=0$ or $\operatorname{ch}(k)\nmid \#G$. Then, the dual group $\hat{G}$ is defined by $\hat{G}=\{\chi\mid {\rm group~...
Yos's user avatar
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0 votes
1 answer
65 views

Determining which characters split or fuse

The ATLAS seems to suppose that in the case $G.2$ it is easy to determine which irreducible characters of $G$ will split and which will fuse in the extension. Is there is a source which can explain ...
NewViewsMath's user avatar
1 vote
1 answer
79 views

How to read labels of character tables in SageMath (and GAP)?

Note: SageMath's Group.character_table() method is built as a wrapper for GAP's CharacterTable() function. Let $G = \text{Sym}_4$...
Gutiérrez's user avatar
4 votes
1 answer
182 views

The character table of $G/Z(G)$ and $G/N$ given knowledge of the character table of $G$

Suppose for a group $G$, of which we completely understand its conjugacy class structure and character table, is there a simple algorithm to deduce the character table of $G/Z(G)$? Or more generally, $...
Robin's user avatar
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