Questions tagged [characters]

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

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28 views

How can we show that the closure property holds in the character group of a finite abelian group?

Let $G$ be a finite abelian group of order $n$. Let $f_1, f_2, \dots, f_n$ be the characters of $G$ where $f_1$ is the principal character and others are non-principal characters. So we now have a set ...
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21 views

Group operator vs. complex number multiplication operator in the multiplicative property of a character of a group

A character $f$ of a group $G$ is defined as a complex-valued function defined on $G$ that has the multiplicative property $f(ab) = f(a)f(b)$ for all $a, b$ in $G$, and if $f(c) \ne 0$ for some $c$ in ...
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18 views

An example to illustrate a complex-valued function $f$ that is a character of a finite group $G$

I am learning this theorem: If a complex-valued function $f$ is a character of a finite group $G$ (i.e. $f$ has the multiplicative property $f(ab) = f(a)f(b)$) with identity element $e$, then $f(e) = ...
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21 views

2-dimensional representation of modular group

My question probably follows from something basic in character theory, but I don't see what I'm missing. Let $\rho: \Gamma \to \mathrm{GL}(2,\mathbb C)$ be a two-dimensional representation of the ...
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30 views

Sum of squares of the dimension of irreducible characters and conjugacy classes

While working through a prior problem on classifying a group of order $625$, I stumbled upon (by virtue of a mistaken answer by a user), the following problem. Since the dimension of irreducible ...
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1answer
37 views

Real characters of odd degree

Suppose that $G$ is a finite group and $\chi$ is an irreducible real character, namely that $\chi(g) \in \mathbb{R}$ for every $g \in G$. Is it true that if $\chi(1)$ is an odd number, then $\chi$ is ...
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27 views

Character triples isomorphism and real characters

A character triple is a triple of the form $(G,N,\theta)$ where $G$ is a finite group, $N$ is normal in $G$, $\theta \in Irr(N)$ and $\theta$ is $G$-invariant. For the concept of character triple ...
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17 views

Ergodicity of surjective continuous endomorphism of compact abelian group (confused about a step)

Let $G$ be a compact abelian group with Haar measure and $A$ surjective continuous endomorphism. Then $A$ is ergodic $\iff$ $\phi(A^n)=\phi$ ($\phi$ are characters) for some $n>0$ implies $\phi$ is ...
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20 views

Primitive characters on S-ideals

Reading across Milne's notes on Class Field theory, I came across the following definition on page 165. Here $S$ is a finite set of non-archimedean primes (of a number field $K$), and $I^S$ is the ...
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1answer
37 views

Group of characters $q$ modulus $\sum_{\chi\pmod q}\chi(n)$

I was solving the following question (in fig), with the hint that if $n \neq 1 \pmod q$ then there exist a character $\phi$ mod q such that $\phi(n) \neq 1$. Show that $$ \sum_{\chi\pmod q}\chi(n) = \...
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22 views

What's wrong with this proof that irreps over $\mathbb{C}$ are dimension $1$?

Let $V$ be an irreducible representation of a finite group $G$ over the complex numbers $\mathbb{C}$. Then $[V, V]_{\mathbb{C}[G]} \cong \mathbb{C}$ since $\mathbb{C}$ is algebraically closed. It ...
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32 views

Exercise 8.8 in Isaac's Character Theory of Finite Groups

I am trying to solve the problem in the title. First I will provide some background information. Definition (Elementary Group) A group $E$ is $p$-elementary ($p$ is a prime) if $E$ is the direct ...
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46 views

Exercise 7.10 from Isaac's Character Theory of Finite Groups

I am having difficulty solving a problem in Martin Isaac's Character Theory of Finite Groups. Could someone please give me a hint?Thanks in advance! The problem is as follows: Let $K$ be T.I.F.N in $G$...
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31 views

Can two non-isomorphic $G$-sets have the same character?

Let $G$ be a finite group, and let $H_1$, $H_2$ be two non-conjugate subgroups. Can we have $\text{Ind}_{H_1}^G(\mathbf{1})\cong \text{Ind}_{H_2}^G(\mathbf{1})$ as representations? Using characters, ...
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45 views

Average irreducible character of $O(N)$

Let $O(N)$ be the group of $N\times N$ real orthogonal matrices, and let $\chi_R(g)=\sum_i M^{(R)}_{ii}(g)$ denote the character of $g\in O(N)$ in the irreducible representation $R$, i.e. the trace of ...
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25 views

Character table-related notation $\mathbf Z_m$ in Curtis–Reiner

This is a question regarding a seemingly unexplained notation in Curtis–Reiner: Methods of Representation Theory, volume 1. In subsection §09D on the character table, on page 215, between equations (9....
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39 views

Two representations of a finite group have no irreps in common if and only if their characters are orthogonal.

We want to prove that two representations of a finite group $\Gamma_{1}$ and $\Gamma_{2}$ have no irreducible representation in common if and only if their characters are orthogonal, i.e., $\sum_{k=1}^...
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135 views

Exercise 6.17 from Isaac's Character Theory of Finite Groups.

I am trying to solve the following problem in Isaac's Character Theory of Finite Groups and I am really sorry if this is a very simple question. 6.17: Let $N$ be a normal subgroup of $G$ with $G/N$ ...
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25 views

Counting characters $\bmod q$ under some divisibility conditions

Let $N$ be fixed. I would like to estimate the number of primitive characters $\chi_1$ modulo $q_1$ and $\chi_2$ modulo $q_2$ such that $$q_1q_2 \mid N$$ How can I tackle this?
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35 views

Showing properties of a character $\widetilde{\chi}_\rho$

Let $G$ be a finite group and $H \le G$ such that $gHg^{-1} \cap H = 1$ for any $g \in G \setminus H$. Let $\pi$ be the permutation representation of $G$ associated to the left regular action of $G$ ...
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99 views

Problem 7.3 I. Martin Isaacs's Character Theory Of Finite Groups

Let $H\subseteq G$ and $\xi\in Irr(H)$. Suppose $(\xi-\xi(1)1_H)^G=\theta$ and $[\theta,\theta]=1+\xi(1)^2$. Show that there exists $N\lhd G$ with $N\cap H=\ker\xi$ and every $x\in G-N$ conjugate to ...
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1answer
46 views

Same character table implies two groups have the same cardinality?

Suppose $G_1,G_2$ are two finite groups and their character tables are the same. Can I prove or disprove that $|G_1|=|G_2|$? I have just started on character theory and I have been thinking proving ...
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69 views

Exercise 6.21 Isaacs's Character theory of finite groups

Let $1= H_0\lhd H_1\lhd\cdots\lhd H_n=G$. Assume that $H_i/H_{i-1}$ is nonabelian. Show that there exists $\chi\in Irr(G)$ with $\chi(1)\ge 2^n$. Hint Use Corollary 6.17(Gallagher): Let $N\lhd G$ and ...
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72 views

Normal Hall subgroups of $M$-groups are $M$-groups.

Dornhoff proved this theorem(Theorem 4.1. on page 250) in the paper https://link.springer.com/article/10.1007/BF01109806. The first few lines of the proof go as follows: On the contrary, assume $G\...
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1answer
27 views

Extending a Group Character to a Dirichlet Character

In https://math.mit.edu/classes/18.785/2015fa/LectureNotes17.pdf, just after Definition 17.4 on page 7, the author states that "Every $m$-periodic Dirichlet character $\chi$ restricts to a group ...
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1answer
45 views

Question about absolute value of complex representation [closed]

Let G be a finite group and V be an n-dim complex representation of G (not necessarily irreducible). What is the maximum of the absolute value of $\chi(g)$? Here $\chi(g)$ is the character i.e the ...
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1answer
59 views

Sum of squares of characters

Given a finite Abelian group $G$, I am interested in the sum $$ \sum_{\chi} \chi^2(g) $$ for some element $g \in G$ and the sum runs over the characters of $G$. I tried using the identity $\chi^2(g) = ...
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1answer
79 views

Exercise 5.24, Isaacs's Character Theory

Let $G$ be a doubly transitive permutation group on $\Omega$ and let $\alpha,\beta\in \Omega$ with $\alpha\neq \beta$. Let $\phi\in Irr(G_\alpha)$ and assume that $\phi_{G_{\alpha\beta}}\in Irr(G_{\...
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52 views

Character of Wedge Product of Reducible Reps

I'm trying to deduce the character of $\wedge^3 V_{perm}$ and then decompose it as irreps, where $V_{perm}$ is the $S_5$-representation $V$ and I would like someone to hopefully check my work and, if ...
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1answer
90 views

Intuition of Artin's Linear Independence Theorem

Let $G$ be a group and $k$ be a field. A character of $G$ is a group homomorphism $G \to k^{\times}$. Theorem (Artin). Distinct characters $\chi_1, \dots, \chi_n : G \to k^\times$ are linearly ...
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21 views

Semisimple algebra F[G] and restriction to subgroups

Let A be a Dedekind domain of characteristic $p$ whose fraction field is $F$. Let $G$ be a finite group (not neccesarily abelian) whose order is not divisible by $p$. Then the group algebra $F[G]$ is ...
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133 views

8.10 Apostol Analytic number theory

I am self studying analytic number theory from Tom M Apostol and got stuck on this problem on page176. It's image: Problem is in question 10 only. I have proved that $f(\chi) $divides $f(\chi_{1})... ...
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24 views

Generating representations of U(n) using Schur functions.

We know that Schur functions $S_\lambda$ are related with irreps. of $U(n)$ and that there is an associated branching rule for the subgroup chain $U(1)\subseteq \ldots \subseteq U(n-1)\subseteq U(n)$, ...
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50 views

Given a subgroup H, show there are distinct characters which are same on H.

Given a subgroup $H\subset G$ and a conjugacy class $C$ in $G$ disjoint from $H$, show that there are distinct characters $\chi_1,\chi_2$ that agree on $H$. Somebody else answered this question (see ...
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77 views

Group of dirichlet characters mod 8

The following is group of dirichlet characters taken from Wikipedia and I have a question in that. Question: why in this table there are no terms involving i, -i(iota) . I think there must as there ...
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27 views

A question in table of Dirichlet characters mod 7 ( verification)

The following is table of Dirichlet characters mod 7. Question: In table of $\chi_{2} (n) $ can I write 1 in box 21 , -1 in box 22 , 1 in box 23 , 1 in box 24 , -1 in box 25 , -1 in box 26 , 0 in ...
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31 views

Unclear idea about conductor and primitives

I have the following theorem: Let $q \in \mathbb{Z}_{\geq 2}, \chi$ a character mod q. Denote by f the conductor of $\chi$. (i) There is a unique character $\chi^{∗}$ mod $f$ that induces $\chi$...
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75 views

A question in the proof of Aramata-Brauer theorem in representation theory

On page 30 of Murty's book "Non-vanishing of L-Functions and Applications", he gives a proof of the following theorem, maybe it is called Aramata-Brauer theorem. Theorem: Let $G$ be a ...
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42 views

Schur-Positivity of a simple polynomial

Let $\chi_{d,p;f}$ be the following symmetric polynomial, $$\chi_{d,p;f}(x)=\prod_{l=1}^d\sum_{k=0}^p x_l^{f_k},$$ where $f=\lbrace f_0,\ldots,f_p\rbrace$ is a set of integers. I need to identify for ...
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15 views

Sum of certain integer numbers related to the elements of a finite abelian group and to its group of characters

We consider a finite abelian group $G$ and its group characters $G^*$. For each $g\in G$ and $\chi\in G^*$ we define $0\leq r_g^\chi< o(g)-1$ such that $\chi(g):=e^{\frac{2\pi i}{o(g)}r_g^\chi}$. ...
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35 views

Non-negative determinant for a matrix of Kronecker deltas

Let $M$ be a $d\times d$ matrix with entries $$(M)_{ij}=\sum_k \delta(f_k-\lambda_i+j-i),$$ where $f_k$ are non-negative integers, the sum is finite and $\lambda_i$ an element of a partition $\lambda=(...
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1answer
52 views

An elementary proof of $\sum_{l=0}^{n-1} \cos(\frac{2\pi kl}{n})^2=\frac{n}{2}$? (without character theory)

Recently I have stumbled upon the following identity: if $n>2$ and $1\leq k\leq\lfloor\frac{n-1}{2}\rfloor$, then $\sum_{l=0}^{n-1}\cos(\frac{2\pi kl}{n})^2=\frac{n}{2}$. There is a lovely proof of ...
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63 views

A question based on linear of a group and Characterstic subgroup of a group

This question was asked in my abstract algebra exam of previous year which I got from a senior and I couldnot solve this particular question. Question is : (a) Let G be a group , $\mathbb{F}$ be a ...
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1answer
65 views

Problem in representations of $D_4$

Problem Let $G=D_4=\langle a,b\ |\ a^4=b^2=1,\ ab=ba^{-1}\rangle$ and $V=\mathbb{C}^2$ with bases $\{e_1,e_2\}$. We see $V$ as a $\mathbb{C}G-$module with action $ae_1=e_2,\ ae_2=-e_1$ and $be_1=e_1, \...
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90 views

Sum of products of multiplicative characters of $\mathbb{F}_q$

This is problem 5.30 from the book "Introduction to Finite Fields" by Lidl and Niederreiter. Let $\lambda_1, \lambda_2, \lambda_3$ be nontrivial multiplicative characters of $\mathbb{F}_q$ ...
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1answer
48 views

Reference request for value of characters of the symmetric group on various conjugacy classes

Let $\chi^{\lambda}$ and $C_{\mu}$ denote the character of the conjugacy class of the symmetric group $S_n$ induced by partitions $\lambda,\mu \vdash n$ I'm interested in where I can find a table that ...
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1answer
165 views

How to prove column orthogonality of character table

This is a standard result in representation theory of finite groups. A poof of the column orthogonality of the character table can be found here. By orthogonality of columns I mean that $$\sum_{V}\...
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27 views

Show that the determinant of the character table for the symmetric group is $\prod_{\lambda \vdash n} \prod_{\lambda_i\in \lambda}\lambda_i$.

Show that the determinant of the character table for the symmetric group is $\prod_{\lambda \vdash n} \prod_{\lambda_i\in \lambda}\lambda_i$. Here, $\lambda \vdash n$ means $\lambda=(\lambda_1,\ldots,\...
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64 views

Every irreducible character of $S_n$ is an integer valued function.

Every irreducible character of $S_n$ is an integer valued function. Why is this true? I have seen this question posted before, but I did not understand the solution. The only solution I can find deals ...
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34 views

Bounding the character sum $\sum e((ax^k+bx)/p)$

The following question is from Carlos Moreno's "Algebraic Curves over Finite Fields": I am trying to solve part (i), but I am keep getting stuck at roughly the same place. Write $C: y=x^d$ ...

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