Questions tagged [characters]

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

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Induced rep of restriction of irreducible rep

Let $H\leq G$ be finite groups and $\chi$ an irreducible character of rep $\rho$ of $G.$ Decompose $Res_H^G\chi=a_1\psi_1+\dots+a_r\psi_r$ with $\psi_i\in Irr(H).$ I want to show $a_1^2+\dots+a_r^2\...
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Positive column in the character table of finite groups

By going through the character table of small finite groups here, we can observe that there always have a single column where all the entries are positive numbers (i.e. the one giving the dimension of ...
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If $G$ is a finite group, $g \in G$, and $\chi$ is a character of $G$, what are the definitions of "$\chi(g)$" and "$\overline{\chi(g)}$"?

I'm looking at lemma $2.15$ on page $20$. https://www.cefns.nau.edu/~falk/classes/511/Isaacs_Character_theory.pdf
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2 answers
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Application of character theory to structure of groups

One application of character theory in the investigation of structure of finite groups is for Burnside's theorem. Can one mention some other results in Group theory, whose proofs are elementary from ...
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Induced representation coefficients

Let $H \leq G$ be a subgroup, where $G$ is finite. Suppose $(\sigma, W)$ is an irreducible representation of $H$. Let $\sigma^{\circ}$ be the induced representation on $G$. Maschke's theorem tells us ...
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2 votes
1 answer
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Method Verification for Finding Irreducible Characters

I am studying the character table of $S_5$ and in this YouTube video https://www.youtube.com/watch?v=Zj5PE6r_Oeo&t=427s (video will start at ~7:05), a method for finding an irreducible was used ...
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3 votes
1 answer
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Extension in semidirect case

Suppose $G=NQ$, where $N$ is normal in $G$ and $(|N|,|Q|)=1.$ In other words $G$ is a semidirect product of $N$ and $Q.$ Can we say every irreducible character of $N$ is extendable to $G?$ Suppose $\...
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Orthogonality relations for matrix elements of irreducible representations

I am reading Howard Georgi's "Lie Algebras in Particle Physics" and have a question concerning the presented orthogonality relations for matrix elements of irreducible representations. To ...
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7 votes
1 answer
134 views

Computing $\chi(1)$ and $\chi(s)$ for $\chi\in\widehat{\mathrm{GL}_2(\mathbb{F}_q)}$ and semisimple non-regular $s$ using formulas of Deligne-Lusztig

Let $G=\mathrm{GL}_2$ and $s=\left(\begin{smallmatrix} a & \\ & b \end{smallmatrix}\right)$ be semisimple and non-regular in $G(\mathbb{F}_q)=\mathrm{GL}_2(\mathbb{F}_q)$ (i.e. $a\neq b$ and $...
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Relation between characters and coordinate ring

I am reading about algebraic groups. I don't fully understand the purpose of the coordinate ring, but I feel this is a way of "parametrizing" characters on the group. Here is an example to ...
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Direct sum decomposition of $\mathbb CG$ module into simple modules given the character table of $G$ (regular representation)

Suppose I have obtained the character table of a group like $S_3$. After that how can I obtain the simple submodules of $\mathbb CS_3$? It's not clear to me how to go from the irreducible characters ...
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  • 317
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Characters of irreducible representations of a finite and semisimple algebra

To clarify: I use $\chi_V$ for the character of the finite representation $V$ of $A$ which is an algebra on a generic field $K$. Then you can consider the idel $[A,A]$ generated by commutators, which ...
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Showing that if a class function $\gamma$ is orthogonal to all irreducible representations of a finite group $G$ then $\gamma \equiv 0$.

Let $G$ be a finite group and $A = \mathbb{C}[G]$ its group algebra with a basis $(u_g)_{g\in G}$. Assume it to be known that $\sum_{g\in G}\gamma(g)u_g$ is in the centre of $A$ iff $\gamma(gh) = \...
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3 votes
1 answer
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Intuitions behind Frobenius' generalization of characters to nonabelian finite group given the historical context

I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius. According to most of the related papers (e.g. Pioneers of ...
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Showing that $\sum_{\rho}\chi_\rho(g)\overline{\chi_\rho(g)} = \frac{|G|}{|C(g)|}$ for a finite group $G$, $g \in G$ and $C(g)$ a conjugacy class

Suppose that it is known that a finite group $G$ has equally many conjugacy classes as it has irreducible representations $\rho_1,\dots,\rho_k$. How can we then show that $\sum_{\rho}\chi_\rho(g)\...
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Proving that the number of conjugacy classes of a finite group $G$ equals the number of $G$'s irreducible representations

Preamble: I want to understand why the number of irreducible representations of a finite group equals the number of conjugacy classes of the group. The only direct proof I managed to find is in the ...
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2 votes
3 answers
69 views

Let $\chi$ be a character of a finite group $G$ which is constant on $G\backslash\{1\}$. Show that $\chi = a1_G + b\rho_G$ where $a,b\in\mathbb{Z}$.

I have been working on this equality for a while, and am a bit stuck. In the case that $\chi$ is identically $0$ on $G\backslash\{1\}$, I have shown that $\chi$ must be an integer multiple of $\rho_{G}...
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About homogenous completely irreducible module

I'm stuck on something, I'd appreciate it if you could help :) Let $A$ be a $F$-algebra, $V$ be a completely reducible $A$-module and $M$ is an irreducible $A$- module. Now let's consider $M(V)$, ...
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The character table of an abelian group

I am attempting to construct the character table for $\mathbb{Z}_8$. I know a few things off the bat: Since $\mathbb{Z}_8$ is abelian, its conjugacy classes are singletons (i.e. we have eight classes)...
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1 vote
1 answer
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Additive characters modulo reducible polynomial in $\mathbb{F}_q[T]$

Suppose we have a prime power $q$ and a polynomial $F \in \mathbb{F}_q[T]$. I am wanting to know what the additive characters look like modulo $F$. If $F$ is irreducible then $\mathbb{F}_q[T]/(F(T)) \...
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1 vote
1 answer
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Character table of modular Heisenberg groups

Let $p$ be a prime number and let $G$ be the modular Heisenberg group of order $p^3$ $$ G = \left\{\, \begin{bmatrix} 1 & b & c \\ 0 & 1 & a \\ 0 & 0 & 1 \end{bmatrix} : a, b, ...
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Are two representations $V, W$ isomorphic if $\left<\chi_V, \chi_W\right> > 1$ for the inner product of their characters?

I know from Schur's lemma(s) that if $V, W$ are two isomorphic irreducible representations of a group $G$, then $\left<\chi_V, \chi_W\right> = 1$. This is because the dimension of the space of ...
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Basic question about defining a character.

I am reading Character Theory of Finite Groups, by Martin Isaacs. If $G$ is a fixed finite group, and $M_i$ is an irreducible $\mathbb{C}[G]$-module, then the book defines $\mathfrak{X}_i$ by choosing ...
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3 votes
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27 views

Representations of $SO_3(\mathbb{R})$ and their characters

I am trying to understand the representations of $SO_3(\mathbb{R})$. Consider the space $P_n$ of homogeneous polynomials of degree $n$ in $(x,y,z)$. I want to understand the characters of $V_n = \ker (...
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Value on $\mathbb{F}_{q}^{*}$ determines character of $\mathbb{F}_{q^2}^{*}$ up to composition of Frobenius automorphism

It is stated in Exercise 3.43 in Amritanshu Prasad's notes that For two characters $\omega$ and $\eta$ of $\mathbb{F}_{q^2}^{*}$, their restrictions to $\mathbb{F}_{q}^{*}$ are equal if and only if ...
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2 votes
1 answer
106 views

Characteristic Polynomial and Group Characters

TLDR: Group characters and characteristic polynomial have a very similar function but are introduced in very different terms. Can characteristic polynomial be understood through the lens of character ...
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(Inverse?) Fourier transform for finite abelian groups

Let $G=\mathbb{Z}_{n_1}\times\cdots\times \mathbb{Z}_{n_m}$ be a finite abelian group and $C(G)$ the algebra of complex-valued functions on $G$ with pointwise-multiplication. Let a basis be given by: $...
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  • 7,780
0 votes
1 answer
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Characters of complex $S_5$-representation

Let the symmetric group $S_5$ act by permutation on the set $X:=\{S \subset \{1,2,3,4,5\}:|S|=2\}$ and denote by $V$ the associated complex $S_5$-representation, with $\chi:S_5 \rightarrow \mathbb{C}$ ...
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  • 463
1 vote
1 answer
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Minors of the character table of a finite group $G$ to solve a system of linear equations

I am working with a system of linear equations $Ax=b$ where $A$ is given by the character table of a finite group $G$ (actually, the group of permutations $\Sigma_k$). In other words, the equations ...
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  • 148
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Eigenvalue argument to determine 5-dimensional irreducible representation of S5.

I was reading this expository paper by Chris Blair on finding the irreducible representations of $S_5$. In Section 2.5 of the paper, the author makes an elementary argument using eigenvalues to ...
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  • 11
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1 answer
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Character induced from a faithful character is faithful

In my last algebra exam, I had this exercise that I wasn't able to solve: Let $H$ be a subgroup of the group $G$. Let $\chi$ be a character of $H$. Assuming that $\chi$ is faithful , prove that the ...
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4 votes
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Characters of Lie algebra representations

1.Three definitions Let $\mathfrak g$ be a Lie algebra over a field $k$. Let $(V, \rho)$ be a $\mathfrak g$-representation. In class I was presented with various definitions of characters of $(V, \rho)...
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Unramified character of the decomposition group leaves no unit fixed

Let $A$ be a local Artinian ring and $D_{p}$ a decomposition group at the prime $p$. If $\phi\neq 1:D_{p}\longrightarrow A^{\times}$ is an unramified character, then no unit of $A$ is left fixed by $\...
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4 votes
1 answer
98 views

Zeros of characters

I was wondering if the following is true: Let $\chi$ and $\chi'$ be two irreducible characters over $\mathbb{C}$ of a finite group with same degree. Suppose that $\chi(g) = 0 \implies \chi'(g) = 0$. ...
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1 answer
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Character Cartan-Killing metric

I am trying to identify the normal real forms of some classical semisimple Lie algebras. These are defined as those real forms whose character $\chi:=n_+-n_-$ equals the rank of the Lie algebra. Here $...
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1 answer
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Explicit construction of induced character from cyclic subgroup of symmetric group

I have been computing some characters by hand, but just can't seem to figure out how they relate to the standard "induced character" constructions I can find. Small example: For $G=S_4$ and ...
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Computing the character of $M\otimes_{\mathbb{C}H} N$ for a $\mathbb{C}[G\times H]$ module $M$ and a $\mathbb{C}[H\times K]$ module $N$.

Let $k = \mathbb{C}$ and let $G$ and $H$ be finite groups. It is well-known that we can view a $k[G\times H]$-module as a $(kG,kH)$-bimodule by defining the action $g \cdot m \cdot h : = (g,h^{-1})\...
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$p$-modular Brauer character table of an extension by a $p$-group

Suppose $G$ is a finite group, and that I know the $p$-modular Brauer character table for $G$. If I take a non-split extension of $G$ by a $p$-group $P$, is the $p$-modular Brauer character table for $...
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1 answer
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Counterexample in Serre about Artin's induction theorem with $\mathbb{Z}$ coefficients

After reading the exposition about Artin's induction theorem for $\mathbb{Q}$-representations, for instance in Serre's Linear Representations of Finite Groups, 12.5, prop 25, one can be tempted to ...
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1 vote
1 answer
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Reference Request for Orbits of Group Representation

Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
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If a representation is irreducible iff $(\chi_V,\chi_V)=\sum_{i=1}^k a_i^2$, wouldn’t that mean that there’s only one $a_i$ and that $a_i=1$?

If a representation $V$ of a group $G$ is irreducible iff $(\chi_V,\chi_V)=\sum_{i=1}^k a_i^2$=1, where $a_i$ is the multiciplity of each $V_i$ in the isotypic decomposition of $V$, wouldn’t that mean ...
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Why are “isomorphic representations given in compatible basis by identical matrices”?

I wanted to understand what is the exact reason why two isomorphic representations have equal character, and found this question here where it is said that the reason is that they’re given by ...
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  • 77
1 vote
1 answer
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Absolute value of Jacobi sum $J(\chi,\psi)$ for Dirichlet characters

We know that a character $\chi$ on a finite field $\mathbb{F}_q$, $q$ being a power of a prime, is a group homomomorphism $\chi:\mathbb{F}_q^*\rightarrow \mathbb{C}$. For characters $\chi,\psi$ on $\...
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3 votes
0 answers
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Question about Suzuki's Theory of Exceptional Characters

As elegant as Suzuki's theory is, the set up requires that the number of conjugacy classes of $p$-elements in a cyclic T.I. (as an example) Sylow $p$-subgroup $P$ of $G$, $e$, is at least 2, in order ...
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  • 89
1 vote
0 answers
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Characters of SU(2) and group elements squared.

SU(2)-group elements can be written in terms of Euler-angles $\alpha,\beta,\gamma$ (using the convention from this wikipedia article, i.e. z-y-z convention). Now, it is stated that the characters of ...
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  • 1,542
3 votes
1 answer
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Any function $f:\mathbb{F}_q \rightarrow \mathbb{C}$ has a unique representation $f(x)=f_{\delta}\delta(x)+\sum_{\chi} f_{\chi}\chi(x)$

While going through an article I have come across the following fact: Any function $f:\mathbb{F}_q \rightarrow \mathbb{C}$ has a unique representation $$f(x)=f_{\delta}\delta(x)+\sum_{\chi} f_{\chi}\...
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  • 975
-1 votes
1 answer
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How to find out if a semilinear representation is irreducible (possibly with gap)

Let be $\Gamma=\Gamma(2^6)$ the semilinear group on $GF(2^6)$, namely the group of semilinear mappings $\tau_{a,\sigma}\colon x\to ax^\sigma$ for $x \in GF(2^6)$ as a vector space over $GF(2)$, $a \in ...
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  • 1,481
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why $\sum_{g∈G}\overline {h(g)}χ_v(g)$ is scalar of identity map of $V$ and what is the scalar?

Let ($χ,V$) be a irreducible representation and $h$ be class function. Then, why $\sum_{g∈G}\overline {h(g)}χ_v(g)$ is scalar of identity map of $V$ ? (I try to prove find the relation between $\sum_{...
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2 votes
1 answer
50 views

Compute character table of $S_4$ in Fulton-Harris

This is an example in Fulton-Harris 2.3 at p.g. 18: computing the character table of $S_4$. I have computed $\chi_{trivial},\chi_{sgn},\chi_{std},\chi_{std\cdot sgn}$. The last character, denoted as $\...
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  • 195
4 votes
1 answer
49 views

Orthogonality of characters for powers of a character

Suppose $G$ is a finite cyclic group, and let $\chi: G \to \mathbb{C}^*$ be a character of order $n$. That is, $\chi^n$ is the identity homomorphism. I came across the following relation in a paper ...
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