Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
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39 views

Maximal subgroup and representations (dual part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Let $g \in G$, $K = H \cap ...
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38 views

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
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103 views

Prove that every 2 dimensional FG-Module with gh not equal to hg is irreducible.

Basically the questions is as follows. Suppose that $V$ is a 2-Dimensional $FG$-Module where F=The complex numbers and that there exists $g,h$ elements of $G$ and $v$ an element in $V$ such that ...
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169 views

$\chi(g)$ group character $\Rightarrow$ $\chi(g^m)$ group character

Let $G$ be a group of order $n$ and and $\gcd(m,n)=1$. Let $\chi:G\rightarrow\mathbb{C}$ be a class function and define $\chi^m\!: g\mapsto\chi(g^m)$. How can one show that $\chi^m$ is a character iff ...
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50 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
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115 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Introductory Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the ...
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1answer
87 views

Irreducible representations of finite lamplighter group

Let $G = \mathbb{Z}_2 \wr \mathbb{Z}_n$ be the finite lamplighter group. What are the irreducible representations of $G$ - can anyone provide a clear reference? Austin, Naor and Valette list ...
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34 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the ...
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1answer
71 views

Modular function of the unipotent radical of a parabolic subgroup of a reductive group

Let $G = \text{GL}_n(\mathbb{R})$. For a partition $\underline{n} = (n_1,\ldots,n_t)$ of $n$, let $P = P_{\underline{n}}$ denote the standard, block-upper-triangular parabolic subgroup of $G$ ...
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26 views

derived equivalence of coalgebras

let $C$ and $D$ be two coalgebras over a field, $C$ and $D$ are called derived equivalent if the derived categories $D(C-comod)$ and $D(D-comod)$ are equivalent as triangle categories. if $C$ and ...
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1answer
82 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
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1answer
66 views

What am I missing about Schur functions?

Let's say I only know the following about Schur functions: you give me a partition $\lambda$ of $d$ such that $\lambda$ has $n$ parts $\lambda_1,\ldots,\lambda_n$, and I can compute the Schur function ...
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73 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
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43 views

A question in the proof that the weight of a finite dimensional module is W-invariant

Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which ...
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1answer
147 views

Finding irreducible representations

This might be a very elementary question in representation theory, but I dare to ask Suppose I am asked to complete the character table of $S_5$, I know it has 7 conjugacy classes as follows : ...
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2answers
76 views

Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
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30 views

representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
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211 views

Representation theory over $\mathbb{Q}$

I am looking for books or papers which tell me something about representation theory of finite groups over $\mathbb{Q}$ (or finite extensions thereof which are not splitting fields of the group ...
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2answers
300 views

Representation theory of infinite groups?

I am familiar with the representation theory of finite groups (at least of the symmetric groups over the field of complexes) And I know that the group algebra of an infinite group is not semisimpe ...
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1answer
41 views

Questions about cuspidal representations of $GL_2(\mathbb{F}_q)$.

All representations of $GL_2(\mathbb{F}_q)$ are classified in the book. They are principal series representations, complementary series representations, 1-dimensional representations. They form all ...
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question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
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71 views

Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
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1answer
26 views

Questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$.

I have some questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in GL_2(\mathbb{F}_q)$ has eigenvalues which ...
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416 views

About the converse of Maschke's theorem

The Maschke's theorem say that\ Let $G$ be a finite group and $F$ a field whose characteristic does not divide $\mid G \mid$. Then every $FG$-module is completely reducible (I'm using the notation of ...
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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1answer
57 views

Why does the universal cover of $GL^+_n$ not admit finite-dimensional representations?

Let $GL^+_n \subset \mathbb{R}^{n \times n}$ be the subgroup of real matrices with positive determinant and $\widetilde{GL}^+_n$ be its universal cover. Why does $\widetilde{GL}^+_n$ not admit ...
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1answer
56 views

$1$-dimensional representations of $GL_2(\mathbb{F}_q)$.

I have some questions about $1$-dimensional representations of $G=GL_2(\mathbb{F}_q)$. I need to show that there are $q-1$ $1$-dimensional representations of $GL_2(\mathbb{F}_q)$. I am able to show ...
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2answers
94 views

Action of a group G on $\mathbb{C}$[G] makes $\mathbb{C}$[G] a G-module

I am trying to show that the action of the group G on the vector space of functions $\mathbb{C}$[G] defined by [g.$\psi$] (h)=$\psi(g^{-1}h$) makes a G-module. My main problem is that I can't get any ...
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Meromorphic Continuation of Intertwining Operator Identities for K-finite vectors

I have a question about a "well-known result" about intertwining operators. I will restrict myself to $\text{SL}_2$ for simplicity. Let $$w = ...
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34 views

weight of a group G

how can I found the weight of a group G ( I find the fundamental weights, but I don't know how found the linear combination of fundamental weights that give me the weights ). so we found the ...
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1answer
98 views

$\Psi_g(A)=\Phi(g)A^t\Phi(g)$; express $\chi_\Psi$ through $\chi_\Phi$

Let $\Phi$ be a matrix n-dimensional representation of the group G. We construct a representation $\Psi$ of $G$ on the space of square matrices of order n, such that ...
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1answer
58 views

Complete reducibility of a field extension of an lie algebra representation

Let $\mathfrak{g}$ be a lie algebra over a field $k$ with characterstic $0$ and $k\subset k'$ a finite field extension. Suppose $\mathfrak{g}\otimes k'$ has the property, that all finite dimensional ...
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Show $\mathbb F_{p}[x]/(x^{p}-1)$ is indecomposable as a representation of $\mathbb Z/ p\mathbb Z$

Let $R=\mathbb F_{p}[x]/(x^{p}-1)$. $R$ has both ring and vector space structure. I am trying to show that, given a representation $\rho : \mathbb Z/ p\mathbb Z\rightarrow GL(R)$, any invariant ...
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75 views

Questions about isotypic subspaces and co-isotypic subspaces of $V$.

Let $G$ be a finite group and $\chi: G \to \mathbb{C}^*$ a character. Let $(\pi,V)$ be a representation of $G$. The $(G, \chi)$-isotypic subspace of $V$ is defined by $$ V^{\chi}=\{v \in V: ...
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Representation theory approach VS Module theory approach?

Given an associative algebra $A$, there is a correspondence between representations of $A$ and left $A-$ modules. Thus, one can study the representation theory of an associative algebra via its left ...
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Fulton-Harris Lemma 3.35

In the proof of Lemma 3.35 in Fulton--Harris, Representation Theory, it is claimed that the identification $H(\phi^2(x),y)=H(x, \phi^2(y))$ implies that $\lambda$ is a positive real ($\phi^2$ is known ...
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44 views

Classification of $G$-modules

Suppose that I work only on vector spaces over $\mathbb C$. If I want to classify all $n$-dimensional modules over a finite group $G$, is it enough to choose a vector space $V$ with dimension $n$ and ...
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1answer
390 views

What is a regular FG-module?

What is meant by a regular $FG$-module. $G$ is a group and I believe $F$ is supposed to be a field. I'm completely confused by this concept on a question sheet and I can find lots of uses of the ...
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1answer
83 views

Homomorphic image of an irreducible representation

Let $H$ be a group. Let $V$ be any representation of $H$ and let $\sigma$ be an irreducible representation of $H$. Let $\varphi \in \text{Hom}_H(\sigma,V)$. I keep reading that the homomorphic image ...
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278 views

representation theory and schur's lemma [closed]

I was doing the exercises in serre's book on representation theory in p.26: Show directly , using Schur's lemma, that irreducible representation of an abelian group, finite or infinite, has degree 1. ...
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1answer
1k views

Proof of Schur's lemma

Can someone give me a simplified proof of Schur's lemma in group theory. Sorry if the question looks a standard textbook proof. But I find the proof complicated in books. It would be helpful if ...
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72 views

Regular representation

This is indeed a very simple question in representation theory, but I can not see why the Regular representation of a group G, satisfies the requirement of being a homomorphism Let $G$ be a group, ...
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127 views

show that the following three statements are equivalent

Let $\Phi$ be an irreducible complex representation of the group $S_n$ and $\Phi'(\sigma)=\Phi(\sigma) \operatorname{sgn}(\sigma).$ $(\sigma \in S_n).$ Prove that 1) $\Phi'$ is ...
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241 views

Decomposing the tensor product representation of $S_3$ in terms of irreducibles

I have a theorem which says that: If $\rho_1,...\rho_n$ are a complete set of irreducible $K$-representations of $S_n$ then we have that: $V^{\otimes n}=\bigoplus_1^k(V^{\otimes n}_{\rho_i})$ as ...
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Proving facts about groups with representation theory.

I was enrolled in a representation theory (of finite groups) course in the fall and throughout the class we focused on properties of representations and paradigms built around them. The whole time, I ...
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92 views

Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$

Let $G$ be a non-trivial finite group. Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$ This is my attempt so ...
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2answers
59 views

Why $\dim V^{G} = \dim\operatorname{Hom}_{G}(\mathbb{C}, V)$?

Let $G$ be a finite group. Let $V^{G} = \{v\in V: \pi(g)v=v, \forall g\in G\}$ be the subspace of invariants, where $(\pi, V)$ is a representation of $G$. Why $\dim V^{G} = \dim ...
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71 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
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1answer
45 views

Why $\dim V^G = Trace(\varphi)$?

Let $G$ be a finite group. Let $V^G = \{v \in V: \pi(g)v = v, \forall g \in G \}$. Here $(\pi, V)$ is a representation of $G$. Let $\varphi = \frac{1}{|G|} \sum_{g \in G} \pi(g)$. How to show that ...