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

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478 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
50 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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50 views

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 ...
5
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0answers
95 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. ...
2
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1answer
28 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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3answers
690 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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0answers
84 views

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 ...
2
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1answer
66 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
63 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 ...
4
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2answers
108 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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0answers
28 views

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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0answers
38 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
99 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 ...
3
votes
1answer
73 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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0answers
60 views

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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0answers
135 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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1answer
173 views

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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60 views

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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1answer
48 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 ...
1
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1answer
689 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 ...
2
votes
1answer
99 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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2answers
423 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
2k 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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1answer
96 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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1answer
160 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 ...
3
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1answer
440 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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3answers
391 views

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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2answers
107 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 ...
2
votes
2answers
69 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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0answers
143 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
50 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 ...
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0answers
90 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
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1answer
41 views

How to show that $V=\oplus_{\chi\in Hom(G, \mathbb{C}^{\times})} V^{\chi}$?

Let $G$ be a finite cyclic group and $Hom(G, \mathbb{C}^{\times})$ be the set of all characters of $G$. For $\chi: G \to \mathbb{C}^{\times}$ and $(\pi, V)$ a representation of $G$, let $$ V^{\chi} = ...
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1answer
64 views

Notation in Fulton-Harris for dual representation

In Fulton-Harris's Representation Theory, on page 4 they write This in turn forces us to define the dual representation by $$\rho^*(g) = \,\,^t\rho(g)^{-1}:V^* \to V^*.$$ I'm confused by this ...
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2answers
201 views

The average value of irreducible character of a non-trivial finite group

Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$ I try. But I think that I am wrong. ...
4
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0answers
201 views

Confused about Borel-Weil theorem

I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...
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1answer
138 views

Reference for Weyl Character Formula

I am reading Lie algebra book by James E.Humphreys. This book giving enough discussion about Weyl Character Formula and its proof, still I would like to know what are the other books or lecture notes ...
3
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1answer
89 views

Irreducibility of complex 2-dimensional character of the group $ S_3 $

Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
3
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2answers
56 views

Isomorphism $\text {Rep}_{G,k}\cong \space \text {Mod}_{k[G]} $

Let $G$ be a group and $k$ a field. Proposition: There is an isomorphism of categories $F:\text {Rep}_{G,k}\rightarrow \space \text {Mod}_{k[G]} $. I begun by proving that there is a functor ...
2
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1answer
365 views

Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
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2answers
262 views

History of representation theory

For a student's journal I want to write a short article about history and importance (applications) of representation theory. Are there some accesible literature about this?
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2answers
78 views

Prove that a subset $\{a \in A \mid \chi (a) = 1\}$ is an $(n - 1)$-dimensional subspace of $V$.

Let $A$ be the additive group of n-dimensional vector space $V$ over the field $\mathbb{F}_p$. Let $\chi$ - a nontrivial irreducible complex character of A. Prove that a subset $\{a \in A \mid ...
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1answer
37 views

Why elements in a non-archimedian local field $F$ are of the form $\pi^n u$?

Let $F$ be a non-archimedian local field with valuation $\nu$. Then $\mathcal{O}=\{x\in F: \nu(x)\geq 0\}$ is the ring of integers of $F$. $\mathfrak{m}=\{x\in F: \nu(x)> 0\}$ is the maximal ideal ...
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51 views

Why will we never be able write down all automorphic representations “explicitly”?

In a recent article (p. 7), J. Arthur writes In fact, it is pretty clear that we will never be able to write down all automorphic representations explicitly. I have often heard similar remarks ...
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52 views

Monoid Ring of a Commutative Cancellative Ordered Monoid

Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by $$ m \preceq m' \text{ if and only if there ...
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55 views

What does “The Hilbert space carries a representation of […] group” means?

Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta). Effectively, I know that this means ...
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1answer
69 views

if $g^k=e$ then $\chi(g)=\sum_j^n \zeta_k$

Let $G$ be a group. Let $g \in G$ and $g^k=e.$ Let $\chi$ be an $n$-dimensional character of the group $G.$ Let $\zeta_k$ be $k$-th root of unity. Prove that $\chi(g)$ is equal to sum of a ...
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75 views

Characterisations of the RSK correspondence

I know of the following three definitions of the RSK correspondence: (i) Row insertion (or more generally, plactic insertion) (ii) Viennot's construction (iii) Fomin's growth diagrams However, all ...
4
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1answer
494 views

Commuting matrices and simultaneous diagonalizability

It is a known fact from linear algebra that if a set of matrices is pairwise commutable then they are simultaneously diagonalizable. A problem in the book I am currently studying asks to prove this ...
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2answers
44 views

Why does $F\langle g\rangle\cong F[x]/(x^n-1)$?

This is example 1 on page 842 of Dummit and Foote. If $g$ is the cyclic group of order $n$, then the group ring over a field $F$ $$ F\langle g\rangle\cong F[x]/(x^n-1) $$ by the surjective ...