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

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Show that irreducible standard cyclic module is finite dimensional

I had problems understanding the following proof. Maybe someone could help me with this? Let {$\ x_i, y_i $} be the standard generators of the Lie Algebra L. Let $\ ...
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2answers
129 views

Lie Algebras: (ad $\ y)^4(z)=0 $, since root strings have length at most 4

Can somebody please explain this to me? (ad $\ y)^4(z)=0 $, since root strings have length at most 4. Note: y and z are root vectors belonging to two negative roots.
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218 views

fundamental representation of $\ sl(l+1,F)$

This problem concerns the topic representation theory of Lie Algebras. The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra. I would be very ...
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1answer
183 views

What is meant by “direct summand in a tensor product”?

I am currently working on the topic of Lie - Algebras and I have stumbled a few times over the expression "direct summand in a tensor product". The text says that $\ V(\lambda) $ as an ...
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1answer
179 views

What is meant by $\sum_{p + q = v + w} {\dim V_p * \dim W_q}$?

I am currently working on the topic of Lie - Algebras. What is meant by $\displaystyle\sum_{p + q = v + w} {\dim V_p * \dim W_q}$ ? $\ V_v $ and $\ W_w $ denote weight spaces I don't know how to ...
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203 views

Questions about Freudenthal's formula

I am reading the book Introduction to lie algebras and representation theory. I have some difficulty in understanding some parts of the book for Freudenthal's formula. Page 120, line 6, why ...
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459 views

How to draw a weight diagram?

Given a weight, say $\omega=3\lambda_1+4\lambda_2$, where $\lambda_1, \lambda_2$ are fundamental weights (type A Lie algebra). How to draw the weight diagram of the irriducible representation with ...
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Is the dual representation of an irreducible representation always irreducible?

Let $G$ be a group and let $V$ be a complex vector space which is a representation of $G$. Let's write the (left) action of $g\in G$ on $v\in V$ as $gv$. The dual vector space of $V$ is the set of ...
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263 views

Universal Casimir element

On page 118 of J.E. Humphreys' book Introduction to Lie algebras and representation theory, paragraph 3 of section 22.1, what is the motivation of the definition of $c_{ad}$ in this way? Why we ...
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105 views

The notation wt in representation theory

Let $\lambda$ be a weight and $P^+$ be the set of all positive roots. Define $wt(\lambda) = \{ w(\mu) \mid w\in W, \mu \in P^+, \mu \leq \lambda \}$, where $W$ is the Weyl group of a Lie algebra. So ...
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1k views

Unitary representations of non-compact Lie groups

This question is somewhat of a continuation of this question that I had asked earlier - Representations of a non-compact group are labeled by its maximal compact subgroup? I want to know when or is ...
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1answer
351 views

induced homomorphism from a group action

let $X$ be a topological space on which a group $G$ acts. 1) is it true that this action always induces an homomorphism $G\rightarrow Aut(X)$? My guess is no. because i think the induced ...
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6k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
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3answers
112 views

group action in orthogonal decomposition

let $V$ be an inner product space. Let $X$ a subspace of $V$ and $X'$ its orthogonal complement i.e, $V=X\oplus X'$. Let $G$ be a group $G$ acting on $V$. an element in $X\oplus X'$ is ...
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1answer
94 views

Where can I find rigorous statements about the spectral decomposition of reductive groups?

Given a global field $F$ and a reductive group $G$, where can I find the spectral decomposition of $$ L^2( Z(\mathbb{A}) G(F) \backslash G( \mathbb{A})).$$ I will need the result in this generality, ...
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730 views

The mathematics behind Clebsch-Gordan Coefficients

In quantum physics we have to work a lot with Clebsch-Gordan coefficients and generalizations like the Wigner 3j,6j, and 9j symbols. In our coursework we are taught that the coefficients are coupling ...
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781 views

Does regular representation of a finite group contain all irreducible representations?

I know that every irreducible representations of $S_n$ can be found in $\mathbb{C}S_n$. I wonder how can I prove that irreducible representations of a finite groups $G$ can be found in $\mathbb{C}G$. ...
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80 views

Eisenstein spectrumfor $GL(n)$

Fix a global field $F$. Does every automorphic representation of $GL(n)$ appear as an arbitrary twists in the continuous spectrum of $GL(m)$, $m>n$? What happens for the automorphic ...
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124 views

Is it possible to compute the “gcd character” of two representations of a finite group?

I have two reducible representations of a finite group $G$ of Lie type, $\rho, \pi$. They both have multiplicity one, and I know that they share exactly one irreducible subrepresentation. Is there a ...
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0answers
112 views

image of symmetric matrices under representation of $GL_2(\mathbb{R})$

Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally ...
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73 views

$\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(Ae,N), N\otimes_{eAe}A=N\otimes_{eAe}eA$.

Let $e$ be an idempotent of a ring $A$ and $N$ is an $A$-module. Why $\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(Ae,N), N\otimes_{eAe}A=N\otimes_{eAe}eA$? Can you prove this explicitly? Is the ...
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277 views

Newbie: Group Representation $\Leftrightarrow$ Left Module over the Group Ring

I am trying to understand the equivalence between group representations, $(V, \rho)$, and left modules over the group ring $F[G]$. Can you explain explicitly why it is the same? My progress: Consider ...
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3answers
515 views

how to compute the Euler characters of a Grassmannian?

Let $G(n,m)$ be the Grassmannian of all n-dim subspaces of an m-dim vector space over $\mathbb{C}$. How to compute the Euler characters of $G(n,m)$? For example, $G(1, 2)$ is $\mathbb{C}P^1$ which is ...
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301 views

Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...
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289 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
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83 views

Construction of representations

Is there an example, where given a conjugacy class in a finite group, can we construct an irreducible representation from it?
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105 views

Computing the dimension of representations in a reducible induced representation

I'm looking at the induction of representations of a parabolic subgroup of $Sp_4$ into the whole group. There are some cases that the result is reducible, and I need to compute the dimensions of the ...
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1answer
33 views

generalized eigenspaces for many operators III

Let $\phi_1, \cdots, \phi_n$ be commutative linear operators on a vector space $V.\,\,$ Then we have $$V=\oplus V_{(a_i)}, \text{ where } V_{(a_i)} = \{x\in V \mid \exists p \,\,\text{ such that }\,\, ...
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1answer
107 views

generalized eigenspaces for many operators II

Let $\phi_1, \cdots, \phi_n$ be commutative linear operators on a vector space $V.\,\,$ Then we have $$V=\oplus V_{(a_i)}, \text{ where }\, V_{(a_i)} = \{x\in V \mid \exists p \,\,\text{ such that ...
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2answers
261 views

Reference for a “wild” problem

I am currently working on something related to the character theory of the group of unipotent upper triangular matrices with elements in a finite field. I have seen in many papers on the topic the ...
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1answer
182 views

generalized eigenspaces for many operators

Let $\phi_1, \cdots, \phi_n$ be commutative linear operators on a vector space $V.\,\,$ Then we have $$V=\oplus V_{(a_i)}, \text{ where }\, V_{(a_i)} = \{x\in V \mid \exists p \,\,\text{ such that ...
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869 views

Restriction to a normal subgroup

More exam preparation. Let $A$ be a normal subgroup of a finite group $G$ and $V$ an irreducible representation of $G$. Show that either $\text{Res}_A^G V$ is isotypic (a sum of copies of one ...
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1answer
176 views

Finding double coset representatives in finite groups of Lie type

Is there a standard algorithm for finding the double coset representatives of $H_1$ \ $G/H_2$, where the groups are finite of Lie type? Specifically, I need to compute the representatives when ...
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1answer
131 views

$2×2=3+1$ for $\operatorname{GL}_2$

If $V$ is the natural representation for $\operatorname{GL}(2,q)$, then $V⊗V$ appears to decompose into the direct sum of a (strange?) one-dimensional module and a three dimensional module. I've ...
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634 views

finite subgroups of PGL(3,C)

The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, ...
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Shortest way of proving that the Galois conjugate of a character is still a character

Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ ...
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Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?

I have a question about the some rings and fields. Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?
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371 views

On “complexifying” vector spaces

I think this question should be quite trivial. For some reason I did not really get the author's argument. I shall use the symbols from the book to avoid ambiguity. In the book "Lectures on Lie ...
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1answer
775 views

Why does the trivial representation have degree 1?

If you have a representation from $G \to Aut(V)$, it has degree $1$ if $V$ is a 1-dimensional vector space over $F$. The trivial representation sends any element of $G$ to the trivial automorphism $v ...
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1answer
59 views

How do you find the matrices of a representation given the matrices of subrepresentations?

Specifically, there is a passage in Dummit and Foote that says Suppose $V$ is a finite-dimensional $FG$-module and $V$ is reducible. Let $U$ be a $G$-invariant subspace. Form a basis of $V$ by ...
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1k views

Irreducible representations of a semidirect product

I have two finite groups. The irreducible representations of their product are given by tensor products of the irreducible of representations of the groups. Is there a way to build the irreducible ...
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1answer
93 views

questions about the paper: Affine quivers and canonical bases

I am reading the paper Affine quivers and canonical bases. I have a question on page 114 of the paper. In the proof of property (b), line 6 of page 114, why "for each $\gamma \neq 1$, $tr(\gamma, ...
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1answer
700 views

relations between root lattice and weight lattice

Let $Q$ and $P$ denote the $\mathbb{Z}$-span of the simple roots and fundamental weights respectively. What are the relations between $Q$ and $P$? Does $P$ contain $Q$? Thank you.
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956 views

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
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182 views

clarification on the definition of a group C*-algebra

I've been trying to understand the definition of a group C*-algebra. Given a topological group $G$ and a C*-algebra $A$, let $u: G \to A$ define a unitary representation $U(G)$ of $G$ on $U(A)$, the ...
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477 views

How can there be multiple irreducible representations of a group each having distinct dimension?

The matrix elements for the $(2l+1)$-dimensional irreducible representation of SO(3) are given by: $D_{m',m}^l(\phi_1,\Phi,\phi_2)=[i^{m'-m}\sqrt{(l+m')!(l-m')!(l+m)!(l-m)!}$ ...
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322 views

How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
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352 views

Field of definition of representations of symmetric groups

Can one show in an elementary way, without recourse to Young tableaux etc., that the complex representations of symmetric groups are realisable over $\mathbb{R}$? It is easy to show that they are all ...
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What is the meaning of an “irreducible representation”?

What does it mean to talk about the "irreducible representatives of SO(3)"? I'm struggling to understand the concept of irreducible representations. Could someone give a concrete example for someone ...
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392 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...