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

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irreducible highest weight modules

Let $\mathfrak{g}$ be a simple Lie algebra. Let $M_{\lambda}$ be the Verma module over $\mathfrak{g}$ of highest weight $\lambda$ and $L_{\lambda}$ be the irreducible $\mathfrak{g}$-module of highest ...
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Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$. For a definition and discussion of ...
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modern treatment of the triviality of the multiplicator of the finite special linear group

Given your favorite version of the Heisenberg group one can prove the Stone-Von Neumann theorem. It is then not to hard to construct a family of representations on a central extension of $Sp\left(2n, ...
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Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to ...
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228 views

Reference request: Deligne's reconstruction theorem

I've heard this result referenced a few times on MO now. It is supposed to be a theorem of Deligne that gives some natural conditions under which an (abelian?) tensor category $C$ is the category of ...
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4answers
502 views

How to interpret the phrase “transforms under the irreducible representation”?

I'm reading Robert Gilmore's "Lie Groups, Physics, and Geometry," and trying to understand his brief presentation of Galois theory. I think I get the gist of the method, but would be grateful for help ...
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Isomorphism between group algebras

I am starting to study group algebras and I am stuck on the following problem. The first part is easy, but I copy it in case it helps to prove the second part. This exercise is taken from ...
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175 views

Computation of Clebsch-Gordan cofficients for point groups

In my work I have to deal with space and point groups and their representation. A lot of computations need Clebsch-Gordan coefficients. I am aware of the fact that these coefficients may be found in ...
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1answer
317 views

Algorithms to compute the orbit of the action of the Weyl group of a semisimple Lie algebra on a given weight?

Given a simisimple Lie algebra $\mathfrak{g}$ and a weight $\lambda$. Let $W$ be the Weyl group of $\mathfrak{g}$. Is there an algorithm (reference or software) to compute the set $W \cdot \lambda$ ...
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354 views

A common mistake (?) on group algebras

I have just started studying group representations with the book Representations of Groups by Lux and Pahlings (published by Cambridge). I have tried to solve some exercises to understand the concept ...
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94 views

Questions about Wronskian

I am reading a paper Bispectral and ($\mathfrak{gl}_n, \mathfrak{gl}_m$) dualities. I have some questions about some computations with Wronskian and dimensions of some vector spaces. On page 9 (line ...
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245 views

Representation which have no unique decomposition into irreducible

What kind of examples of groups and representations should I keep in mind, which do not uniquely decompose into irreducible representations? I am mostly interested in characteristic zero ...
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161 views

Schwartz kernel theorem for induced representation/ Schur algebra for locally compact groups

Given a finite group $G$ and subgroups $H$ and $K$, and representation $$\sigma: H \rightarrow GL(V_\sigma), \qquad \pi: K \rightarrow GL(V_\pi).$$ Now consider the space of functions $f: G ...
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277 views

Representation of Finite Groups

Is it true that any finite group determined by representation over closed field? In other words, are there exists two different groups with the same representations? For example, any non-abelian group ...
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1answer
352 views

Centre of symmetric group algebra

I'd like to know a reference for a simple proof that $\{c_\mu\mid \mu\vdash n\}$ is a basis for the centre of the symmetric group algebra $\mathbb{C}\mathfrak{S}_n$, where $c_\mu$ is the sum of all ...
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285 views

Showing $\mathbb C G$ modules are projective

Let $\mathbb C G$ be the group ring of a finite group $G$ and let $V$ be an irreducible $\mathbb C G$ module. I'm having trouble showing that $V$ is projective. Any help? Thanks!
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149 views

Compare of coefficients of two formal power series

Define $$ \Lambda_{i}(u) = \sum_{r=0}^{\infty} \Lambda_{i,r}u^{r}, \Psi_{i}(u) = \sum_{m=0}^{\infty} \psi_{i,m}u^{m}=k_i\frac{\Lambda_{i}(uq_i^{-1})}{\Lambda_{i}(uq_i)}. $$ How can we show that $$ ...
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131 views

Fourier analysis on groups and paths in a Cayley graph

If one takes a cyclic group and a function on this group, and performs harmonic analysis on it (classical Fourier analysis), the result is a set of coefficients, each one of them corresponding to ...
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565 views

Highest root, highest weight and highest short root

Are highest root and highest short root the same? Are there some example to show that the highest root and the highest short are not the same? Are there some example to show that the highest root and ...
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481 views

Action of dihedral group on polynomials on the plane

I am studying the action of the dihedral group on polynomials and I cannot find an answer to the following question. Let $V=\mathbb{R}^2$ be the standard representation of the dihedral group where ...
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359 views

Irreducible finite dimensional representations of $SL(2,\mathbb C)$

Is there a book that finds all the irreducibile finite dimensional representations of $SL(2,\mathbb C)$ without considering the Lie algebra $sl(2,\mathbb C)$? For example, how can I show "directly" ...
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323 views

Invariant inner product $\langle\,,\rangle$ on a Lie algebra

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ...
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519 views

Induction from normal subgroups

Let $G$ be a (finite) group and $N$ a normal subgroup. Given an irreducible representation $\pi$, how can I decompose $Ind_N^G \pi$? I'd be happy also about a good reference for this.
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143 views

Reference to a list of affine Cartan matrices

I can find a list of affine Dynkin diagrams in some books but cannot find a list of affine Cartan matrices. We can write down affine Cartan matrices using affine Dynkin diagrams. But are there a list ...
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194 views

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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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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227 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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188 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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218 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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518 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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271 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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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
388 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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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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114 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
95 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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809 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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849 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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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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128 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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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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1answer
74 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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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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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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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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296 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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1answer
84 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?