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

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Are spinors, at least mathematically, representations of the universal cover of a lie group, that do not descend to the group?

Following on this question about how to characterise Spinors mathematically: First, given a universal cover $\pi:G' \rightarrow G$ of a lie group $G$, is it correct to say we can always lift ...
7
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1answer
434 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
7
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1answer
1k views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
7
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4answers
370 views

What makes irreducible representations nice?

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi,\Omega)$ a cyclic representation. What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations ...
2
votes
2answers
95 views

Is every linear representation of a group $G$ on $k[x_1,\dots,x_n]$ a dual representation?

Let $\rho\colon G\to GL(V)$ be a linear representation of $G$ on a $k$-vector space $V$. The dual representation is $$G\to GL(V^*),\quad g\mapsto(\varphi\mapsto\varphi\circ\rho(g^{-1})).$$ By the ...
2
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4answers
215 views

Irreducible representation of dimension $5$ of $S_5$

i am searching for a concrete as possible description of the (there are two but the are obtained from each other by tensoring with the signature representation) irreducible representation of dimension ...
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vote
2answers
78 views

Continuous group representation

Suppose you have a topological group $G$ , a normed $k$- vector space $V$ and a group homomorphism $\rho:G\longrightarrow GL(V)$. How do you define the topology on $GL(V)$ to make this map ...
3
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1answer
102 views

Representation of topological groups

I am looking for a good book of topological representation. I have a very good insight of representation theory of finite groups, and I want to explore topological representations. I saw a book by ...
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1answer
52 views

$d\pi(X)$ is skew-symmetric. What does it mean?

This is from a lemma in Lang $SL_2$ If $\pi$ is a unitary representation of G, and $X \in \mathfrak g$, then $d\pi(X)$ is skew symmetric on $H_\pi^\infty$ What does skew symmetric mean here? And ...
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1answer
116 views

lagrangian subspace and Heisenberg group

Let $(V,\omega)$ be a symplectic vector space. Also we assume $L\subset V$ be a Lagrangian subspace., and $H(V)$ be Heisenberg group, then why $L\bigoplus U(1)\subset H(V)$ is maximal abelian ...
3
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1answer
36 views

Inverse of spherical transform

For further notation see unit vector of induced representation in $SL_2$ Let $f\in C_c^\infty(G/\!/K)$ and set $$f_x(y) \colon= \int_K f(xky) \, dk.$$ Let $$\phi(x,s) \colon= \int_K p(kx)^{s+1} \, ...
3
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2answers
117 views

Equivalent definitions of Verma modules

This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ ...
2
votes
1answer
42 views

What are modules in $\operatorname{add} T$ explicitly?

Let $A$ be a $K$-algebra and $T$ an $A$-module. The category $\operatorname{add} T$ is defined as the smallest additive subcategory of the category $\operatorname{mod} A$ (the category of all finite ...
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1answer
27 views

Lang $SL_2$ two formulas for Harish transform

Let $G = SL_2$ and give it the standard Iwasawa decomposition $G = ANK$. Let: $$D(a) = \alpha(a)^{1/2} - \alpha(a)^{-1/2} := \rho(a) - \rho(a)^{-1}.$$ Now, Lang defines ($SL_2$, p.69) the Harish ...
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1answer
203 views

How to compute the ordinary quiver of $B = \operatorname{End}_A(T_{A})$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
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1answer
46 views

How to show that ${}_{B}T_{A} \otimes DM \in \operatorname{Gen}({}_{B} T)$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
0
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1answer
73 views

How to show that $DA\cong D\operatorname{Hom}_{B}(T, T) \cong DT \otimes_{B} T$?

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski. Let $A$ be a ...
3
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1answer
68 views

Where does a modular tensor category come from?

I have studied the definition of a modular tensor category. I jumped into this subject and almost have no background. My question is: what kind of mathematics does a modular tensor category ...
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1answer
30 views

unit vector of induced representation in $SL_2$

Consider the Iwasawa decomposition $SL_2 = ANK$, and let $P = AN$. Consider the modular function *Serge Lang $SL_2 p. 46$:*$$\Delta(p) = \Delta(an) = \alpha(a),$$ let $$\rho(a) = \alpha(a)^{1/2}$$ ...
7
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4answers
180 views

In how many ways can a group element in a finite group be written as a commutator?

It seems there is a result by Frobenius that states that the number of ways an element $g$ of a finite group can be written as a commutator ($\phi(g) = | \{(x,y) \in G \times G: g = [x,y]\}|$) is ...
3
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1answer
237 views

Measures on Iwasawa decomposition

In the following I present two results, which look very similar, but require different proofs. I'd like to know why the second result doesn't admit the same proof as the first. Lang $SL_2$ p39: ...
3
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1answer
104 views

What is an awesome C*-algebras result and/or theory derived from C*-algebra Theory

First of all, I'm deeply sorry this isn't a real math. question, but 'meta' didn't seem like the right place to ask this either. So there goes: I'm studying representation theory and operator theory ...
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0answers
91 views

Fixed subspace of an complex orthogonal representation

Let $V\cong\mathbb{C}^n$ be a finite dimensional complex vector space. We have essentially one degenerate symmetric bi-linear form $(-,-)$, namely the one corresponding to the identity matrix. Let ...
4
votes
1answer
314 views

Cyclic vectors of an irreducible representation of a C*-algebra

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi)$ an irreducible representation of $\mathcal{A}$. I want to prove the statement: all $\xi \in H$ are cyclic or $\pi(\mathcal{A})=\{0\}$ and ...
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1answer
75 views

Lang $SL_2$: fin-dim irreducible subspace for abelian group has dim < 2

Lang $SL_2(\mathbb R)$ p. 24, Theorem 2 : Let $\pi$ be an irreducible representation of G on a Banach space H. Let $H_n$ be the subspace of vectors v s.t. $$\pi(r(\theta))v = e^{in\theta}v.$$ If ...
2
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1answer
203 views

Understanding the definition of the represention ring

In Fulton, Harris, "Representation Theory. A first Course" there's the following paragraph which I don't really understand: The representation ring $R(G)$ of a group $G$ is easy to define. First, ...
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2answers
120 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...
2
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2answers
236 views

Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$, where $N \unlhd G$ and $ \theta \in Irr(N)$.

Let $N \unlhd G$ and $ \theta \in Irr(N)$. Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$. Where $I_G(\theta)$ is the stabilizer of $\theta$ in the action of $G$ on $Irr(N)$ defined by ...
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0answers
123 views

Hamiltonian reduction for unit sphere

Let $(M, \omega)$, be Symplectic Vector space and $N\subset M$ be unit sphere. Then why $N/ker\omega\mid _N$ is naturally $\mathbb{P}^{n-1}(\mathbb{C})$ Here ker$\omega$=$\{ y\in M: \omega(x,y)=0, ...
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461 views

Simultaneously (generalized) diagonalizable matrices

I heard the following theorem from our textbook: Given $A,B$ are two commuting ($AB=BA$) real normal matrices. There's some real orthogonal matrix $P$ such that $P^{-1}AP$, $P^{-1}BP$ are ...
2
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1answer
115 views

$\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ .

Let $G = H \times K$. Let $\phi \in \operatorname{Irr}(H)$ and $\theta \in \operatorname{Irr}(K)$ be faithful. Show that $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$. Problem 4.3 of ...
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0answers
62 views

writting a code for finding the Kostant partition function

How to write a code in sage for finding the Kostant partition function for the elements of root lattice of rank 1 affine lie algebra $A_{1}^{(1)}$ which is defined as follows: $K(\beta)$ = the ...
8
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1answer
191 views

The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group

Question What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group? The trivial corepresentation is given by $\Delta_{|W}$ where $W$ is just the one dimensional ...
2
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1answer
46 views

How to show that $K[t]/(t^d)$ is indecomposable as a $K[t]$-module and $\operatorname{End}_{K[t]} (K[t]/(t^d)) \cong K[t]/(t^d)$?

How to show that (1) $K[t]/(t^d)$ is indecomposable as a $K[t]$-module? (2)$\operatorname{End}_{K[t]} (K[t]/(t^d)) \cong K[t]/(t^d)$? I think that if (2) is true, then $\operatorname{End}_{K[t]} ...
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1answer
54 views

How to show that the index $[G: H]$ is invertible?

I am reading the book Elements of representation theory of associative algebras, volume 1. I have a question on page 176, the proof of Corollary 5.2. In order to applied (5.1)(b), we have to show that ...
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0answers
75 views

Extending isomorphisms in the semi-simple case.

Is there some proposition saying how to extend an isomorphism of $k$-vector spaces where $k$ is a field of characteristic $p$ to an isomorphismus of $k[H]$-modules where $H$ is a group of order prime ...
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0answers
273 views

Character Table of the *finite Heisenberg Group* $H_2$

My task is to compute the character table of the finite Heisenberg Group $H_2$: $$ \left\lbrace\ a=[a_1,a_2;a_3]:= \begin{pmatrix} 1 & a_1 & a_3 \\ 0 & 1 & a_2 \\ 0 & 0 & 1 ...
1
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1answer
254 views

Intertwiner of symmetric group representations (Basic)

I am preparing for an exam and there is an excercise which I have to solve but I got stuck. The excercise states: Let $V=\mathbb{C}^3$ be the permutation representation of the symmetric group $S_3$. ...
6
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1answer
261 views

Question on the proof that exterior powers of standard representations are irreducible

I am reading Fulton and Harris representation theory Chapter 3.2 right now and I got a question on the proof of the fact that exterior powers of standard representations are irreducible. Let ...
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1answer
87 views

How to prove this statement on finite groups?

I Fulton and Harris Chapter 3.2 we have that if $\mathbb{C}^n$ is the permutation representation of $S_n$ (symmetric group) then we can write $\mathbb{C}^n=V\oplus U$, where $U$ is the trivial ...
3
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1answer
43 views

Restriction of the Steinberg representation

Let $G_{n}=GL(n,F)$, where $F$ a locally compact non-Archimedean field, $St_{G_{n}}$ the Steinberg representation of $G_{n}$, and $B$ the standard Borel subgroup of $G_{n}$. We denote $\pi_{n}$ the ...
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1answer
253 views

Computing eigenvalues from characters

This is a question in Representation theory, a first course, where the authors try to explain why character theory turns out to be so effective for the study of representations of finite groups. In ...
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131 views

Comparing two character tables

Suppose that you are given two finite groups, for example, via their Cayley tables. One can efficiently compute their character tables (efficiently = polynomial time in the order of the group), this ...
6
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2answers
792 views

Relation between root systems and representations of complex semisimple Lie algebras

I'm trying to understand the machinery of root systems for the purpose of classifying complex semisimple Lie algebras. During this process i lost the overview, espacially when it came to highest ...
3
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1answer
302 views

Irreducible representations over $\Bbb R$

How to prove that all irreducible representations over $\mathbb{R}$ of finite abelian group have dimension 1 or 2?
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2answers
201 views

How important are automorphic representations among admissible ones?

I'm currently studying automorphic representations on Bump's book "Automorphic forms and representations" and on Gelbart's "Automorphic forms on Adele groups". And I have some problems in ...
4
votes
3answers
114 views

Fourier analysis on finite abelian groups

Can someone help me show if $f$ is a character of a finite abelian group then for all $a\in G$, $$\sum_{[f]}f(a)\stackrel{}{=} \begin{cases} |G| & \text{if $ a$ is the identity} \\ 0 & ...
5
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0answers
128 views

On Applications of the Murnaghan-Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
3
votes
1answer
48 views

Number of indecomosbale $\mathbb{Z}_p[G]$ modules finite

Is there a theorem like those of Jones, which tells if the number of different $\mathbb{Z}_p[G]$ modules is finite, where $G$ is a finite group and $\mathbb{Z}_l$ the $p$-adic ring?
5
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1answer
92 views

Very basic question on continuous representations

Suppose $G$ is a topological group, i.e. $$G \times G \to G, (g,h) \mapsto gh ~~~~ G \to G, g \mapsto g^{-1}$$ are continuous. Suppose $K$ is a field with norm $|\cdot|$ on it and $V$ is a normed ...