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

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Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable ...
2
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1answer
147 views

Proof: Clifford-Algebra representations are semisimple / completely reducible

There is a theorem: Every finite-dimensional Clifford-Algebra representation $V$ is semisimple / completely reducible, which means that it's a direct sum of irreducible subrepresentations. How this ...
10
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1answer
179 views

Theorem 1 chapter 8 of Fulton's Young Tableaux

I am reading Theorem 1 on page 110 of Fulton's Young Tableaux and have several questions on it. Let $E$ be a free module on $e_1,\ldots,e_m$ (for our purposes $E$ being a finite dimensional complex ...
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2answers
341 views

The unique representation of Heisenberg Group.

How can we construct the unique-up-to-isomorphism irreducible representation of Heisenberg Group.
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0answers
63 views

If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
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0answers
71 views

Group's algebra of finite group (representation theory)

Let $G$ be a finite group, and let $r$ - the number of it's conjugate classes, so our group has $r$ classes of irreducible representations. $\varkappa_1,..,\varkappa_r$ - irreduciible characters, and ...
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2answers
531 views

Find all idempotent elements in the group algebra $\mathbb CC_3$

Sorry but I'm quite new to group algebras and even Latex so if this is all wrong I apologize. By $\mathbb CC_3$ I mean the group algebra of the cyclic group of order 3 in the complex numbers A group ...
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0answers
85 views

Counting Young tableaux

Let's say we have some shape $\lambda$ and we want to fill this shape with numbers $\{1, .., m\}$ in non-decreasing order in rows and columns. How many such numberings do we have? I can not find ...
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1answer
64 views

The tangent space of $\mathrm{Aut}(T_eG)$

Let $G$ be a Lie group and $e \in G$ be the identity. I want to understand the following sentence. " $\mathrm{Aut}(T_eG)$ being just an open subset of the vector space of endomorphisms of $T_eG$, its ...
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40 views

Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
3
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1answer
121 views

Number of ways a group element of a finite group can be written as a given word

I had previously asked about the number of ways a group element in a finite group could be written as a commutator (the question is still open for a proof, by the way) In how many ways can a group ...
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1answer
101 views

representations and modules

I am reading representations of Lie Algebra in Humphreys.He is defining representation as $L$-modules. In case of group representation we have the correspondence between representation and modules ...
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1answer
283 views

the representation on the regular representation is faithful

I am reading the proof of the following proposition. Proposition. As algebras, $\mathbb{C} G \cong \bigoplus \mathrm{End}(W_i),$ where $G$ is a finite group and $W_i$ are irreducible representation ...
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1answer
124 views

A character of an induced representation

I want a help to solve the following exercise from the book, Representation Theory, by Fulton and Harris. Exercise 3.19 (p.34) Let $H$ be a subgroup of a finite group $G$. Let $W$ be a representation ...
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0answers
139 views

Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel: Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ ...
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1answer
404 views

The number of irreducible representations

I am reading a textbook "Representation theory" by Fulton and Harris and I have a question. They proved the following theorem on page 16. With an Hermitian inner product on a set of class function, ...
2
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2answers
208 views

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 ...
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1answer
397 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 ...
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1answer
981 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?
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349 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 ...
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2answers
90 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 ...
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4answers
197 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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2answers
76 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
100 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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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 ...
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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} \, ...
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2answers
113 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 $ ...
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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
173 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 ...
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1answer
70 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
67 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 $$\rho(a) = \alpha(a)^{1/2}$$ ...
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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 ...
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1answer
223 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: ...
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1answer
100 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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82 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 ...
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1answer
272 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
73 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 ...
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1answer
184 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
107 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 ...
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224 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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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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453 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 ...
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1answer
109 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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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 ...
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1answer
187 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]} ...