# Tagged Questions

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

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### Representation theory via examples.

I have referred to some books on representation theory and somehow I cannot get a feel out of this subject. I guess the main reason is every book talks big and lacks examples. I am looking for best ...
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### The Number of Standard Young Tableau of a Frame

Suppose $\mathbb{F}$ be a frame corresponding to a partition $m_1 \geqslant m_2\geqslant...\geqslant m_r>0$ and $f(\mathbb{F})$ represents the number of standard young tableaux. Then i want to ...
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### What are the irreducible representations of a direct sum of Lie Algebras? [duplicate]

If I have two semisimple Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, then I have read that the irreducible representations of $\mathfrak{g} \oplus \mathfrak{h}$ are precisely the tensor products ...
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### Representation of $\mathbb{R}$, drop continuity assumption, Axiom of Choice.

A representation, for instance, of $\mathbb{R}$ is a group homomorphism $f: \mathbb{R} \to \text{GL}_n(\mathbb{R})$. If we assume that $f$ is continuous, then there is a very nice formula for all such ...
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### One dimensional representations of the plane orthogonal group $O(2)$.

I recently thought about representations of the orthogonal group $O(2)$ and found the one dimensional representations a bit confusing. We have that $SO(2)$ is a normal subgroup of $O(2)$. In fact, we ...
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### The $e$ in the operator $exe'$

I am learning the group algebra. A way to study the irreducible representations of a finite group is to use the idempotent operators in the group algebra. Suppose $e$ and $e'$ are the primitive ...
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### Dimensions of various lie groups by counting parameters of matrix representations

Show that the groups $\text{GL}(N,R), \text{GL}(N.C), \text{SL}(N,R), \text{SL}(N,C), O(N), SO(N), U(N),SU(N)$ have dimension $N^2, 2N^2, N^2-1, 2(N^2-1), \frac{1}{2}N(N-1), \frac{1}{2}N(N-1), N^2$ ...
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### Representation Theory: homomorphism of representation

I am working on the following lemma. Lemma Let $G$ be a finite group, $V,W$ vector spaces over a field $k$. Let $\rho_1 :G\rightarrow GL(W)$ and $\rho_2:G\rightarrow GL(\oplus_{h\in G} V)$ be ...
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### Constructing faithful representations of finite dim. Lie algebra considering basis elements

Well, I'm studying (engineering-) quantum mechanics dealing with representation theory of Lie algebras. The books I read introduce irreducible representations of $su(2)$ which is heavily related with ...
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### Endomorphisms of an ideal

If $R$ is a ring (with identity, but not necessarily commutative), and $I$ is an ideal of $R$, then thinking of $I$ as an $R$-module, what is $$\mathrm{End}_R(I)$$ Is there some way to see what this ...
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### A question about the proof of a theorem in representation theory

My question is about some parts of the proof of Theorem 8.1.10 from the book "A Course in the Theory of Groups" by Derek J.S. Robinson. To prove Theorem 8.1.10, we want to prove that there is a ...
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### Doubt while proving that an irreducible matrix lie algebra representation implies an irreducible matrix lie group representation

Let $G$ be a connected matrix Lie group with Lie algebra $g$. Let $\Pi$ be a representation of $G$ acting on $V$ a finite dimensional vector space, and $\pi$ the associated representation of $g$. ...
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### Archaic terminology: “algebra with order $n$”

I've been reading what's available as a Google books preview of "The Theory of Group Characters and Matrix Representations of Groups" by D. E. Littlewood. This book is from the 1950 and contains a lot ...
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### Invariant subspace under representation $\phi$

Let $\phi: \Bbb{Z}/n\Bbb{Z}\to GL_{2}(\Bbb{C})$ be the representation which takes $\bar{m} \to \begin{bmatrix} \cos (\theta) & -\sin (\theta) \\ \sin (\theta) & \cos (\theta) \end{bmatrix}$ ...
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### Polynomials in two variables over $\mathbb{F}_p$ fixed by $\operatorname{SL}_2(\mathbb{F}_p)$

Let $A=(a_{ij})\in\operatorname{SL}_2(\mathbb{F}_p)$. Consider the ring map $A:\mathbb{F}_p[x,y]\to\mathbb{F}_p[x,y]$ defined by $$A(x)=a_{11}x+a_{21}y$$ $$A(y)=a_{12}x+a_{22}y$$ and extended ...
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### Show that $\mathbb CG$ is isomorphic to $\mathbb CH$

Let $G$ be a group generated by $x,y$ with relation $xyx=yxy$. Let $H$ be a group generated by $a,b$ with relation $a^2=b^3$. Show that the group algebra $\mathbb CG$ given by the presentation for ...
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### Two-Component Spinor Index Placement

This may ultimately be a silly question, but a pedantic mind like mine gets tied into knots over differing notation. Let $\mathbb{W}$ be a complex two-dimensional vector space which carries the ...
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### Representations of Spin(2n)

Consider $G=Spin(2n)$. It is known that its representation ring is \begin{eqnarray} R(G)\cong\mathbb{Z}[V, \bigwedge\nolimits^2 V, \cdots, \bigwedge\nolimits^{n-2}V, S^+, S^-] \end{eqnarray} where ...
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### $\mathfrak{gl}$-style vs $\mathfrak{sl}$-style fundamental weights for $A_{r}$

I've recently been trying to brush up on the representation theory of Lie algebras of type A, and I'm running into some $\mathfrak{gl}$ vs $\mathfrak{sl}$ confusion. That is, when I compute the ...
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### Decomposition of group algebra $\mathbb{C}[G]$ [duplicate]

Let $G$ be a finite group. By Wedderburn theorem, we see that $\mathbb{C}[G]$ is product of matrix algebras. "Linear Representations of Finite Groups" by Serre has an explanation for this result ...
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### 3dimensional irreducible representation in $A_4$

I am trying to understand the $3$-dimensional irreducible representation in the alternation group $A_4$. $(\rho, \mathbb{C^4})$ is a $4$-dimensional representation (reducible) where for any element ...