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

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How is a vector a representation?

I am working on a homework problem that gives the character table for the octahedral group O, and then asks to ``decompose the vector (x,y,z) into irreps of O''. What does this mean? How can a vector ...
4
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77 views

What are all representations of the quiver $Q$?

Let a quiver be $Q=(Q_0, Q_1, s, t)$, where $Q_0$ is $\{1, 2\}$. The quiver has only one arrow: $\alpha: 2 \to 2$. What are all representations of $Q$? Thank you very much. In my original ...
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93 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
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75 views

Induced representation is isotypical?

Is there a theorem like this for the induced representation? Let $N$ be a normal subgroup of a finite group $G$ and $\rho$ be an irreducible linear representation over any field $k$. Then one of ...
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39 views

Different induced representations - same simples?

is the following case possible: $\pi_1, \pi_2$ two simple representations of the same subgroup over an arbitrary field. $\operatorname{Ind}(\pi_1)$ and $\operatorname{Ind}(\pi_2)$ have equal ...
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57 views

Give a bijection between unitary, degree one representations of Z and elements of T.

Definition: My book defines $\mathbb{T}$ as the unit circle in $\mathbb{C}$, i.e. $\mathbb{T}=\{z \in \mathbb{C} : |z| = 1\}$ I'm trying to answer this question: "Give a bijection between unitary, ...
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93 views

Representations of $\text{GL}_2(\mathbb{Q})$

Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
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584 views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
3
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104 views

Unitary Equivalence of Two Irreducible $ * $-Representations of a GCR $ C^{*} $-Algebra that Have the Same Kernel.

In general, if two irreducible $ * $-representations of a $ C^{*} $-algebra $ A $ have the same kernel, then we can say that they are approximately unitarily equivalent. When $ A $ is GCR, how can we ...
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156 views

Completing a Character Table for a Group of Order 18

I have the following homework question: A group of order 18 has the following partial character table, where $y=-\frac{1}{2} + xi$: \begin{array}{c | c c c c c} \hline\hline & g_1 & g_2 ...
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113 views

Some questions about representation theory in the modular case

I'm working on a paper which uses representation theory in order to compute some characters and deduce arithmetical statements about certain field extensions. Let $\Delta$ be a group of order prime ...
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121 views

Question about the radical of the Jacobson radical.

I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
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Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
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2answers
129 views

Isomorphism of faithful representations

Let $G$ be a group and $f,g: G \rightarrow GL(V)$ be two faithful representations over some field $K$ with $f:x\mapsto f(x)$ and $g:x \mapsto f(x^{-1})$. I would like to find out if $f$ and $g$ are ...
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103 views

For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?

I'm reading the Atiyah-MacDonald book on Commutative Algebra. At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is: $G$ = finite group, ...
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95 views

highest weight module correspondence with irreducible representation

Let g be a simple Lie algebra. L(λ) be the irreducible g -module of highest weight λ . are all highest weight modules irreducible ?
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Subspaces stabilized by representations of $\mathrm O(9)$

I am trying to figure out what representations of maximal subgroups of $\mathrm{GL}_{n^2}$ stabilize one dimensional subspaces in $\mathrm{GL}(\mathrm{Sym}^n(\Bbb C))$. More precisely, let the setup ...
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73 views

Representations over $\mathbb{Q}_p$

I would like to understand representations over the $p$-adic field $\mathbb{Q}_p$ and find simple $\mathbb{Q}_p[G]$ modules for a finite group $G$. Is there some famous literature like for ...
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408 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
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59 views

Do we have $e(\operatorname{rad}^2 A)=\operatorname{rad}^2 (e A)$?

Let $e$ be a primitive idempotent of $A$, where $A$ is a finite dimensional algebra over an algebraically closed field $K$. Do we have $e(\operatorname{rad}^2 A)=\operatorname{rad}^2 (e A)$? Here ...
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122 views

Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
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1answer
47 views

Surjective map results in subrepresentation

I need to prove that a surjective homomorphism of finite $\mathbb{F}_p[\Delta]$-modules $$A \twoheadrightarrow B$$ results in $B$ being a subrepresenation of $A$ of the group $\Delta$ of order prime ...
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70 views

Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
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263 views

Good book on representation theory after reading Rotman

I'm about to finish Rotman's "Introduction to the Theory of Groups" and I would like to continue my study of group theory with a book on representation theory. The book should give a broad overview ...
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1answer
89 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
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367 views

Decomposing tensor product of lie algebra representations

I'm given a lie algebra representation $\pi$ of some semi-simple algebra and that it decomposes into a sum of irreducible representations. What technique should I use to show the decomposition of ...
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27 views

Tensor product of an irreducible $G$-representation and a one-dimensional representation [duplicate]

If $G$ is a finite group, $V$ is an irreducible $G$-representation and $W$ is any 1-dimensional $G$-representation (both over an algebraically closed field of characteristic zero), show that $V ...
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1answer
112 views

How does the holonomy act on the tangent space at a point?

Suppose $(X,h)$ is a compact $n$-dimensional Hermitian manifold, with holonomy group $H$. Now we know,since $X$ is a complex manifold, that $H\subset U(n)$, and that there is a representation of $H$ ...
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1answer
121 views

Artinian ring with zero finitistic dimension

Let $R$ be a left artinian ring with identity. Suppose $R$ contains copies of all its simple right $R$-modules. Is it true that every left $R$-module of finite projective dimension is projective (so ...
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35 views

concerning coadjoint representation

Let $\xi $ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
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Induced representation - Workout for $S_3$

I want to give an induced representation of $S_3$. Elements of $S_3$ are given by: $$e,\ c=(123),\ c^2=(132),\ r=(12),\ rc=(23),\ rc^2=(13)$$ with the following relations: $$r^2=c^3=e,\ rcr^{-1}=c^2$$ ...
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139 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
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86 views

Linearizations induced by the trivial one?

Let $k$ be a field, algebraically closed for simplicity. $G=GL_{n+1}(k),$ $X=\mathbb{P}_k^n$ and consider the action $G\times X\rightarrow X$ given by $(g,x)\mapsto gx$ (thus induced by usual left ...
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Why is the dual space of Cartan subalgebra an irreducible representation of Weyl group

it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved ...
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Nonabelian group with all irreducible representations one-dimensional

All irreducible representations of an abelian group are one-dimensional. For a finite group, the coverse is also true - if all irreducible representations are one-dimensional then the group is ...
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100 views

Induction from normal subgroup, problem with degrees

Suppose $K$ is an arbitrary field, $G$ a finite group and $N$ a normal subgroup. If one know all the irreducible representations of $N$ and then form the induced representations, one can use Mackey ...
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60 views

Standard representation of $\frak S_4$

On p. 18 of Representation Theory: A First Course, Fulton and Harris write The character of the standard representation is $\chi_V = (3, 1, 0, -1, -1)$. Note that $|\chi_V| = 1$ so $V$ is ...
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1answer
47 views

Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$. Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?
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action of orthogonal group on the space of antisymmetric bilinear forms

What is the natural action of orthogonal group on the space of antisymmetric bilinear forms.
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1answer
75 views

irreducible representation of a group

Reduced group $C^\ast$-algebra of group $G$ is defined to be $G^*_{r}(G)=\overline{\lambda(L^1(G))}$ where $\lambda$ is left regular representation. My question is how to get a irreducible ...
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203 views

Real representations of Lie algebra $\mathfrak{so}(3)$

How does one construct an $n$-dimensional, irreducible, real-valued and non-zero representation of the three generators of the Lie algebra $\mathfrak{so}(3)$ for a given value of $n$?
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198 views

Show $U \otimes V$ is an irreducible G-module

Let $G$ is some group and $U$ is an irriducible $G$-module over the complex numbers. Now if $V$ is a $G$-module of dimension 1, I would like to prove $U \otimes V$ is an irriducible $G$-module. My ...
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1answer
43 views

Meaning of representation

In mathematics, what does it mean by "one object is represented by another object"? Here is my guess. Given a mapping $f: X\to Y$. If $f$ preserves some structures when mapping from $X$ to $Y$, then ...
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Complex finite dimensional irreducible representation of abelian group

I'm supposed to show that each Complex finite dimensional irreducible representation of an abelian group is one dimensional. For any map $\phi: V \rightarrow V$ it holds that $\phi(\rho(g)v) = ...
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208 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
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1answer
74 views

Induced representation, isomorphism between vectorspaces

I want to prove the following statement: The mapping $Ind_H^G{V}\rightarrow V^m: \psi\mapsto(\psi(g_1),\ldots,\psi(g_m))$ is a isomorphism. Here is $g_1,\ldots,g_m$ a representing system of $G/H$ ...
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39 views

Why $A/\operatorname{rad}A$ is generated by $e_a$?

Let $A$ be an algebra over an algebraically field $K$ and $(Q_A)_0$ be its ordinary quiver. Let $\{e_a \mid a \in (Q_A)_0\}$. Then $\{e_a \mid a \in (Q_A)_0\}$ is a complete set of primitive ...
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1answer
97 views

How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem?

Let $A$ be a $K$-algebra and $K$ an algebraically closed field. How to show that $A=(A/\operatorname{rad}A)\oplus \operatorname{rad}A$ using Wedderburn-Malcev theorem? Thank you very much. ...
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Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
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Irreducibility of $Sym^2$ and $\Lambda ^ 2$ representations

I'm given a representation $\Pi : \mathrm{Gl}(n,\mathbb{C}) \rightarrow \mathrm{Aut}(\mathrm{Mat}_{n\times n}(\mathbb{C}))$ by $\Pi(g) = gXg^T$ (does it have a name?). Then the representations ...