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Questions tagged [representation-theory]

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

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A difficulty in understanding an example in Vinberg.

The example is given below: But I have difficulties in understanding the following: 1- why $V_{0}$ is called $(n-1)-$dimensional subspace, I want a concrete example please? 2- Why if the ...
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0answers
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Let $V$ and $W$ be real representations of a torus $T$ s.t. $\dim V^H=\dim W^H$, $\forall H<T$. Show that $V\simeq W$

$V^H:=\{v\in V:hv=v,\,\forall h\in H\}$ is the fixed point set. I'm trying to show this result first for the irreducible real representations, which are the trivial (one dimensional) ones and those ...
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1answer
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Any quotient representation of completely reducible is completely reducible. [on hold]

Prove that every quotient representation of a completely reducible representation is completely reducible. Could anyone give me a hint for this?
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1answer
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For two invariant complements in the space of representation $T$. Prove that $T_{W_1}$ isomorphic to$ T_{W_2}$

Let $W_{1}$ and $W_{2}$ be two invariant complements of the invariant space $U$ in the space of the representation $T$ prove that $T_{W_{1}}$ equivalent to $T_{W_{2}}$. Invariant Complement ...
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For $U$ a $G-$invariant subspace of $V$ a $G-$representation, show that $T_g(U)=U$

Prove that if the subspace $U$ of the space of the representation $T:G\to GL(V)$ of $G$ is invariant, then $T(g)U = U$ for all $g \in G.$ Could any one give me a hint how to solve this question ?
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Complex Representations of A4

Im asked to obtain the number of irreducible complex representations of the group $A_4$ and their repesctive dimensions. I know that the number of irreducible representations is going to be the number ...
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A difficulty in understanding the solution of #2 section 1 Vinberg.

The question and its answer is given below: But I do not know how I should think to find all the invariant subspaces and why the answer is as mentioned above, could anyone explain this for me please?
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1answer
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decompositions of a representation

I am reading J.P Serre's book on Linear representations of finite groups. In chapter 2.6 it states: Let $\rho: G \rightarrow GL(V)$ be a linear representation of $G$. We are going to define a ...
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Explain projective representation vs faithful representation

This is very basic. I am trying to explain projective representation vs faithful representation in a most naive way to a class of middle school students. My formal way of understanding is that ...
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26 views

How exactly does character inflation work?

Let $A_4$ be the alternating group and let $V = {\{(1), (12)(34), (13)(24), (14)(23)}\}$ be a normal subgroup of $A_4$. Then $A_4/V \simeq C_3$, so $A_4$ has $3$ one dimensional representations, which ...
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Finite-dimensional representations of the integers (2)

I was reading the answer of the question from this link: Finite-dimensional representations of the integers But I have 2 things that I do not understand in the solution and the comments of the ...
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What is the dimension of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
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1answer
50 views

Show that $\exp(C^{-1} AC ) = C^{-1} \exp(A C)$

Show that $\exp(C^{-1} AC) = C^{-1} \exp(A C)$ for any matrices $A \in L_{n}(\mathbb{R})$ and $C \in GL_{n}(\mathbb{R})$. The hint of the question is given below: Consider the linear operator $\...
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3answers
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irreducible representations and character table of $D_6$

Let $$D_6=\langle a,b| a^6=b^2=1, ab=ba^{-1}\rangle$$ $$D_6=\{1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}$$ I would like to compute its character table and its irreducible representations. I will ...
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Equivalent definitions of Clifford algebra, verification

Let $(V,B)$ be a finite dimensional $k$ vector space $V$ with an associated quadratic form $Q$. $char \, k \not= 2$. Let $X:= \{e_i \}_{i=1}^n$ be a set of basis for $V$. Construct $k\langle X \...
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1answer
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Converse of Schur's lemma for Lie algebras. [duplicate]

The statement of Schur's lema for Lie algebras says that. Let $(\rho,\mathcal{V})$ is a complex irreducible finite-dimensional representation of a Lie algebra $\mathfrak{g}$. If $T$ conmmutes with $\...
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1answer
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Direct sums and products of SU(2) representations

I am reading the book on group theory and stuck with a simple problem. Why $$(2\bigotimes2)\bigoplus(2\bigotimes1)\bigoplus(1\bigotimes2)\bigoplus(1\bigotimes1)=3\bigoplus1\bigoplus2\bigoplus2\...
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1answer
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Irreducibility of the derived representation

Let $G$ a linear Lie group with lie algebra $\mathfrak{g}$. If $(\pi, \mathcal{H})$ is a irreducible representation of $G$. Does the irreducibility of $\pi$ imply the irreducibility of derived ...
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Equivariant morphisms to $G/H$ are locally trivial?

Let $X$ be an algebraic variety over $\mathbb{C}$ and $G$ is a complex algebraic group which acts on X. Fix also an algebraic subgroup $H \subset G$ and consider a $G$-equivariant morphism $f\colon X \...
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1answer
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$GL_2(\mathbb R)$ acting on $\hat{\mathbb R}=\mathbb R\cup \{\infty\}$. [closed]

The question and its hint is given below: But I could not understand what the question is trying to teach me, could anyone explain this for me please? Also I could not understand how the hint could ...
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1answer
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The set of real structures on $\mathbb{C}^n$ is isomorphic to $GL(n,\mathbb{C})/GL(n,\mathbb{R})$ as a $GL(n,\mathbb{C})$-space.

In representation theory a real structure on a $G$-module $V$ (finite dimensional complex vector space, in which the group $G$ has a linear action on it) can be definide as a conjugate-linear $G$-map $...
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Functor category of the module category $\mathrm{Vect}(\mathbb{C})$

My current exercise is the following: Take $\mathcal{M}=\mathrm{Vect}(\mathbb{C})$ as a module category over $\mathcal{C}=G\text{-}\mathrm{Vect}(\mathbb{C})$, the $G$-graded vector spaces over $\...
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1answer
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Showing that a matrix F is a matrix representation of $\mathbb{R}$

The question is given below and its answer are given below: 1-I know that according to Vinberg book which is called "Linear representations of groups" all finite dimensional differentiable ...
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0answers
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Determining maximal Cohen-Macaulay modules over an invariant ring

Suppose that $G$ is a finite small (i.e. reflection-free) subgroup of $\text{GL}(n,\mathbb{C})$ acting on $S := \mathbb{C}[x_1, \dots, x_n]$. Set $R := S^G$. By 5.20 Corollary of this, the maximal ...
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general linear group over a vector space

Let $V$ be a vectors space with given basis, say $a_1, a_2, \ldots a_n$ and let $\mathbb{L}$ be an algebraic closure of a field $\mathbb{K}$. What should $GL(V_{\mathbb{L}})$ mean? and also, up to ...
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2answers
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When is the character of a representation a character?

Let $G$ be a group with a complex, finite dimensional representation $r:G \to \operatorname{GL}(V)$. What is the condition to make $\operatorname{trace}(r(g))$ also a character of $G$, i.e., group ...
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0answers
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Is there any relation between real and complex character functions of irreducible representations of compact lie groups?

Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $\chi_U^\mathbb{R}:G\to\mathbb{R}$ as $\chi_U^\mathbb{R}(g)=\operatorname{Tr}(l_g)$. If $V$ is a complex $...
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The Kernel is inside the radical when we have an essential epimorphism

This is a proposition in Auslander's book (Representation Theory of Artin algebras). I want proof that: If $f$ is an essential epimorphism then Ker$f \subset rad A$, where $f: A\rightarrow B$, and $A$...
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Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup: Let $A$ be a finite-dimensional $k$-algebra over some field $k$. Let $\mathcal{B} = Hot^-(Proj A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
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Prove that $U=P*\overline{P}$ iff $P$ is uniform (group representation exercise.)

This exercise comes right after the introduction of Fourier Inversion Theorem and Plancherel’s Formula in Diaconis’s Group Representation in Probability and Statistics. Let $P$ be a probability on $...
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1answer
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Can the fix point set of a nontrivial irreducible complex representation of a finite odd order group be non trivial?

I'm trying to show that if $G$ is a finite odd order group then, all of its nontrivial complex representations are of complex type (i.e., it is not realisable over the reals). (I have answered it ...
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1answer
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Almost having invariant vectors vs having almost invariant vectors?

Let $\Gamma$ be a discrete and countable Group and let $\pi:\Gamma\to \mathcal{B(H)}$ be a unitary representation. We say that $\pi$ almost has invariant vectors if for every compact (=finite) subset ...
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Section 0 in Ernest B. Vinberg, “Linear Representations of groups” Q.7(e)

I know from the answers at the back of the book that the following map: $$(S(t)f)(x) = e^tf (x + t),$$ where $S: \mathbb{R} \rightarrow S(V)$ and $V$ is the subspace of all polynomials with real ...
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1answer
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Can a finite group have 2D and 3D faithful irreducible representations?

I am looking for finite groups that have a 2D (complex matrix) faithful irreducible representation and a 3D faithful irreducible representation. Up to order 1023 GAP found none. Other combinations of ...
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1answer
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What is the difference between a map being linear in linear algebra and a map being linear representation in linear representation theory?

What is the difference between a map being linear in linear algebra and a map being linear representation in linear representation theory? I know from the answers at the back of the book that the ...
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Geometric intuition for exceptional divisor of resolutions of Kleinian singularities

In this blog post by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a Kleinian singularity $\mathbb{C}^2/G$ is given by Nakajima's ...
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1answer
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Isomorphism of representation induced by a morphism of groups

Let $G_1$,$G_2$ be groups. The question is to prove that if $$u: G_1 \rightarrow G_2$$ induces an isomorphism $$\overline{u}: \operatorname{Hom}(G_2,Gl(M)) \rightarrow \operatorname{Hom}(G_1,Gl(M))$...
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1answer
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Dimension of an irreducible representation and the index of the group's center

Let $(\pi,V_\pi)$ be an irreducible representation of finite group $G$, over algebraically closed field $F$ s.t. $char(F)$ is coprime to $|G|$ (for example $\mathbb{C}$). Prove that $dim(\pi)^2 \le [...
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Induced characters of $G$ from a normal subgroup $H$

Let $H \lhd G$ and let $\chi$ be a character of $H$. Let $g \in G$ and let $H^g = gHg^{-1}$. Define $\chi^g$ to be the class function on $H^g$ given by $\chi^{g}(x) = \chi(g^{-1}xg)$. Suppose that $\...
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Why does the Frobenius-semisimplicity of a Weil representation not depend on the choice of the Frobenius element?

Definition: Let $K$ be a (non-Archimedean) local field and $k$ its residue field. A Frobenius element of the absolute Galois group $G_K$ is any element of $G_K$ which is a lift of the Frobenius ...
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Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\...
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1answer
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Unique faithful $7$-dimensional representation of semisimple Lie Algebra with $G_2$ root system

I am asked to show that if $\mathfrak{g}$ is a semisimple Lie Algebra with root system of type $G_2$, then it has a unique, $7$-dimensional faithful representation. To start, let $\omega_1, \omega_2$...
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1answer
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One dimensional representations of a simple group

I keep seeing this following fact stated; If $G$ is a simple group, then the only $1$ dimensional representation of $G$ is the trivial representation. But I’ve not seen a proof and I can’t seem to ...
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Is there a difference between the definition of the group $L_{n}(R)$ and that of $GL_{n}(R)$?

Is there a difference between the definition of the group $L_{n}(R)$ and that of $GL_{n}(R)$? I already know the definition of $GL_{n}(R)$.
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Whittaker model equation

This is from Cogdell and Piateski-Shapiro's paper Derivative and L-functions for $GL_n$. Let $\lambda$ be a non-trivial $\psi$-Whittaker functional. The Whittaker model is defined as$W_v(g)=\lambda(\...
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How to formulate Langlands correspondence over a finite field?

Langlands correspondence is a conjectural correspondence between automorphic forms over a reductive group and Galois representation. This is widely open subject in number theory and there are some ...
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2answers
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How orbits of $G$-sets and these characters are related

I've been learning about induced representations recently and I've come across something which I'm very confused about; For any $G$-set $X$, the number of orbits is equal to $(1_G, \chi_{\mathbb{C}[...
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35 views

orthogonality relation for characters

Let $\rho : G \to GL(V)$ be a representation on G. Then, its character is defined as $\chi_V(g) := Tr(\rho(g)) $, where $Tr$ denotes the trace function. For an exercise I am trying to solve, I would ...
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1answer
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Motives and representations

Assuming standard conjectures, the category of motives is equivalent to the category of representations of a certain group over $\mathbb Q$, but I don't understand the abstract construction. Now, if ...
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1answer
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Highest Weights of Defining and Adjoint Representations of $\mathfrak{so}_5$

I am asked to describe the defining representation of $\mathfrak{sp}_4$ in terms of highest weights, and then I am asked to repeat this process for the defining and adjoint representations of $\...