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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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Conjugacy Classes and Irreducible Representations

For finite groups number of inequivalent irreducible complex representations equals the number of conjugacy classes. When the group is $S_n$ it is easy to see that both are available one each for ...
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Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of ideles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
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Centralizer of one element on a compact connected Lie group

Exercise 16.2 from Daniel Bump - Lie Groups. Let $G$ be a compact connected Lie group and let $g\in G$. Show that the centralizer $C_{G(g)}$ of $g$ is connected. I have some problems verifying this, ...
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What is the definition of semi-simplication of a group representation?

When learning Galois representations , it often happens to take the semi-simplication of a 2-dim representation, but I can't find the definition. Could anyone help me?
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What is $\ell^2(\Gamma)$ for a discrete group $\Gamma$?

I am trying to get my head around the left regular representation of a group, and I am not sure of the definition of the space $\ell^2(\Gamma)$ if $\Gamma$ is a discrete group. To quote what I am ...
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Fundamental, Regular, and Defining representations

Are these three representations for compact Lie groups and its Lie algebra mean the same thing? Let's focus on "classical" cases like $SO(N)$ and $\mathfrak{so}(n)$ for concreteness (unless I need to ...
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Will any modulo operation of relatively prime numbers be related to cyclic group of same order?

I am engineer/applied mathematician who is quite new to most things number theory and also groups so please bear with me if this is obvious. Consider the $b\times b$ matrix constructed by: $$M_{ij} =...
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Local system associated to monodromy representation

How can I associate a local system to a representation $\rho: \pi_1(X) \to \mathbb C^*$? I have seen some construction, but it doesn't click for me. I know that the idea is to use a diagonal action ...
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coadjoint orbits and symmetry

Suppose we have some Lie group $G$, and we consider a co-adjoint orbit $\Omega$, which is a symplectic manifold. Does $G$ act on $\Omega$ by symplectomorphisms? As in, is $G$ a symmetry of $\Omega$, ...
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centralizers of nilpotent element in simple Lie algebra and associated Levi subalgebra

Let $\{e,h,f\}$ be a $sl_2$ triples in simple Lie algebra $\mathfrak g$ with usual relations $[h,e]=2e,~ [h,f]=-2f,~[e,f]=h$. Then the centralizer of $e$ is $\mathfrak g_e=\{b:[b,e]=0\}$ and the ...
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Two definitions of torsion pairs/theories. How are they equivalent?

Let $\mathcal{A}$ be an abelian category, $\mathcal{T}, \mathcal{F}$ two strictly full additive subcategories of $\mathcal{A}$. Then according to nLab and other sources including Constructing Torsion ...
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Under what condition is the group G isomorphic to the direct product of irreducible representations in a block diagonal representation? [on hold]

Given an injective completely reducible representation D(g) of G. Using a smiliarity transformation S, $SDS^{-1}$ is in block diagonal form with ireducible representaion $D_1 , D_2, D_3 .... D_n$. ...
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quantization clarification

I'm wondering something small about the link between representations and quantization. For quantization you start with some phase space (symplectic manifold) $M$, and you have a classical observable $...
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Endomorphism rings of indecomposable modules

This is a more structured reformulation of this question. Let $k$ be a field and $A$ be a commutative $k$-algebra, say $A=k[x_1,\dotsc,x_n]$, and $M$ be a finite dimensional (ungraded or $n$-graded) $...
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A certain decomposition of a semisimple Hopf algebra

$\newcommand{\Irr}{\mathrm{Irr}}$ Let $H$ be a finite-dimensional semisimple Hopf algebra over an algebraically closed field $k$, and let $\Irr(H)$ be the set of (choices of representatives of) ...
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Irreducible Characters

I tried to solve the following questions but I got stuck. I need to some hints to continue. If the induced character of an irreducible character of $H$ is again irreducible, then center of the ...
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Regular representation, two viewpoints, the isomorphism

Let $G$ be a group. Let $G^* = \{\phi: G\to \mathbb{C}\}$ be the complex functions defined on $G$. We have the representations $$\rho: G \to GL(G^*), \space \rho(g) = \phi \mapsto (h\mapsto \phi(hg))...
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Deligne-Lusztig theory and cuspidal representations of $\mathrm{GL}_{2}(\mathbb{F}_{q})$

I heard that Deligne-Lusztig theory gives us geometric way to construct representations of finite algebraic groups over finite fields, such as $\mathrm{GL}_{2}(\mathbb{F}_{q})$, which arises from ...
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Recover a matrix from an irreducible representation.

Let $G$ be a finite group and $X: G\rightarrow\text{GL}_3(\mathbb{C})$ be an irreducible $3$-dimensional complex matrix representation of $G$. Suppose that $$ B=\frac{1}{|G|}\sum_{g\in G}X(g)AX(g)^{...
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if $A$ is a finite dimensional commutative $\mathbb{C}$ algebra with dimension $n$, must $A$ have $n$ simple modules up to isomorphism?

I'm trying to prove the question above. I'm not sure whether this is true or not, but I'm trying to figure it out. If $A$ is semi-simple I don't think it's too hard to see that it's true, but ...
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How to decompose a group representation which is a direct sum of copies of one irreducible representation?

Let $G$ be a finite group. Let $F$ be a field of characteristic zero. Let $V$ and $W$ be finite dimensional representations of $G$. Assume that $V$ is irreducible and $W$ is isomorphic to $nV$ for ...
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Does usual Jordan decomposition be preserved by finite dimensional representation?

If $\mathfrak g\subseteq \mathfrak{gl} (V)$ is a finite dimensional Lie algebra, then we have a usual Jordan decomposition in $\mathfrak{gl} (V)$, $$ \forall x \in \mathfrak{g}\ \exists x_s, x_n\in \...
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Unitary representations of $U(3)$

We know that the unitary dual of $SU(2)$ is given by the symmetric powers of the standard representation. Is there anything similar in the case of $U(3)$ or $SU(3)$? In general, is there a reference ...
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Real representation of $\langle \sigma_1,\sigma_2\rangle$ where $\sigma_i\in S_n$ are n-cycles

$S_n$ acts on $\mathbb{R}^n$ by permuting the coordinates, giving us the permutation representation. This representation can be decomposed into $\mathbb{R}^n=V\oplus W$, where $V$ is the trivial ...
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Is this intertwiner a scalar, working over a field which is not algebraically closed?

Let $G$ be a finite group. Let $F$ be a field of characteristic zero. Let $(V,\rho_V)$ be a finite dimensional irreducible representation of $G$ over $F$. Define $P:V\rightarrow V$ by $P=\frac{1}{|G|}...
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What does it mean for two group action to be conjugate?

I was reading a book on symplectic topology and get confused of the following sentence. The action of $S^{1}$ on $T_{p}M$ is conjugate to an n-fold product of circle actions on the complex plane by ...
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Example of a bialgebra which is Frobenius but not Hopf

Let $\Bbbk$ be a field. A well-known result (see Larson, Sweedler, An Associative Orthogonal Bilinear Form for Hopf Algebras and Pareigis, When Hopf algebras are Frobenius algebras) states that a ...
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Automorphism group of the projective unitary group PU(N) and SO(N)

I would like to determine the automorphism group of the projective unitary group $G=PU(N)=PSU(N)$ and $G=SO(N)$. We also knew that $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$...
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Intuition behind this q-binomial formula counting sums of subsets

We have these transparent, motivating interpretations for binomial coefficients and their $q$-analogue. $$ \binom{n}{j} = \begin{matrix} \text{"The number of subsets of size $j$}\\ \text{of a set ...
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Schur's Lemma in Infinite Dimensional Lie Algebras

Let $\mathfrak{g}$ be a $\mathsf{k}$-Lie algebra, with $(\rho, V)$, $(\sigma, W)$ irreducible $\mathfrak{g}$-representations. Then the easy part of Schur's Lemma states that a $\mathfrak{g}$-linear ...
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Existence of a subgroup

I want to show that a group has no subgroup of index 4 and 6 by using character table. But, I didnt find which argument must be used. I show that it is simple using KerX. Then, what should the ...
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Artin-wedderburn and simple modules

Given a finite-dimensional semisimple algebra $A$ over algebraically-closed field $k$, Artin-Wedderburn says it is isomorphic to the direct sum of matrix rings $M_{n_i}(k)$. If $\dim(A)=n$ then $\...
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Homology of solvable Lie algebras

Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the ...
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classification of representations of $D_{1009}$

A follow-up of this question To fix ideas, take $n=1009$. $D_n$ has $2$ irreducible representations of degree $1$ and $504$ representations of degree $2$. Are the degree 1 representations all ...
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Tensor product representation of $SO(3)$

Let $D^{(1)}$ be the representation of $SO(3)$ such that for $g \in SO(3)$: $D^{(1)}(g)=g$. I need to find an invariant subspace of dimension $1$ (of $\mathrm{Sym}^2(V)$, where $V$ is the ...
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Using the Peter-Weyl theorem and Schur orthogonality relations

I am reading a paper and I will explain the setup below. Let $G$ be a compact group with Haar measure $\mu$ and suppose $\pi:G\to \mathbb{R}^{d,d}$ is a representation of $G$. We get a $G$-invariant ...
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Quotient of semi-simple representation is semi-simple

Prove that Quotient of semi-simple representation is semi-simple. Take $V=\oplus_i V_i$ a semi-simple representation of finite dimension of a finite group. For a fixed j, we have $V/V_j=\oplus_i (...
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characters of symmetric and exterior algebras [duplicate]

Let $\rho,V$ be a representation of $S_3$. In my lecture notes, It says that we can deduce the characters of $V\otimes V$ and $Sym^2(V)$ and $\wedge^2V$ only from the character table of $S_3$. I ...
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Decomposition of a specific $SO(n)$-module

I need to decompose a specific $SO(n)$-representation into irreducible ones, but my background on representation theory is rather poor, so I would like to post the problem here. Let $V=\mathbb{R}^n$ ...
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Character of induced representation: explicit formula

I'm trying to work through Exercise 7.1 in Serre's "Linear Representations of Finite Groups" but I'm having trouble finishing my proof. The problem reduces to the following. Let $N\subset H$ be a ...
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Connection between a representation and its isomorphic dual

Suppose we have a complex irreducible representation $(V, \phi)$ of a finite group $G$. (Where $\phi : G \rightarrow \text{Aut}_{\mathbb{C}}(V)$). Suppose $V$ it is isomorphic to its dual $V^*$, and ...
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Why $(x v_{i}, v_{j})(x w_{i^{\prime}}^{*}, w_{j^{\prime}}^{*})=((P\left(v_{i} \otimes w_{i^{\prime}}^{*}\right), v_{j} \otimes w_{j^{\prime}}^{*})$

When I was learning a theorem of Orthogonality of matrix elements, I faced a step I could not understand. (,) is the Hermitian inner product, P=$\frac{1}{|G|} \sum_{x \in G} x $, we have $\left(t_{i j}...
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Irreducible representations of a finite group are equivalent?

Given a finite group $G$. If $D(g)$ and $D'(g)$ are injective irreducible representations of $G$, do $D(g)$ and $D'(g)$ have to be equivalent? That is $SD(g)S^{-1} = D'(g)$? We assume here complex ...
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what does this Dirac notation mean?

Here,$D_a(g)$ and $D_b(g)$ don't have to have the same dimension. The image is from page 17 of the book Lie Algebra by Georgi. In one of the theorem used, $D_a(g)$ and $D_b(g)$ are blocks in the ...
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trivial modules of group rings

Let $R=\mathbb{F}_p[D]$ where $D$ is a finite group of order prime to $p$. Let $M$ be any simple $R$-module. If one knows that $H^0(D,M)=0$, is $M=0$? If not, under what further conditions can one ...
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Block-diagonalize group representation with two copies of the same irrep

Lets say we have a finite group $G$ with irreducible representations: $\mathbf{D}^{(1)},\,\mathbf{D}^{(2)},\dots$. Let us also say that we have a reducible representation of that group over the ...
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What's the time complexity of character computational algorithms?

I couldn't find any information on the time complexity of the Burnside and Dixon-Schneider algorithms and I was wondering if this is well-known? In his paper, Schneider actually includes a table ...
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Are groups completely determined by their representations?

Recently, I became interested in representation theory, and I found out a natural philosophical (vague) question: are groups completely determined by their representations? To be specific, I want to ...
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dimension of standard representation of $\mathfrak{sl}_2(\Bbb C)$

In my lecture notes I read that the standard representation of $\mathfrak{sl}_2(\Bbb C)$ is of dimension 2. $\mathfrak{sl}_2(\Bbb C)$ is the set of matrices of $M_2(\Bbb C)$ of trace equal to 0. ...
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A Reducible Real Representation

Given a finite group $G$. Let $V, \mathbb{p}$ be a representation of $G$. i.e. $$\mathbb{p} : G \rightarrow GL(V)$$ is a homomorphism. Prove that $V \otimes V^{*}$ is a real representation. Where $V^...