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

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Indecomposable representations of an algebra

Let $\rho$ be an indecomposable representation of Algebra A on a finite dimensional vector space V. Let $a \in Z(A)$, how do I show $\rho(a)$ has exactly one eigenvalue. $Z(A)$ represents the centre ...
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52 views

Outer Automorphisms of PSL2(R)

As far as I've been able to tell, a description of Out$(PSL_2(\mathbb{R}))$ isn't available online. I also looked in Lang's $SL_2(\mathbb{R})$ but it's not discussed. I guess my first question would ...
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Irreducible characters of the direct product of two groups

I am studying representation and character theory because of my field of research. So, my question is not a homework. I want to solve a problem of the book "character theory of finite groups" by M. ...
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25 views

Continuous complex finite dimensional irreducible representation of $GL_n( \mathbb C )$

What are all the continous finite dimensional irreducible representation of $GL_n( \mathbb C )$? I tried the following since the continuous irreducible representations of $GL_n( \mathbb C )$ are in ...
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Canonical representation for lines

I came across this statement "by means of canonical representation for lines" in a paper (Excerpt). By searching around, although, I did understand what canonical representation means but couldn't ...
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Two different definitions for Lie Algebras for closed subgroup of $GL_n(\mathbb R)$

Let $G$ be a closed subgroup of $GL_n(\mathbb R)$. There are two definitions for $\mathrm{Lie}(G)$ $\mathrm{Lie}(G) = \{ \gamma'(0) : \gamma : (-\epsilon, \epsilon) \rightarrow G \text{ is ...
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Are $h$-eigenspaces of infinite-dimensional $sl(2,\mathbb C)$-modules of dimension at most $1$?

Let $(\pi,V)$ be an irreducible representation of the Lie algebra $sl(2,\mathbb C)$ on a possibly infinite-dimensional complex vector space $V$. Further let $h,e^+, e^-$ be the usual standard basis ...
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Finite-Dimensional Representations of The Classical Groups in Tensor Spaces: Invariant Theory

I. When we study finite-dimensional irreducible representations over the space of general tensors (e.g.,Chapter 13, Group Theory in Physics by Wu-Ki Tung), is it enough to obtain all the ...
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Indecomposable representations of poset (1,1,m), m>=1

The problem says : Find indecomposable representations of poset (1,1,m) where m $\geq 1$. I've tried to find the the associated matrix problem for this poset but got nowhere. Any help would be ...
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Shared representation of $SU(2)$ and $SO(3)$

I'm reading a famous book, "Group Theory and Quantum Mechanics" written by Michael Tinkham. In Chapter 5 of this book, he introduces the relationship between $SU(2)$ and $SO(3)$ and I already ...
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Compact objects in categories of comodules.

Are the finite dimensional comodules the compact objects in the category of comodules over a Hopf algebra? If yes, is there a reference? If no, which are the compact objects? Here by compact I mean ...
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An irreducible representation of a complex Lie algebra is the product of a 1-dim rep'n and a semisimple one

I am reading on p.128 of Fulton and Harris's Representation Theory the proof of the following fact about lie algebras Let $\mathfrak{g}$ be a complex Lie algebra, and set $\mathfrak{g}_{ss} := ...
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35 views

Quadratic casimir of a representation of SO(N)

For $SO(N)$ the quadratic Casimir for the spinorial representation is $N(N-1)/8$ and that of the vector representation is $N-1$, but what is the quadratic Casimir of the spin $2$ representation? or ...
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Representations of Lie groups

It's not at all obvious to me why a connected and simply connected Lie group has only single valued linear irreducible representations. This would come as a particular case to a more general ...
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Relation between Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$.

What is the relation between the definitions of Yangian $Y(sl_2)$ and quantum affine algebra $U_q(\widehat{sl_2})$? There are two definitions of $U_q(\widehat{sl_2})$. The following is Jimbo ...
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Relation between quantum affine algebra $U_q(\widehat{sl_2})$ and the affine Lie algebra $\widehat{sl_2}$?

The relation between the definition of quantum group and correpsonding lie algebra is discribed here. Are there some similar relation between $U_q(\widehat{sl_2})$ and $\widehat{sl_2}$? There are two ...
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42 views

Reduced Trace, Central Simple Algebra

Let $A$ be a central simple $K-$algebra and $A = M_n(D^{o}), Z(D) = K, [D:K] = m^2$. Let $\psi(a)$ be the reduced trace of an element of $A$. I am not sure if the definition I ...
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Advanced beginners textbook on Lie theory from a geometric viewpoint

There are several questions resembling this one but none of them are quite the same I believe. I have a background in differential geometry and topology, as well as analysis (locally convex spaces). ...
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Clebsch-Gordan Coefficients for 8 in $3 \otimes \bar 3 $ and the $6$ in $3 \otimes 3$ of $SU(3)$?

Do Tables for the Clebsch Gordan coefficients for the decomposition of the $8$ dimensional irrep of $SU(3)$ into $3 \otimes \bar 3 $ and the $6$ in $3 \otimes 3$ (in the Dynkin basis) exist somewhere? ...
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Why do we represent groups in the category of vector spaces?

For any category $\mathcal{C}$ and any object $X$ in $\mathcal{C}$, $\operatorname{Aut}_{\mathcal{C}}(X)$ is a group. Thus, given any group $G$, it makes sense to talk about representing $G$ in ...
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Does the character with the following properties exist?

Given a proper subvariety $V\subseteq G_m^n$ (V is also a subgroup). Can we always find a character $\chi:G_m^n\to G_m$ which is a morphism and a group homomorphism such that $V\subseteq \ker\chi$?
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Definition of parahoric subgroup

Let $G$ be a connected semi-simple group over the ring of integers $O_F$ of a non-archimedean local field $F$. Let $\varpi_F$ be a prime element of $O_F$. Usually parahoric subgroups of $G$ are ...
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Representations of a subgroup of a finite grup [closed]

Let $H$ be a subgroup of a finite group $G$. Let's say that $H$ has $h$ irreducible representations $\psi_1,\psi_2,...,\psi_h,$ and $G$ has $g$ irreducible representations, ...
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Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
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Tensor product of irreducible representations

Let $\mathfrak{g}$ be a complex simple finite dimensional Lie algebra and $V,W$ two irreducible finite dimensional representations. When is $V\otimes W$ irreducible?
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Fundamental representations of lie algebras

All the lie algebra considered are over $\mathbf{C}$. I know that for the lie algebra $\mathfrak{sl}_{n+1}$ the fundamental representations $L(\omega_k), k \in \{1,\cdots,n\}$ are the $\Lambda^k V$ ...
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7 views

Are Whittaker functions for lie algebras the same as Whittaker functions for corresponding Lie groups?

Some papers call some Whittaker function the Whittker function for some Lie group $G$. Some other papers call some Whittaker function the Whittker function for some Lie algebra $g$. Is the Whittaker ...
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Method to decompose Product Representation in terms of Subgroup Irreps?

The tensor product of a representation $R$ of a Lie group $G$ with itself is in general a reducible representation of $G$, i.e. a sum of irreducible representations $$ R \otimes R = R^1 \oplus R^2 ...
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*-representations, unitary representations, and adjunctions

I've been reading Folland's Abstract Harmonic Analysis, and I am currently in the section on the correspondence between unitary representations of a locally compact group $G$ and $*$-representations ...
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Representation of a Kac-Moody algebra

Let $n$ be an integer $\geq 3$ and let $\mathfrak{g}$ be the Kac-Moody algebra with cartan matrix $C$ given by $C_{ij} = 2 \delta_{i,j} - \delta_{i,j+1} - \delta_{i,j-1} - \delta_{|i-j|,n-2}$. For ...
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Scalar-Matrix multiplication, representation theory of fields and Kronecker products

Sorry for potentially posting a confusing post. I've read some very basic algebra and about matrix representation of groups. So consider the $2 \times 2$ matrix ${\bf A}$ with elements from the ...
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Representations of the symmetric group $S_3$

I'm considering the following example/application. Let $k$ be a field, such that $char(k)\nmid\mid G\mid$.Let $G=S_3$ then we have the following representation on it: TRIVIAL REPRESENTATION: ...
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Invariant theory and irreducible representation of a group

I am now reading the book 'Invariant theory' by Neusel. It seems that the logic is like this: You have a representation of a group $G$ on some vector space $V$, this might be a reducible ...
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locally algebraic representations

Let $K$ be a number field. Consider $$ \rho_{\ell}: \mathrm{Gal}(\bar K/K) \longrightarrow \mathrm{GL}(V)$$ an $\ell$-adic Galois representation. Assume it is semi-simple rational and abelian. Is ...
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Short exact sequence of linear representations of a group

Let $G$ be a group with $V',V,V''$ being representations of $G$. Let $$0 {\longrightarrow}V' {\longrightarrow}V\overset{v}{\longrightarrow}V''\longrightarrow 0\tag{1} $$ be a short exact sequence of ...
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A question on representation of commutative groups

I am recently reading Naimark and Stern's book "Theory of Group Representations". On page 37, an exercise is to show that a finite dimensional representation of a finite commutative group is a direct ...
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39 views

Definition of Regular Representation

I have troubles understanding the following definition. I post two pictures of the same definition, one is of my notes. Definition - Regular Representation In my notes, in the second picture, ...
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Pair of nonzero continuous functions $\text{Aff}(\mathbb{R}) \to \mathbb{R}^\times$, left and right invariant measures.

For $a \in \mathbb{R}^\times$ and $b \in \mathbb{R}$, let$$g_{a, b} : \mathbb{R} \to \mathbb{R}, \text{ }x \mapsto a \cdot x + b$$be an affine linear map. Let$$\text{Aff}(\mathbb{R}) = \{g_{a, b} : a ...
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Peter-Weyl theorem versions

Let $G$ be a compact group. I learned the version of the Peter-Weyl theorem which says: the matrix coefficients of $G$ are dense in $L^2(G)$. Call this Peter-Weyl I. Apparently there is another ...
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Affine linear map, finite-dim. reducible rep. but can't be decomopsed as a direct sum of irreducible subreps?

For $a \in \mathbb{R}^\times$ and $b \in \mathbb{R}$, let$$g_{a, b} : \mathbb{R} \to \mathbb{R}, \text{ }x \mapsto a \cdot x + b$$be an affine linear map. Let$$\text{Aff}(\mathbb{R}) = \{g_{a, b} : a ...
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1answer
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Abel transform a topological isomorphism?

Given a noncompact semisimple Lie group $G=NAK$ with Weyl group $W$, consider the symmetric space $X=G/K$. Let $f$ be a function in $D(X)^K$, the space of $K$-invariant functions on $X$. Then the Abel ...
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Definition of auxiliary function in proof of Maschke's theorem

Let $K$ be a field and $G$ be a finite group, and denote by $K[G]$ its group ring, then Maschke's theorem is: Suppose $\mbox{char}(K)$ does not divide $|G|$. Let $V$ be a $K[G]$-module. If $W ...
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Intuition for decomposition of regular representation

Working over $\mathbb{C}$ it is well-known that if $G$ is a finite group and $V$ is its regular representation then every irreducible representation $V_i$ of $G$ occurs as a summand of $V$ with ...
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How can I express pre- and post-multiplication as a single linear operator?

An example is probably the best way to illustrate what I'm asking. Let $G_1$ be $SO(3)$ and $G_2$ be $SO(3)$ as well. Given a matrix $A \in \mathbb{R}^{3 \times 3}$, the combined action of $g_1 \in ...
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Reducible and unfaithful module

Is this kind of module possible? I have a homework in which i must give an example or prove its not. I can do neither. We know the following about a reducible unfaithful module: $$ V = U_1 \oplus U_2 ...
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54 views

Vectors fixed under compact subgroup

Let $G=SL(2,\mathbb{R})$ and let $K=SO(2)$ be our maximal compact subgroup. Let $(\pi,V)$ be a real irreducible representation of dimension $d$. Apparently one has that the set of $K$-fixed vectors ...
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Formulas give irreducible representation, $SL(2, \mathbb{C})$.

Let $\text{U}$ be an associative $\mathbb{C}$-algebra with three generators $E$, $H$, $F$, and three defining relations$$HE - EH = 2E,\text{ }Hf - FH = -2F,\text{ }EF - FE = H.$$Let ...
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For simple $\mathbb C[G]$-modules is the representation unique

Let $R$ be a ring, a $R$-module is called simple if it has no proper, nontrivial submodules. Let $G$ be a finite group, and denote by $\mathbb C[G]$ the free vector space over $G$, with the product ...
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Character representation of right regular representation as sum of irreducible characters

Let $G$ be a finite group acting on itself by the action $g \ast x := xg^{-1}$. Then this corresponds to an representation $\rho : G \to GL(L^2(G))$, where $L^2(G)$ denotes the space of function on ...
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Proof that the induced class function $\theta^G$ is a character if $\theta$ is a representation on subgroup

In these lecture notes by Daniel Bump on Induced Characters I have a question on the proof of Theorem 2.5.1. If $H$ is a subgroup of the finite group $G$ and $(\pi, V)$ a representation of $H$, i.e. a ...