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

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Probability distribution on finite group

I'm preparing for finals and this is a practice question. I'm not really sure how to start, so any solutions/hints/starting points are appreciated. Suppose $P = \sum a_gg$ were a probability ...
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22 views

The property of a module that has a simple socle

Let M be a module that has a simple socle.I can get that M is indecomposable and all submodules of M contain socM. Are there any other properties of M? Can we character the structure of M? And is ...
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30 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
6
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1answer
65 views

$SL_2(\mathbb{F})$, decomposing $\mathbb{C}\{X\}$ into irreducible $G$-representations and dimensions

Let $\mathbb{F}$ be a finite field with $q$ elements and $H = \mathbb{F}^\times$, the multiplicative group of $\mathbb{F}$. It is known that $H$ is a cyclic group of order $q - 1$, so $\widehat{H} = ...
5
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1answer
29 views

Name for the module corresponding to a square matrix

I recently learned that for each $n \times n$ matrix $A$ with entries in some field $F$, there is a corresponding $F[x]$-module $M_A$. Namely, $M_A$ is the set $F^n$ with vector addition defined as ...
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40 views

Decompose $Sym^2 (V)$ into direct sum of irreducible $S_n$-subrepresentations, where $V$ is the $2$-dimensional representation

Decompose $\operatorname{Sym}^2 (V)$ into direct sum of irreducible sub representations. (Hint: Again consider the action on basis vectors.)" Here, $V=\Bbb C^2$, with its standard basis, and the ...
2
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1answer
28 views

Coprime submodules: Quotients, intersections and direct sums

Let $R$ be a unital ring, and let $M$ be an $R$-module. Suppose $M_1, \ldots, M_n$ are $R$-submodules of $M$ such that $M_i + M_j=M$, for $i \neq j$. Are the modules $M/(\cap_{i=1}^n M_i)$ and ...
2
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1answer
43 views

G-invariant projector of irreducible representation

In the proof of Theorem 3.8 Etingof's notes (page 37, orthogonality of characters) there is the following claim. Let $G$ be a finite group, and let $P = \frac{1}{|G|}\sum_{g \in G} g \in \mathbb ...
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24 views

For any positive integer $n$, is there an irreducible representation of degree $n$?

I want to prove a problem: Prove that $Z(M_n(\Bbb{C}))=\{\lambda I\mid \lambda\in \Bbb{C}\}$. If $M\in Z(M_n(\Bbb{C}))$, let $\varphi:G\to GL_n(\Bbb{C})$ be an irreducible matrix representation of ...
0
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1answer
36 views

A question about the representation theory of finite dimensional algebra

Let $A$ be a finite dimensional algebra, $M$ be a finite dimensional module of $A$.The socle of $M$, $\mathrm{soc}(M)$, is the maximal semisimple submodule of M. The top of $M$ is ...
2
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1answer
33 views

What is the definition of $\ell^2(G)$ where $G$ is a group?

First I'll give some context for my question. I'm learning about crossed products of dynamical systems involving $C^*$-algebras and I've just seen the definition of a covariant representation. I have ...
1
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1answer
25 views

Motivation of the definition of principal series.

I am reading the book representation theory of semisimple groups. On page 33, the principal series representation $\mathcal{P}^{k,iv}$ is defined as follows. What are motivations of the above ...
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0answers
14 views

Why unitary principal series is unitary?

I am reading the book representation theory of semisimple groups. On page 33, I tried to verify that $\mathcal{P}^{k,iv}$ is unitary. We need to verify that $$ \left|\left| ...
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0answers
39 views

Category of Morphisms Between Modules

Let $A$ be a connected finite dimensional basic $k$-algebra with $k$ an algebraically closed field, and denote by $mod(A)$ the category of finite dimensional left $A$-modules. We define the category ...
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36 views

Fourier transformation of the symmetric group $S_3$

I am trying to compute the Fourier transformation of the symmetric group $S_3$ following the section 4 of Quantum Computing and the Hunt for Hidden Symmetry. The multiplication table of $S_3$ is as ...
0
votes
1answer
46 views

Derived equivalences and complexes of injective modules

I have a question about derived equivalences: Let $k$ be a field and $A$ and $B$ two finite dimensional algebras over $k$. Let $F : D^-(A) \to D^-(B)$ be an equivalence of triangulated categories. ...
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41 views

Representation of the automorphism group of a graph is reducible

We can define a representation of the automorphism group $H$ of a $n$-vertex graph $\Gamma$ as the map $\rho : H \to M$ where $M$ is the set of all $n \times n$ binary matrices. What is the ...
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1answer
32 views

Contraction of representations of universal enveloping algebra

$\quad$ (Following, e.g. SBBM) Given a Lie algebra contraction $\mathfrak{g}\xrightarrow{t(\epsilon)}\mathfrak{g}_0$, one can contract a family $\{\rho_{\epsilon}:\mathfrak{g}\rightarrow ...
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120 views

$k[x_1, \dots, x_n]$ free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism.

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
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23 views

Finite Dimensional representations as Langlands Quotient

Let $G$ be a real reductive group. For $P=MAN$ a parabolic subgroup of $G$, $(\sigma,W)$ is a unitary representation of $M$ and $\gamma \in \mathfrak{a_{\mathbb{C}}}^*$, denote by $J(P,\sigma,\gamma)$ ...
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15 views

Functions commonly listed in chemistry character tables for point groupss.

Is there a good resource for explaining why certain functions are associated with certain irreducible reps listed in character tables for common chemistry point groups? I can reason through the ...
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1answer
26 views

Do we have a $g \otimes g'$-action on $V \otimes V'$?

Let $g, g'$ be Lie algebras. Let $V$ (resp. $V'$) be a $g$-module (resp. $g'$-module). Do we have a $g \otimes g'$-action on $V \otimes V'$? In particular, when $g=g'$ and $V = V'$, do we have a $g ...
3
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1answer
36 views

Do we need transpose in the definition of a dual representation?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. There is an action of $G$ on itself given by left multiplication: $G \times G \to G$, $(f,g) \mapsto fg$, $f, g \in G$. There is a ...
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1answer
25 views

Unitary representation with non-closed invariant subspace

What would be an easy example of a unitary representation of a group on a Hilbert space that is topologically irreducible(has no closed invariant subspaces)) but not algebraically irreducible (has no ...
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19 views

$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$. My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
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1answer
26 views

Trivial representation of a lie Algebra?

Can someone explain why if $\rho:L \rightarrow \text{End}(\mathbb{C})$ is a lie algebra representation then it must be that $\rho(x)=0\ \forall \ x\in L$.
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2answers
41 views

What is the diagonal $\mathfrak{g}$-action on $V \otimes V^*$?

Let $\mathfrak{g}$ be a Lie algebra and $V$ a left $\mathfrak{g}$-module. Then the dual vector space $V^*$ is a right $\mathfrak{g}$-module with right $\mathfrak{g}$-action given by $(f.g)(v) = ...
1
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1answer
34 views

Do we have $End(V \otimes V) = End(V) \otimes End(V)$?

Let $V$ be a finite dimensional vector space. Do we have $End(V \otimes V) = End(V) \otimes End(V)$? Any help will be greatly apprciated!
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2answers
28 views

Exactly two irreducible characters of dimension 1

I've been working through Artin's Algebra on my own time, and I'm stuck on one of the questions, namely 10.5.3: Suppose that a group G has exactly two irreducible characters of dimension 1, and ...
2
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25 views

Another Stable Category

Let $A$ be a finite dimensional $k$-algebra. For $M,N \in mod(A)$, define: $$ \mathcal{P}_m(M,N) = \{ f\in Hom_A(M,N) | \exists \ P\in mod(A) \ with \ pd(P)=m \ and \ f \ factors \ through \ P \} ...
2
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1answer
19 views

Why $b: V \times \overline{W} \to \mathbb{C}$ is $G$-invariant?

I am reading the book Representations of Compact Lie Groups. On page 79, in the proof of Theorem 4.6, it is said that $b: V \times \overline{W} \to \mathbb{C}$ is $G$-invariant. We have \begin{align} ...
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40 views

How to show that $v \mapsto \pi(f)v$ is differentiable?

Let $G$ be a compact group. Let $(\pi, V)$ be a representation of $G$ and $f$ a smooth function on $G$. Define \begin{align} \pi(f)v = \int_G f(x)\pi(x) v dx. \end{align} We have \begin{align} & ...
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0answers
19 views

SU(3) tensor methods in representations [duplicate]

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
2
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0answers
30 views

Tensor formula in SU(3) representations

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
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0answers
17 views

Center of $U_q(g)$.

Let $g$ be a complex simple Lie algebra and let $U_q(g)$ be the corresponding quantum group. Is it true that the invariants of $U_q(g)$ under the adjoint action is the center of $U_q(g)$? It seems ...
2
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1answer
77 views

why the column sums of character table are integers?

There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number. It is mentioned in Martin Isaacs Character ...
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27 views

The functor from $sl_2-mod$ to $U_q(sl_2)-mod$.

Let $sl_2-mod$ be the category of all finite dimensional $sl_2$-modules and let $U_q(sl_2)-mod$ be the category of all finite dimensional $U_q(sl_2)$-modules, $q$ is not a root of unity. It is said ...
2
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1answer
17 views

If $w'(\beta)<0$ and $\ell(w)+\ell(w')=\ell(ww')$, then $ww'(\beta)<0$?

There's a small step in a computation with root systems that eludes me. Suppose $w,w'$ are elements of the Weyl group (which is a Coxeter group) such that $\ell(w)+\ell(w')=\ell(ww')$. Suppose you ...
4
votes
1answer
39 views

Infinite-dimensional Unitary representions that are not completely reducible

The Peter-Weyl theorem asserts that for a compact Lie group $G$ every unitary irreducible representation is necessarily finite-dimensional and any unitary representation is a direct sum of ...
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24 views

Verification of Ext groups and projective resolution for S3 over F3

So I've been looking at Ext groups of irreducible representations of $S_3$ over $\mathbb{F_3}$. Specifically, I'm doing a project where I'll be looking at extensions themselves, so am really only ...
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2answers
38 views

Property of minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm stuck on the proof of corollary 2.5.4 : If $M$ is a module for an Artinian ring $\Lambda$ and $S$ is a simple ...
2
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1answer
54 views

Understanding representable functors

I'm trying to wrap my head around the concept of representable functors - even though I know the definitions. I'm referencing the second page here for the example I want to understand about the ...
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1answer
40 views

Why are projective representations of a group classified by the second cohomology group?

I'm reading about the classification of bosonic SPT's, and I came across this statement: projective representations, where $v(g_1)v(g_2)=\alpha(g_1,g_2)v(g_1g_2)$, $v(g_1)$ being the transformation ...
2
votes
2answers
67 views

About the number of inequivalent irreducible representations of a finite group

We know that if $G$ be a finite group and $F$ be an algebraically closed field whose characteristic does not divide the order of $G$, then the number of inequivalent irreducible $F$-representations of ...
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14 views

Questions about vector fields on the upper half plane.

I am reading the lecture notes. On page 46, why $R_{X}$ as a vector field on $\mathcal{H}$ is $L_{pXp^{-1}}X$? Why $R_{\kappa} = 0$, $R_{\alpha}=2y \frac{\partial }{\partial y}$, ...
2
votes
1answer
61 views

A step in proof of Burnside's Theorem

I am reading the proof of Burnside's Theorem and it uses the following lemma in the online notes. page $70$ of - ...
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33 views

Faithful irreducible character of a group with exactly two minimal normal subgroups

Prove that any finite group with exactly two minimal normal subgroups has a faithful irreducible $\mathbb{C}$-character. What I have tried: Let $N_1$ and $N_2$ be two minimal normal subgroups of ...
2
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1answer
48 views

Largest irreducible representation of a finite non-commutative group

Let $G$ be a finite non-commutative group of order $k$. Is there any way to determine a number $m$ such that there will necessarily exist an irreducible representation of $G$ of dimension $d \geq m$? ...
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2answers
50 views

About character table of S3

Could anyone please explain how the character table (of conjugacy classes as column and irreducible representations as rows) gives information about the group? I want to understand this by applying on ...
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15 views

Are there some other non-degenerated natural bilinear forms on $M_n \otimes M_n$?

Let $M_n$ be the space of all $n$ by $n$ matrices. Then a non-degenerated natural bilinear form on $M_n \otimes M_n$ is $tr(AB)$. The reason is as follows. The natural linear form on $M_n$ is $f: M_n ...