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

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12
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138 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 ...
0
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27 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 $J(P,\sigma,\gamma)$ ...
1
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0answers
17 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 ...
1
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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
46 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 ...
0
votes
1answer
29 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 ...
2
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0answers
20 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 $\text{Harm}(\mathbb{R}^n, S_n)$?
-1
votes
1answer
36 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 $\rho(x)=0\ \forall \ x\in L$.
1
vote
2answers
42 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
vote
1answer
36 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!
1
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2answers
41 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
votes
0answers
26 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} ...
2
votes
0answers
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} & ...
1
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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
votes
0answers
32 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 ...
0
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0answers
21 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
votes
1answer
94 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 ...
3
votes
0answers
28 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
votes
1answer
18 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
64 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 ...
0
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27 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 ...
2
votes
2answers
47 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
63 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 ...
1
vote
1answer
45 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
86 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 ...
0
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0answers
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
66 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 - ...
1
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0answers
38 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
votes
1answer
51 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 $d \geq m$? ...
0
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2answers
65 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 ...
1
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0answers
17 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 ...
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0answers
19 views

How to find a highest weight vector in a tensor product of two representations of $sl_2$.

Let $V_\lambda, V_{\mu}$ be two representations of $sl_2$. How to find a highest weight vector in a tensor product of two representations of $sl_2$? Thank you very much.
0
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0answers
36 views

Character Decomposition of Permutation Representation

This question has already been asked on this site before. Here is the link to the question - Regular representation. My doubt is regarding the answer.As far as I know the 6 permutation matrices ...
4
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0answers
44 views

What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
0
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27 views

Homomorphisms of quantum spaces

Suppose we have a look at the Hopf $*$-algebra $U_q(sl(2))$ and the Hopf $*$-algebra $A_q(\widetilde{S}^3)$ introduced in the paper: http://arxiv.org/pdf/q-alg/9605017v1.pdf. I want to find a relation ...
2
votes
0answers
25 views

Tensor products and decomposition of $SU(3)$ representations

For each finite irreducible representation of Lie algebra $su(3)$ one knows that it is characterized by highest weight $(\lambda_1, \lambda_2)$ with integral entries. In this notation, $(1,0)$ is ...
1
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1answer
31 views

Matrix Reps of Associative algebra

It is widely known that non-associative algebras do not possess matrix reps as matrix multiplication is associative. Is the converse true? I.e. do all associative algebras has a faithful matrix rep?
0
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1answer
60 views

Sincere module and Grothendieck Group

Let $A$ be basic finite-dimensional $K$-algebra and $K$ algebraically closed. Let $F$ be the free abelian group generated by representatives of the isomorphism classes of objects in $mod A$. We ...
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0answers
39 views

Decomposing Into Real Representations

Decompose the regular representation of $C^3$ into irreducible real representations. Since $C^3$ is abelian,thus all its complex irreducible representations must be one dimensional.I guess that its ...
-1
votes
1answer
32 views

Tensor Product of irreducible modules

Let $A$ be a $\mathbb C$ algebra. Let $S$ be an irreducible $A$ module? Then what $ S \otimes_A Hom_A(S,S)$? Is it equal to S? I know that $S \otimes_{\mathbb C} Hom_A(S,S)$ is isomorphic to $S$ as a ...
3
votes
1answer
69 views

Example of Minimal Projective Resolutions

I am reading "Elements of the representation theory of associative algebras"'s book of Skowronski, Simson and Assem. I want to compute the global dimension of the example 2.5 c), of chapter 3, page ...
6
votes
1answer
28 views

Banach algebra, map $f \mapsto {1\over{2\pi i}} \int_S f(z) \cdot (z - a)^{-1}dz$ well-defined?

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge0}$ the corresponding multiplicative norm. For any $a \in A$, we define$$\text{spec}(a) = \{\lambda \in \mathbb{C} : \lambda ...
3
votes
0answers
45 views

Why is the image of a $\pmod p$ Galois representation finite?

Let $\overline{\mathbb{F}_q}$ be the algebraic closure of the finite field on $q=p^r$ elements, and $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ the absolute Galois group with the profinite ...
5
votes
1answer
158 views

Irreducible Representation by Restriction

Let $H$ be a subgroup of a finite group $G$.Given an irreducible representation $\pi$ of $G$,we may decompose its restriction to $H$ into irreducible $H$- representations.Show that every irreducible ...
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21 views

Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group. Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional ...
2
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0answers
22 views

Lower bound on multiplicative norm of Banach algebra.

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge 0}$ the corresponding multiplicative norm. For $a \in A$, do we have$$\limsup_{n \to \infty} (N(a^n))^{1\over{n}} \le ...
1
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1answer
54 views

$e_{i}Ae_{j} \neq 0$ implies that $e_{i}rad(A)e_{j} \neq 0$.

Let $A$ be a finite dimensional algebra over an algebraically closed field with $rad^2(A)=0$. Let $\{e_{1},\dots,e_{n}\}$ be a complete set of primitive orthogonal idempotents. Suppose that ...
1
vote
1answer
30 views

Independence of parabolic subgroup in parabolic induction and restriction?

Suppose $G$ is a complex algebraic group, $L$ a proper Levi subgroup, and $\lambda$ an irreducible character of the subgroup $L^F$ of $F$-stable points in $L$, which is contained in $F$-stable ...
0
votes
1answer
21 views

A definition in Character theory?

I would like to know the meaning of the term Character Field used by B. Huppert in his book Endliche Gruppen 1. For example they have used the notation $K(\chi)$. I dont know what it stands for?