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

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(representation theoretic) meaning of sum over even rows of a Young tableau

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define ...
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Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=q\{a,b\}+o((q-1)^2). $$ Is ...
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16 views

Degrees of irreducible complex characters of alternating groups [on hold]

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
2
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25 views

How do we know if a Young Tableau represents $3$ or $\bar{3}$?

Dimension three in the title was just an example...In general I want to know for any dimension $d$ and any SU(n). I am aware of the rule for tensors ; if more lower indices than upper then it is the ...
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22 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
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1answer
30 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$? [on hold]

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
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1answer
25 views

$V^{\oplus3}$, linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
3
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1answer
73 views

Finding $10\otimes 8\otimes 8\otimes 8$ in $SU(3)$

I know that in $SU(3)$ $$8\otimes 8 = 27+10+\bar{10}+8+8+1. $$ How can one use this to compute $$10\otimes 8\otimes 8\otimes 8?$$ Can one start with simplifying $$\tag{1} 10\otimes 8\otimes 8 = ...
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11 views

Dimension of intertwining operators on an irreducible representation

This is an exercise from Serre's Linear representations of finite groups: Let $H_{i}$ be the vector space of intertwining linear operators on the irreducible representation $W_{i}$ in $V_{i}$. ...
3
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1answer
25 views

The irreducibility of representations of parabolic induction over $\mathfrak{gl}_n(\mathbb{C})$

I have the following problem: Let $\mathfrak{g} = \mathfrak{gl}(n)$ be the general linear Lie algebra over $\mathbb{C}$. Denote $\Pi = \{\alpha_i:=\epsilon_i-\epsilon_{i+1}\}_{i=1}^{n-1}$ by the ...
2
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2answers
38 views

Group homology with coefficients vanish

Say $G$ is a group and $M$ is a $\mathbb ZG$-module with the property that $H_i(G;M)=0$ for all $i\ge 0$. Does this happen besides $M=0$?
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38 views

In what way is the representation not continuous

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf pp.7-8 Following the definition of a linear representation it states 'if V (an E-vector space) is equipped with a topological structure and if ...
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1answer
34 views

Representation theory of $\mathbb{Z}_k$ and complex roots of unity

Is there a natural relationship between the (characters?) of irreducible representations of $\mathbb{Z}_k$ and the $k$ complex-roots of unity? Can they be like thought of as characters of its ...
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47 views

Is a matrix element of a norm continuous representation always a trigonometric polynomial?

Let $\pi:G\to {\mathcal B}(H)$ be a unitary representation of a compact group $G$ in a Hilbert space $H$. Consider a matrix element of $\pi$, i.e. a function of the form $$ ...
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63 views
+50

Why a tensor product of 2x2 unitaries cannot implement a 3x3 unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
3
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1answer
29 views

Counterexample for Maschke's lemma

I'm trying to relax conditions and come up with a counterexample to Maschke's lemma in such a case. For example, with $k = \mathbb Z/p\mathbb Z$, I'm considering the two dimensional representation of ...
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22 views

Showing that a subrepresentation is isomorphic to the trivial representation

I'm considering $V$ to be the regular representation of a group $G$ and $W$ to be the 1-dimensional subspace of $V$ generated by the element $x=\sum_{s\in G} e_s$. I'm trying to show that $W$ is a ...
4
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3answers
52 views

if $\rho: H \to \text{GL}_n({\bf C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful

How do I show that if $\rho: H \to \text{GL}_n(\mathbb{C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful?
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2answers
50 views

Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?

Let $G$ be a nonabelian group with center $Z(G)$. Let $\rho: Z(G) \to \text{GL}_n({\bf C})$ be an irreducible representation. Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not?
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2answers
39 views

Frobenius reciprocity, proof if $V$ irrep of $G$ then multiplicity of $V$ in regular rep of $G$ is $\dim(V)$

I know the standard proof of the fact that if $V$ is an irreducible representation of $G$ then the multiplicity of $V$ is the regular representation of $G$ is $\dim(V)$. Does there exist a proof using ...
2
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1answer
30 views

exists homomorphism of $G$-representations $\pi: V \to V$ with image $X$

Let $G$ be a group (not necessarily finite) and $F$ a field. Let $V$ be a $G$-representation. Suppose $V$ is isomorphic to a direct sum of irreducible representations. Let $X \subset V$ be any ...
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1answer
52 views

An example of $_AM$ is simple but $M^\star_A$ is not?

Let $A$ be a finite dimensional $k$ algebra, $k$ is a field. It is evident that the duality functor ($(-)^\star=hom_k(-,k)$) preserves the simplicity in the case of finitely generated $A$- modules ...
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10 views

When are all Gorenstein projective also pure-injective?

For an artin algebra of finite global dimension, each Gorenstein projective module is projective then is pure-injective. Are there any other examples having this property? That is, all Gorenstein ...
3
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1answer
39 views

Fixed point subspaces $V^B$ and $V^G$ for a Borel subgroup $B\subset G$ coincide

Assume that $G$ is a linear algebraic group, and let $B \subset G$ a Borel subgroup of it. Let $(V,\rho)$ a rational $G$-module. Define $$V^G := \{ v \in V \mid g \cdot v = v \quad \forall g \in G ...
5
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2answers
49 views

$n-1$ dimensional permutation module for $S_n$

Say $n \ge 5$. Let $P$ be the $(n-1)$ dimensional permutation module for $S_n$, i.e. the permutation representation on $\{(x_1, \dots, x_n) \in {\bf C}^n: \sum x_i = 0\}$. Prove that: $\wedge^2P$ ...
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61 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.
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45 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
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1answer
30 views

Why it is central in $\mathbb {Z}[G]$?

In proposition 4.17, why is $P$ an central element?
2
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2answers
42 views

extracting the middle term of $ (z \cos \theta + w\sin \theta )^m(- z\sin \theta + w\cos \theta )^m $

Is there a systematic way to extract the middle term of the following expression? $$ (z \cos \theta + w\sin \theta )^m(- z\sin \theta + w\cos \theta )^m $$ This is homogeneous polynomial of degree ...
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1answer
25 views

adjoint representations

I am trying to work out the adjoint representations of $$H=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right), X = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} ...
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31 views

representations of Lie algebras

I am studying irreducible representations of Lie algebras when our filed is of positive characteristic, I need an explicit explanation with example (or an article) which describes the differences what ...
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28 views

An example contradicting the validity of Maschke's theorem for infinite groups.

I am learning representation theory of finite groups and I am in doubt about a homework problem: Let $G = \mathbb{Z}$ and $V = \{(a_1 , a_2 , . . . )|a_i ∈ R\}$ be a vector space of infinite ...
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26 views

On the contragredient representation

Let $\pi$ be a representation of group $G$.Then its contragredient representation $\pi^{\vee}$ is defined by $\pi^{\vee}(g)=^{t}\pi(g^{-1})$. (here $^t$ means the transpose) But I heard that it is ...
2
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1answer
36 views

$E_6$ lie algebra and its representation

I've just started learning about Lie theory (only just finished up to basic classification of semisimple lie algebras) and I've got the following questions: How do I show that the complex lie algebra ...
0
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1answer
54 views

Representations of symmetric groups of order $2n$ and $n$

Background: Denote by $S_n$ the symmetric group of order $n$. There are many ways to embed $S_n$ as a subgroup into $S_{2n}$. Given a symmetric group, we can use Young diagrams to classify all ...
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problem in representation of a finite abelian group

There is problem, asking, find all in-equivalent representations of an abelian group $G$. My attempt: Let $f:G \to GL(V)$ a representation, by maschacke theorem $f_g$ is equivalent to direct sum of ...
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1answer
43 views

2-transitively, formula [closed]

Let $G$ be a finite group and let $X$ with $|X| \ge 2$ be a set on which $G$ acts. Then $G$ acts on $X \times X$ via $g \cdot (x, y) = (g \cdot x, g \cdot y)$. The action of $G$ on $X$ is called ...
4
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0answers
74 views

Every irreducible character of $G$ is an irreducible character of $H$?

Let $H$ be a proper subgroup of $G$ such that for all $\chi\in Irr(G)$, $\chi_H\in Irr(H)$. That is, the restriction of every irreducible character of $G$ to $H$ is an irreducible character of $H$. ...
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How getting the unitarized irreducible representations with GAP?

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example: ...
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1answer
35 views

Fundamental weights of $A_n$

I have the following problem: Let $\mathfrak{g}$ be the Lie algebra of type $A_n$. We choose $e_i^*-e_{i+1}^*$ as simple roots. Is there a closed formula for the fundamental weights? Thank you!
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1answer
35 views

Character Table S4

I am trying to understand how to build a character table of S4. I've already read many articles about it but I am stuck at one point. S4 has 5 conjugacy classes and therefore 5 irreducible ...
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2answers
31 views

Automorphism group of vector space

I was trying to understand definition of representation and trivial representation thus came across the case where $ V= K $ here $V$ is a vector space over a field $K$ and thus $Aut_K (V) \cong ...
3
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1answer
24 views

$G$-invariant subspaces in $K[G]$

Let $K$ be an algebraically closed field and $G$ an linear algebraic group (i.e. a group object of the category of affine varieties over $K$). Denote by $A$ the coordinate ring of $G$. Then the right ...
3
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21 views

Tensor product via the diagonal action of a Hopf algebra

Let $H$ be a Hopf algebra and $V$ and $W$ two left $H$-modules, then $V\otimes W$ is also a left $H$-module via the comultiplication of $H$. I now want to consider the functor $-\otimes_H (V\otimes ...
2
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0answers
12 views

$[L_+^m, L_y^n]$ in the $SO(3)$ Lie Algebra

Let $SO(3)$ be generated by infinitesimal rotations $L_x, L_y, L_z$ such the typical relations $ [L_x, L_y] = L_z $ and similar. Let $L_\pm = L_x \pm i L_y$ be the raising and lowering operators. Is ...
2
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1answer
65 views

Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me ...
6
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3answers
96 views

Textbooks on permutation groups?

I need good texts on group theory that cover the theory of permutation groups. I think there is a book called Wielandt. Is it good? are there newer alternatives? Can I find books that are not ...
0
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1answer
48 views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
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1answer
19 views

Natural Lie algebra representation on function space

There is natural Lie group representation of $GL(n)$ on $C^\infty(\mathbb{R}^n)$ given by \begin{align} \rho: GL(n) & \rightarrow \text{End}(C^\infty(\mathbb{R}^n)) \\ A & \rightarrow ...