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

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2
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1answer
50 views

Branching rule restriction to $\mathrm{O}_9 \Bbb C$ from $\mathrm{GL}_9 \Bbb C$

On page 427 of Fulton and Harris's Representation Theory, the authors give the branching rule for the above restriction as $$ \mathrm{Res}_{\mathrm O_m \Bbb C}^{\mathrm{GL}_m \Bbb C} (\Gamma_\lambda) ...
2
votes
1answer
76 views

Sum of representations and characters of the symmetric group

Hi I was wondering if I could have some help to go in the right direction. I want to show that $\displaystyle\sum\limits_{\sigma \in S_n} (sgn(\sigma)*\chi(\sigma)) =0$ where $sgn(\sigma)$ : $S_n ...
1
vote
1answer
101 views

Generalization of Schur's lemma

I would like to proof a generalization of Schur's lemma for representations. (Schur's lemma) (cfr. Jean-Pierre Serre, Linear representations of finite groups) Let $\rho^1$: G $ \to $ GL($V_1$) and ...
1
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0answers
120 views

Irreducible representations of $SO(5)$

I am looking for irreducible representations of the group $SO(5)$ that can be described by a tensor of at most rank two. My own considerations have brought me to the conclusion that there is a ...
5
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1answer
79 views

Construct a rational matrix $A$ s.t. $A^m = I$

Let $K$ be a field of either $\mathbb{C}$, $\mathbb{R}$ or $\mathbb{Q}$, Let $V$ be a $n$ dimensional vector space over $K$. I want to construct a matrix $A \in GL(V)$ s.t. $A^m = I$ for some $m$ and ...
2
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1answer
47 views

For z in Z(G) show that there exists $\lambda_z$ such that $z.v=\lambda_z v $ for all v in V

Let $V$ be an irreducible $\mathbb CG$ module. We define $Z(G)$ to be the centre of $G$. For $z\in Z(G)$ show that there exists $\lambda_z\in\mathbb C$ such that $z\cdot v=\lambda_z\cdot v$ for all $v ...
3
votes
1answer
93 views

Standard represention of $S_3$

I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in the ...
1
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1answer
76 views

Irreducible Representation and the center of a group

Hi I was wondering if someone could help me/hint along the right path. Let $\rho:G \rightarrow GL(V)$ be an irreducible representation. Let $Z(G)$ be the center of $G$. Show that if $a\in Z(G)$, then ...
1
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2answers
108 views

On the graph of induction-restriction for group-subgroup representations

Let $G$ be a finite group, and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ and $H$ (up to isom.). Consider the graph ...
3
votes
1answer
118 views

How to show that Yang-Baxter equation is the same as braid equation?

The quantum Yang-Baxter equation is $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$. The braid equation is $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. It is said that these two equations are equivalent. How to ...
1
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1answer
91 views

Representation of $SL(3,\mathbb Z)$

I read the paper "Real-analytic actions of lattices", it says that any representation of any finite-index subgroup of $SL(3,\mathbb Z)$ into $GL(2,\mathbb R)$ has finite image, so how to prove it? ...
0
votes
1answer
102 views

Determinant of the irreducible characters table of a finite abelian group

Let $G$ be an abelian group, $|G|=n<\infty$. Let $\Phi$ be an irrep of G. Find the modul of determinant of the characters table. Ther is answer in my book. It is $n^{n/2}.$ If G is cyclic ...
3
votes
2answers
303 views

Reference request for studying Lie group & Lie algebra representations

I am learning representation theory of Lie groups & Lie algebras from the book by Brian Hall. Unfortunately, this does not discuss infinite dimensional representations. Which books should I study ...
3
votes
1answer
93 views

Number of Cocharge Tableaux Summing to Fixed Numbers

First, some background: I will assume the anglophone conventions for Young tableaux in what follows. Given a standard Young tableau $T$ of shape $\lambda$, we can define the cocharge tableau $C(T)$ as ...
11
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191 views

Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters

I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
8
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1answer
100 views

Non-polynomial representations of $GL_n$

Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the ...
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1answer
107 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...
0
votes
1answer
63 views

character of a representation of the group $S_n$

Let $\Phi$ be a representation of the group $S_n$ in the space with basis $(e_1, \ldots, e_n)$ such that $\Phi(\sigma)e_i=e_{\sigma(i)}.$ Find character of $\Phi$ This is looks like a regular ...
2
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0answers
57 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...
3
votes
1answer
322 views

Decomposing the tensor product representation of $S_3$ in terms of irreducibles

I have a theorem which says that: If $\rho_1,...\rho_n$ are a complete set of irreducible $K$-representations of $S_n$ then we have that: $V^{\otimes n}=\bigoplus_1^k(V^{\otimes n}_{\rho_i})$ as ...
1
vote
1answer
148 views

Isaacs exercise 5.4 (Character Theory of Finite groups)

Any advice/hints how to prove the following statement? If $G$ is a finite group and $b(G)=\max\{\chi(1); \chi\in \mathrm{Irr}G\}$ is the maximal degree of irreducible characters and $H\leq G$, then ...
8
votes
1answer
292 views

Decomposition of Permutation Representation

Let $G$ be a finite group acting on a finite set, 2-transitively, and $(\rho,V)$ be the corresponding permutation representation of $G$ over $\mathbb{C}$. Then it is known that $V$ is direct sum of a ...
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1answer
57 views

Show that $W = \langle v_1 + \dots + v_n\rangle $ is a submodule of $V$ .

Let $G = S_n$. Let $V$ be a vector space over a field $F$ with basis $\{v_1,\dots,v_n\}$. Then $V$ is an $FG$-module with action defined by setting $g · v_i = v_{g(i)}$ for all $g \in G$ and $1\leq i ...
3
votes
1answer
59 views

1-dimensional FG-Modules

Suppose $V$ is a two-dimensional FG-Module and there exists $g,h \in G$ $v \in V$ such that $(gh).v \neq (hg).v$. Show that $V$ is irreducible. I can understand the idea of this is to use Maschke's ...
3
votes
1answer
127 views

Modules and submodules

Let $G=S_n = Sym_n$ be the symmetric group and $V$ a vector space with basis $\{v_1,...,v_n\}$, then $V$ is a module with action defined by $g$. $v_i$=$v_{g(i)}$ for 1$\leq$i $\leq$ n and extending ...
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0answers
38 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
2
votes
1answer
62 views

Why does the universal cover of $GL^+_n$ not admit finite-dimensional representations?

Let $GL^+_n \subset \mathbb{R}^{n \times n}$ be the subgroup of real matrices with positive determinant and $\widetilde{GL}^+_n$ be its universal cover. Why does $\widetilde{GL}^+_n$ not admit ...
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2answers
219 views

Let G be a finite group and V be the regular CG-Module. Find a submodule of V which is isomorphic to the trivial CG-Module

As in the question. I have read through different books and articles and they seem to set W=<$\sum_{g \in G}$ g> as a submodule. I can understand that this is unique, but I fail to see how this is ...
3
votes
1answer
130 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Introductory Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the ...
2
votes
1answer
66 views

Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Question: Is $dim(V^H) \le ...
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1answer
44 views

Maximal subgroup and representations (dual part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Let $g \in G$, $K = H \cap ...
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0answers
39 views

Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
2
votes
1answer
42 views

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
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0answers
117 views

Prove that every 2 dimensional FG-Module with gh not equal to hg is irreducible.

Basically the questions is as follows. Suppose that $V$ is a 2-Dimensional $FG$-Module where F=The complex numbers and that there exists $g,h$ elements of $G$ and $v$ an element in $V$ such that ...
9
votes
1answer
178 views

$\chi(g)$ group character $\Rightarrow$ $\chi(g^m)$ group character

Let $G$ be a group of order $n$ and and $\gcd(m,n)=1$. Let $\chi:G\rightarrow\mathbb{C}$ be a class function and define $\chi^m\!: g\mapsto\chi(g^m)$. How can one show that $\chi^m$ is a character iff ...
1
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1answer
53 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
1
vote
1answer
100 views

Irreducible representations of finite lamplighter group

Let $G = \mathbb{Z}_2 \wr \mathbb{Z}_n$ be the finite lamplighter group. What are the irreducible representations of $G$ - can anyone provide a clear reference? Austin, Naor and Valette list ...
3
votes
1answer
103 views

Modular function of the unipotent radical of a parabolic subgroup of a reductive group

Let $G = \text{GL}_n(\mathbb{R})$. For a partition $\underline{n} = (n_1,\ldots,n_t)$ of $n$, let $P = P_{\underline{n}}$ denote the standard, block-upper-triangular parabolic subgroup of $G$ ...
0
votes
0answers
39 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the ...
0
votes
1answer
86 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
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30 views

derived equivalence of coalgebras

let $C$ and $D$ be two coalgebras over a field, $C$ and $D$ are called derived equivalent if the derived categories $D(C-comod)$ and $D(D-comod)$ are equivalent as triangle categories. if $C$ and ...
4
votes
1answer
80 views

What am I missing about Schur functions?

Let's say I only know the following about Schur functions: you give me a partition $\lambda$ of $d$ such that $\lambda$ has $n$ parts $\lambda_1,\ldots,\lambda_n$, and I can compute the Schur function ...
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89 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
2
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0answers
52 views

A question in the proof that the weight of a finite dimensional module is W-invariant

Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which ...
4
votes
2answers
84 views

Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
4
votes
1answer
122 views

An analogy between group actions and group represenations

I was trying to make a 'dictionary' between group action and group representation terms using the $\mathbb{C}[-]$ functor. I immediately found that if the set $Y \subset X$ is invariant under the ...
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1answer
222 views

Finding irreducible representations

This might be a very elementary question in representation theory, but I dare to ask Suppose I am asked to complete the character table of $S_5$, I know it has 7 conjugacy classes as follows : ...
2
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1answer
74 views

$\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(Ae,N), N\otimes_{eAe}A=N\otimes_{eAe}eA$.

Let $e$ be an idempotent of a ring $A$ and $N$ is an $A$-module. Why $\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(Ae,N), N\otimes_{eAe}A=N\otimes_{eAe}eA$? Can you prove this explicitly? Is the ...
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0answers
33 views

representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
6
votes
1answer
347 views

Representation theory over $\mathbb{Q}$

I am looking for books or papers which tell me something about representation theory of finite groups over $\mathbb{Q}$ (or finite extensions thereof which are not splitting fields of the group ...