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

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Suppose f:V->W is an FG homomorphism. Show that ker(f) is a submodule of V.

I think I have done this I just want to check that this is enough to show that it is true, given it is worth 12 marks. First we recall from Linear Algebra that her(f) is a subgroup of V. Suppose that ...
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35 views

Show that $U=\langle v_1-v_2,v_2-v_3,…,v_{n-1}-v_n\rangle$ is a sub module of $V$.

$G=S_n$ and $V$ is a vector space over a field $F$, with basis $\{v_1,....v_n\}$, then $V$ is an $FG$ module with action defined by setting $g · v_i = v_{g(i)}$ for all $g\in G$ and $1 < i < ...
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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 ...
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1answer
157 views

Heisenberg XXX spin model

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
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34 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 ...
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29 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 ...
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Closed form for 3j-symbol ratio

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch Gordan coefficients. ...
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1answer
95 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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83 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 ...
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20 views

Why $\widehat{G^{\mathbb{C}}}$ can be identified with the space of highest weights

Let $G$ be a compact connected Lie group and $G^{\mathbb{C}}$ be the complexification of Lie group $G$ and we denote $\widehat{G^{\mathbb{C}}}$ the set of isomorphism classes of irreducible rational ...
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39 views

Reference Request: Characters of Finite General Linear Groups

I've been looking at J.A. Green's article The Characters of Finite General Linear Groups and it seems that Green in this article comes up with a way of calculating all irreducible characters of a ...
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18 views

How to decompose a representation of $so(n)$ into representations of a subalgebra

In some cases, it is possible. For instance the representation $16$ of $so(9)$ decomposes as $8_c+8_s$ of $so(8)$. Now I would like to do the same with representations of $so(8)$ into a sum of ...
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39 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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24 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 ...
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1answer
26 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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29 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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79 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 ...
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119 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 ...
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35 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 ...
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1answer
63 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 ...
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1answer
38 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 ...
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27 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 ...
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1answer
40 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 $G$ ...
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33 views

Equivalent representations of $\mathfrak{sl}_2$

Hello I have a question about the equivalence of two representations of the Lie algebra $\mathfrak{sl}_2$. The first representation is $(ad,\mathfrak{sl}_2)$ the adjoint representation with map ...
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17 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 ...
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1answer
56 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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1answer
37 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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29 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
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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 ...
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1answer
34 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 : ...
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2answers
55 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 ...
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20 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 $f(r_1, \ldots, r_n)=0$ ...
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1answer
73 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 ...
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43 views

Representation of $\mathbb C$

Let $M_2(\mathbb R)$ be the ring of $2\times 2$ matrices with real entries. Its group of multiplicative units is $GL_2(\mathbb R)$, consisting of the invertible matrices in $M_2(\mathbb R)$. (a) ...
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132 views

Representation theory of infinite groups?

I am familiar with the representation theory of finite groups (at least of the symmetric groups over the field of complexes) And I know that the group algebra of an infinite group is not semisimpe ...
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26 views

Questions about cuspidal representations of $GL_2(\mathbb{F}_q)$.

All representations of $GL_2(\mathbb{F}_q)$ are classified in the book. They are principal series representations, complementary series representations, 1-dimensional representations. They form all ...
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16 views

Multiplicity of the tensor product of $A_n$-modules

In Coutinho, A Primer of Algebraic D-Modules, Theorem 13.4.1 (p. 128), I read: Theorem. Let $M$ be a finitely generated $A_m$-module, $N$ a finitely-generated $A_n$-module. $d(M ...
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42 views

question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
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Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
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25 views

How to show that $Ind_K^G \mathbb{C}_\chi$ is natrally isomorphic to $\mathbb{C}[G]e_{\chi}$?

Let $K \subset G$ be finite groups and $\chi: K \to \mathbb{C}^*$ be a homomorphism. Let $\mathbb{C}_{\chi}$ be the corresponding 1-dimensional representation of $K$. Let $$ e_{\chi} = \frac{1}{|K|} ...
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24 views

Questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$.

I have some questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in GL_2(\mathbb{F}_q)$ has eigenvalues which ...
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2answers
82 views

About the converse of Maschke's theorem

The Maschke's theorem say that\ Let $G$ be a finite group and $F$ a field whose characteristic does not divide $\mid G \mid$. Then every $FG$-module is completely reducible (I'm using the notation of ...
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78 views

representation of abelian which is noncyclic

Let $G$ be a noncyclic abelian group acting by conjugation on an elementary abelian $p$-group $V$, where $p$ is a prime not dividing the order of $G$. (a) Prove that if $W$ is an irreducible ...
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Holomorphic representation of $G$ on vector space $V$.

Let $G$ be a Lie group. What is the definition of holomorphic representation of $G$ on vector space $V$.
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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1answer
38 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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46 views

$1$-dimensional representations of $GL_2(\mathbb{F}_q)$.

I have some questions about $1$-dimensional representations of $G=GL_2(\mathbb{F}_q)$. I need to show that there are $q-1$ $1$-dimensional representations of $GL_2(\mathbb{F}_q)$. I am able to show ...
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72 views

Action of a group G on $\mathbb{C}$[G] makes $\mathbb{C}$[G] a G-module

I am trying to show that the action of the group G on the vector space of functions $\mathbb{C}$[G] defined by [g.$\psi$] (h)=$\psi(g^{-1}h$) makes a G-module. My main problem is that I can't get any ...
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11 views

Meromorphic Continuation of Intertwining Operator Identities for K-finite vectors

I have a question about a "well-known result" about intertwining operators. I will restrict myself to $\text{SL}_2$ for simplicity. Let $$w = ...
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36 views

Weights of group in terms of it fundamental weights

how can I found the weight of a group G ( I find the fundamental weights, but I don't know how found the linear combination of fundamental weights that give me the weights ). so we found the ...