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

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Procedure to construct a map from the automorphism group of a graph to the natural permutation representation

Let $\Gamma$ be a graph with $n$ vertices. Let $\varphi_\Gamma$ be the map from the symmetric group $S_n$ to the space of natural permutation representation $\text{Mat} \left(n, \mathbb{C}\right)$ ...
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Representation of Weyl algebra

Let's consider an algebra $W$, generated by a family of differential operators of type $$\sum_{k=0}^{n}{a_{k}(x) \cdot \frac{d^{k}}{dx^{k}}}$$ (may also known as Weyl algebra). I would like to prove ...
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Weyl group of complex Lie group

Let $G$ be a compact connected Lie group with maximal torus $T$. The Weyl group is defined by $$W:=N_G(T)/T.$$ Now, $G$ has a complexification $G_{\Bbb C}$ with maximal torus $T_{\Bbb C}$ which is the ...
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38 views

How to show that $\frac{1}{u} q^{2u\frac{d}{du}} =q^2 q^{2u\frac{d}{du}} \frac{1}{u} $?

Let $q$ be a complex number. Let $u$ be a variable. How to show that $\frac{1}{u} q^{2u\frac{d}{du}} =q^2 q^{2u\frac{d}{du}} \frac{1}{u} $? I think that it suffices to show that $q^2 u ...
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56 views

Relation between reflection group and coxeter group

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, ...
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34 views

Prove that $(\mathfrak{su}(2))^* \cong \mathfrak{sb}(2)$

Let $\mathfrak{su}(2)$ be the Lie algebra with basis elements $$ e_1=\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} , \quad e_2=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} , ...
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38 views

Finite linear Representations are necessarily distance preserving

Suppose I have a linear representation of a finite group $G$. That is a homomorphism $$ \pi: G \rightarrow GL(\mathbb{R}^n) $$ Meaning a collection of matrices, which under matrix multiplication form ...
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24 views

Ideal as subrepresentation of algebra

I am confused the statement that ideal is a subrepresentation of regular representation of an algebra. Actually I am not very clear about the definition of regular representation of an algebra, ...
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1answer
19 views

How to define a quiver of basic and non-connected associative algebras

Recently, I am reading a book "Elements of the Representation Theory of Associative Algebras". Let $A$ be a basic and connected finite dimensional $~\mathbb{K}$-algebra and $\{e_1, e_2, \cdots, ...
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1answer
37 views

A quotient module of a Lie algebra

Let $L$ be a Lie algebra. If $A$ and $B$ are $L$-submodules of an $L$-module $V$, such that $A\subset B$ and $I\cdot B\subset A$ for some ideal $A$ in $L$. I want to understand why this implies that ...
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62 views

Linear independence for functions on Z/m

Let $p$ be prime and consider the functions $f_k:(\mathbb Z/p)\backslash\{0\}\rightarrow\mathbb R$ defined by $f_k(x)=\csc^2\left(\frac{k\pi x}{p}\right)$. Question: How might I show that the ...
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Haar Measure of a Topological Ring

A topological ring is a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and ...
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73 views

Homorphism from $B(G)$ to $\mathbb{Z}$

Let $G$ be a finite group, and $B(G)$ be its Burnside ring. Show that each ring homorphism $\varphi:B(G)\to\mathbb{Z}$ is the mark of some $H\le G$, i.e. it maps to an equivalent class of finite ...
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Dimension of character in square equals index of center.

Let $G$ be a finite group and let $\chi$ be an irreducible character. Assume that $G/Z(\chi)$ is abelian, how can I prove that then $\chi(1)^2=\mid G:Z(\chi) \mid$? Note that $Z(\chi)= \{g \in G : ...
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19 views

Definition of multiplicity of trivial representation

Suppose we have a group representation $$\rho: G \to GL(V)$$ where $V$ is a finite-dimensional complex vector space and $|G| < \infty$. I have been confusing myself about the definition of ...
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43 views

Confused by two different perspectives on $G$-vector bundles

I'm trying to understand how these two perspectives on vector bundle with a $G$-action come together. Perspective 1: Let $P \to X$ be a principal $G$-bundle. The associated bundle construction gives ...
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Question on irreducible character.

Suppose that $\chi \text{Irr}(G)$, i.e $\chi$ is an irreducible character, and assume that $G/Z(\chi)$ is abelian, where $Z(\chi)=\{g \in G : \mid\chi(g)\mid = \chi(1) \}$. How can I prove thet ...
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57 views

Congruence subgroup of $\mathbb{GL}_n(\mathbb{Z}_p)$

In course of my research I met the following situation : 1) I have a bunch of open subgroup (so of finite index) in $\mathbb{GL}_{n}(\mathbb{Z}_p)$. 2) My groups arises naturally as stabilizers of ...
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57 views

Tensor product group representations and spaces of intertwiners.

Let $V_{1}$, $V_{2}$, $W_{1}$, and $W_{2}$ be the carrier spaces of representations of some finite group $G$. Suppose also that $G$ acts trivially on $V_{1}$ and $V_{2}$. I would like to prove the ...
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29 views

Second symmetric power of sum of representations

Let $\mathfrak{g}$ be a complex Lie algebra of type $A_{n-1}$. Consider representation of $\mathfrak{g}$ on direct sum of complex vector spaces which is given by the highest weight ...
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24 views

Show $\rho_1 \oplus \rho_2 : G → GL(V \oplus W)$ is a homomorphism

Show that the direct sum $V \oplus W$ is a representation of the finite group $G$. Given that $V, W $ are representations. attempt: Suppose that $V, W$ are vector spaces. Then define $\rho_1: G → ...
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35 views

If a group algebra acts regularly on a module, can this module be identified as a left ideal?

To be more specific, I am looking at $F_2[D_p]$, where $D_p$ is the dihedral group of order $2p$. If this group acts regularly on the basis of a vector space $F_2^{2p}$, and there is a subspace of ...
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Sum of elements in row of character table is positive integer.

If $G$ is a (finite) group, how can I prove that in the corresponding character table, the sum of the elements in any row is a non-negative integer? The hint in the book says that I should let $G$ act ...
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Compute the sum of number of fixed points

$\newcommand{\def}{\mathrm{def}}\newcommand{\std}{\mathrm{std}}\newcommand{\triv}{\mathrm{triv}}$Suppose $V_\def, V_\std, V_\triv$ are the defining , standard and trivial representations of the ...
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Show $\phi$ is a isomorphism as a lie algebra homomorphism

Show $\phi$ is a isomorphism as a lie algebra homomorphism $\phi: \textbf{su}_2 \bigotimes_{\mathbb{R}} \mathbb{C}\rightarrow sl_2(\mathbb{C})$ and $\phi: a(I \bigotimes 1)+b(J \bigotimes 1)+c(K ...
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25 views

Schur-Weyl duality for general representations

The classical Schur-Weyl duality deals with the decomposition of $V^{\otimes k}$ into irreps of $S_k\times GL(n)$, where $V=\mathbb{C}^n$ is the defining irrep of $GL(n)$. Is there a version of the ...
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Compute $\Sigma_{\pi \in S_n} f(\pi)$ and $\Sigma_{\pi \in S_n} f(\pi)^2$.

Suppose $V_{def}, V_{std}, V_{triv}$ are the defining , standard and trivial representations of the symmetric group $S_n$. And let $V_{def} \cong V_{std} \oplus V_{triv}$, and suppose the characters ...
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80 views

Contraction map on tensor product of symmetric powers is surjective.

The context is the representation of $\mathfrak{s}\mathfrak{l}_3$ as per Fulton and Harris: The contraction map $i_{a,b}:\mathrm{Sym}^aV\otimes \mathrm{Sym}^bV^*\rightarrow \mathrm{Sym}^{a-1}V\otimes ...
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irreducible representations of $GL_2$ over $p$-adic field

Let $E$ be a finite extension of $\mathbf{Q}_p$. In class we stated the following fact : Every irreducible algebraic representation of $GL_2(E)$ is of the form $$ \mathrm{Sym}^{k-2}(E^2) \otimes_E ...
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48 views

The relation between quasi-permutation matrix and permutation matrix?

We know that a quasi-permutation matrix is a square matrix over the complex numbers with non-negative integral trace. Can anyone tell me why it is called "quasi-permutation matrix"? Is there any ...
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26 views

Understanding Weyl character formula and highest weight integrable representations

Weyl character formula is $\chi=\frac{\sum_{w \in W} \epsilon(w) \exp{(w (\lambda + \rho}))}{\sum_{w \in W} \epsilon(w) \exp{(w ( \rho}))}$ So I understand what is $\epsilon(w)$ but I don't understand ...
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Real topological K-theory of cyclic group

Letting $C_n$ be the cyclic group on $n$ elements we know through the use of the Atiyah-Segal completion theorem that $$ K^*(BG) = \pi_*(KU)[[t]]/((t+1)^n -1) $$ where $\pi_*(KU)=\mathbb{Z}[u^{\pm ...
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How many subalgebras are there in $sl_3$?

The Lie algebra $sl_3$ is 8 dimensional and $B=\{h_1, h_2, e_1, e_2, [e_1, e_2], f_1, f_2, [f_1, f_2]\}$ is a basis of $sl_3$. For every $x \in B$, $\text{Span}\{x\}$ is a one-dimensional subalgebra ...
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dimension of projective modules over KG-algebra

In Aleprin's book "Local Representation Theory" on page 33 the Corollary 7 says: " If a Sylow p-subgroup P of the group G has order p^a then every projective KG-module has dimension divisible by p^a". ...
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46 views

Is there a change of basis that transforms any finite subgroup of $GL_n(C)$ into a subgroup of $GL_n(\bar{Q})$?

I vaguely know that there is a related statement that is true... something like, if G is finite, then every representation of it can defined over some finite algebraic extension F of Q. (By defined ...
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Question on function on $GL(3,\mathbb R)$ invariant under the action of minimal parabolic subgroup

Let $G=GL(2,\mathbb R)$ and $\Phi\in C_c^\infty(\mathbb R\times\mathbb R)$. We can define $f:GL(2,\mathbb R)\rightarrow \mathbb C$ by $$ f(g):=\int_{\mathbb R^\times}\Phi((0,t) g)d^\times t. $$ ...
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How do I find the Irreducible Representations from this Character Table?

I have been wondering how you can get the irredcible representations of the Dihedral group $\mathbb{D}_{8}$ of order 8 from its Character table. $\mathbb{D}_{8}= \left \langle a,x : a^4=x^2=e, \, ...
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Lie group actions

I am looking for a nice reference to study the action of a Lie group $G$ on a smooth manifold $M$, $\psi : G\to\mathrm{Diff}(M)$: Linearization: in a neighborhood of a fixed point, what we can ...
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Representations of $SU_q(n)$

I am searching for a classification of all irreducibel representations of the quantum group $SU_q(n)$ for general $n$. Can someone give referenced or some statements about this? Moreover does one has ...
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40 views

Character of Exterior Product of Standard Representation

Letting $V=U\oplus W$, where $U$ is the trivial representation of $S_n$ and $W$ is the standard representation of $S_n$, I want to find the inner product $\langle \chi_{\wedge^k V}, \chi_{{\wedge^k ...
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Unitary dual of $\{0\}$ and $\mathbb R$

How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively. Thank you ...
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Lowest dimensional faithful representation of a finite group

How does one compute the lowest dimensional faithful representation of a finite group? This question originated in the context of given a finite group G: trying to find the lowest dimensional shape ...
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30 views

R-Module Structure

In Chapter 10 of Dummit & Foote (3rd Edition), the authors make remarks about the "R-module structure" of things. For instance, when talking about an R-Algebra 'A', he says: "If A is an R-algebra ...
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Intuition/Motivation behind Algebras (R-Algebras, Q-Algebras, etc.)

I'm currently reading Ch. 10 in Dummit & Foote (3rd Edition) and towards the end of the first section, it defines an R-algebra. Dummit & Foote do a decent job of motivating certain ...
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Change of basis for irreducible representations of symmetric group

It is known that the group algebra of the symmetric group decomposes into a direct sum of its irreducible representations \begin{equation}K[S_n] = \bigoplus_{\lambda\vdash n} P_\lambda^{\oplus ...
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Exterior power of representation and invariance

Let $G$ be a group, $(\rho,V)$ a finite dimensional real representation of $G$, and $W$ a subspace of $V$ of dimension $k$. Assume that $\Lambda^k W$ is a $G$-invariant (and one dimensional) subspace ...
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One-dimensional L-submodules

If $L$ is a Lie algebra over $\mathbb C$. Consider the representation $\pi: L\rightarrow gl(V)$ where $dim\ V=1$. Can this representation be irreducible?
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The category of Lie algebra representations

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a homomorphism of Lie algebras $\mathfrak{g} \to \mathfrak{gl}(V)$. We define morphisms between representations as ...
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63 views

Why are these definitions of groups of central type equivalent?

Let $G$ be a finite group. In the celebrated paper of Howlett and Isaacs, On Groups of Central Type, Math. Z. (1983)., the group $G$ is called to be of central type if $G$ has an irreducible ...
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prove that there is a group representation of $\Bbb Z$ which is not totally decomposable over $\Bbb C$

prove that there is a group representation of $\Bbb Z$ which is not totally decomposable over $\Bbb C$ what I tried - let $\mu: \Bbb Z \to GL_{2*2} \Bbb (C)$ $$\mu(x)(v) = \begin{pmatrix} ...