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

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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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23 views

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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24 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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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 ...
2
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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 ...
2
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45 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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23 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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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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38 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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1answer
21 views

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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50 views

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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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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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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43 views

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} ...
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28 views

There is a group representation of $\Bbb Z/p\Bbb Z$ that is not decomposable over the field $\Bbb F_p$, where $p$ is prime

let p be prime. prove there is a group representation of $\Bbb Z/p\Bbb Z$ that is not decomposable over a field $\Bbb F_p$ for similar and simpler questions i showed homomorphism $\mu$ such that ...
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Lie group action and Lie algebra action.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $r \in g \otimes g$ and $b \in g$. Consider the adjoint action $g \times g \otimes g \to g \otimes g $ given by $(b, x, y) \mapsto b.(x \otimes y) = ...
5
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Semisimplicity of the induced representation of an irreducible representation

Let $G$ be an arbitrary group, $H$ be a subgroup of finite index $n$ and $k$ be an algebraically closed field of characteristic prime to $n$. Suppose that we have an irreducible representation ...
3
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1answer
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What the expression of a one-dimensional representation of $H$

Let $G= \{ g=(x,y,t); \quad x,y,t \in \mathbb R\}$ be the Heisenberg group and $H= \{ g=(x,y,t) \in G; \quad x=0\}= \{ h=(0,y,t); \quad y,t \in \mathbb R\}$ be a subgroup of $G$. I want to know why ...
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What is $q^{\frac{1}{2}h \otimes h}$?

One can find in some sources (e.g. Pavel Etingof, Lectures on representation theory and KZ equation, page 91) formula for $R$ matrix (for $\mathfrak{sl}_2$ case) $$R = q^{\frac{1}{2} h \otimes h} ...
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1answer
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Schur Index Divisibility Question: Ind A^E divides Ind A

Background notation: If $A\in\mathcal{F}$ is a central simple algebra, $A\cong M_n(D)$, where $D$ is a division algebra. The Schur index of $A$ is defined as $Ind(A)=Deg(D)$. How do we prove $A^E$ ...
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If $P_I$ is a parabolic subgroup in $GL_n(k)$, is $P_I$ conjugate to ${P_I}^-$ under some element of $W$?

Suppose $P_I$ is a standard parabolic subgroup of $GL_n(k)$ (we can assume $k$ is finite, but I doubt it matters). Is $P_I$ conjugate to the opposite parabolic ${P_I}^-$ under some element of the Weyl ...
3
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68 views

How to explain this contradiction?

Let ${\mathbb{F}}_{p^{2n}}$ be a finite field of characteristic $p$, and $1\neq \tau \in Aut({\mathbb{F}}_{p^{2n}})$ such that ${\tau}^2=1$. For every representation $\psi: G \rightarrow GL(V)$ of ...
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30 views

Subspace generated by irreducible characters

It is mentioned in the answer for this question: Perhaps we should also note that the subspace generated by irreducible characters is the same as that generated by their conjugates. Can you ...
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1answer
27 views

What is the name of this function on group-ring modules?

Let $G$ be a finite group. Let $M$ be a $\Bbb{Z}G$-module. What is the name of the map $M\to M$ given by: $$ m \mapsto \sum_{g\in G} g m\, , $$ possibly divided by $|G|$? What is it used for? A ...
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Is an algebra $A$ wild if there exists a representation embedding $\underline{\mathrm{mod}\,B} \rightarrow \mathrm{mod}\,A$, where $B$ is wild?

Let $A,B$ be finite dimensional $K$-algebras. A $K$-linear functor $T:\mathrm{mod}\,B \rightarrow \mathrm{mod}\,A$ is called a representation embedding if $T$ is exact, $T$ maps indecomposable modules ...
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embeddings $SU(2) \to SU(N)$ and representations

How can we prove that group immersions $SU(2) \to SU(N)$ (up to conjugacy) are in 1-1 correspondence with (non-trivial) $N$-dimensional representations of $SU(2)$ (up to equivalence)? Feel free to ...
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A tensor identity - $\text{Hom}_{R}(A,B \otimes_S C) \cong \text{Hom}_{R}(A,B)\otimes _SC$

Let $R,S$ be associative algebras over $\mathbb{C}$. Let $A$, $B$ and $C$ be, a left $R$-module, a $(R,S)$-bimodule, a left $S$-module, respectively. Assume that $B\otimes_S C$ is finite-dimensional. ...
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Central Simple Algebra Tensor Question

Suppose $B$ and $C$ are $F$-algebras. Assume that $B$ is central simple and $C$ is simple. Prove that $B\otimes C$ is simple. I am stuck with how to use the fact that $B$ is central simple. I have ...
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1answer
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Different methods to compute a unitary representation

Given a nilpotent Lie group $G$ (for example the Heisenberg group), what is the most effective method to calculate their unitary representation: The orbit method due to Kirillov; or The induction ...
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Field Extension Embedding into Matrix Conditions

Let $L/K$ be a field extension. How do we prove $L$ is a subfield of $M_n(K)$ (n by n matrices with entries in $K$) if and only if $[L:K]\mid n$? My attempt: I can't prove the forward direction. For ...
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Show that $D^n$ is simple and unique [duplicate]

Suppose $A=M_n(D)$, i.e. n by n matrices over a division algebra $D$. We want to prove that: $D^n$ (viewed as row vectors) is a simple right $A$-module and any other simple $A$-module is ...
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Finite dimensional representation of $SL_{2}$

Let ($\pi , V$) be a finite dimensional representation of $SL_{2}$. Also, let $\alpha$ be highest weight vector. I want to show that for any $m>0$, then the following holds: ...
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Classify all $2 \times 2$ and $3 \times 3$ matrices in $sl_2$ and $sl_3$ respectively.

I would like to classify all $2 \times 2$ and $3 \times 3$ matrices in $sl_2$ and $sl_3$ up to conjugation respectively. We take the ground field to be $\mathbb{C}$. Let $g = sl_2$. A matrix in ...
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How to show that $( \Lambda^3 g)^{g} = \mathbb{C}Z$?

Let $g$ be a simple Lie algebra over $\mathbb{C}$. Let $\Omega$ be the Casimir element of $g \otimes g$ associated to a non-degenerated invariant form on $g$. How to show that $( \Lambda^3 g)^{g} = ...
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prove that $\det X(\pi) = \operatorname{sgn}(\pi)$ for all $\pi \in S_n$.

Let $X:S_n → GL_n(\mathbb{R})$ be the defining representation of $S_n$. Prove that $\det X(\pi) = \operatorname{sgn}(\pi)$ for all $\pi \in S_n$. attempt: I was thinking in trying to use $X(e) = ...
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Proof of Howe-Moore Property for SL(n,R)

On page 210 of Howe and Tan's Non-Abelian Harmonic Analysis there is the following proposition and proof: Let $(\rho, V) $ be a unitary representation of $SL(n,\mathbb{R})$. The following are ...
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1answer
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Express cofundamental weights using coroots.

In type $A_2$ root system, we have $\alpha_1 = 2 \omega_1 - \omega_2$, $\alpha_2 = - \omega_1 + 2 \omega_2$. How to express cofundamental weights $\omega_1^{\vee}, \omega_2^{\vee}$ using coroots ...