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

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Weights in $\mathfrak{sl}(3,\mathbb{C})$

Let $\mathfrak{h} \subset \mathfrak{sl}(3,\mathbb{C})$ be the set of diagonal matricies. Then for $A = \begin{pmatrix} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & a_3 \end{pmatrix} ...
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18 views

A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...
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question on lemma in Bushnell and Henniart, irreducible components of a particular induced representation

I have a question on a lemma that appears in the book "The Local Langlands Conjecture for GL(2)" by Bushnell and Henniart. The setting is as follows: we let $G = GL_2(k)$ where $k$ denotes a finite ...
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About lower bounds on the size of irreducible representations of subgroups of symmetric groups.

Is there a subgroup $G_n$ of $S_n$ (one $G_n$ for each $S_n$) increasing in size such that their permutation representations are such that the smallest non-trivial irreducible size in them is ...
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21 views

Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
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Can the adjoint representation on a discrete centered group merge connected components?

Let $\mathfrak{G}$ be a Lie group with algebra $\mathfrak{g}$ and with in general many connected components but which is either centerless or has at most a discrete center. Then we have an ...
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40 views

Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
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21 views

Representation Theory group element as a vector

This is a very introductory level question. I am reading "The Symmetric Group" by Bruce Sagan and I am stuck on a concept. In Example 1.3.4 he talks about the regular representation of the cyclic ...
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15 views

Killing form of a reductive symmetric Lie algebra

suppose $(g; , k ,p)$ is a reductive symmetric Lie algebra. i.e. $k$ is a sub-algebra of $g$, $[k,p] \subset p$ , $[p,p] \subset k$ and $g= k \oplus p$. this is actually from Lepowsky and McCllum's ...
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Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
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How to compute perverse sheaves?

In the video, from 49:00 to the end of the video, there is an example of computing $IC(S, L)$ and equivariant local systems. I don't understand some parts of the computations. Let $X$ be the variety ...
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Why Demazure operator is an endomorphism of $\mathbb{Z}[P]$?

Let $P$ be the weight lattice of some Lie algebra. Let $$ \Delta_{\alpha}(u) = \frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}, $$ where $\alpha$ is a root, $u \in P$. In the article, it is said that ...
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Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
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30 views

Corollary to Maschke's Theorem.

If in Maschke's Theorem, for group ring KG where K is any field s.t. char $K \nmid |G|$, I take G to be finite , then I know Maschke implies that KG will be semisimple so it is isomorphic to a direct ...
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Expression of character of coset representation

Let $G$ be a finite group with subgroup $H$. Consider the set of all cosets $G/H$ and a corresponding transversal $\{x_1,...,x_m\}$. We now have a representation $\rho$ of $G$ on $\mathbb{C}[G/H]$: ...
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Is trace of regular representation in Lie group a delta function? [duplicate]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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58 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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1answer
55 views

direct sum and tensor product of representation of lie algebra

Let $(p_1,V_1)$ , $(p_2,V_2)$ representation of a lie algebra $g$ on $V_1,V_2$. I have to prove that: $ i) $ the direct sum $p_1 \oplus p_2$ is a representation of $g$ in $V_1 \oplus V_2$ $ ii) $ ...
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32 views

Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$

Let $G=S_3$ and $H=\langle (12) \rangle < G$. Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$. I have computed the character using the induction ...
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30 views

Book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups?

Does anybody have book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups? Preferably, I am looking for a book that is ...
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How to complete this character table of GL(3,2)?

As is well-known, the second smallest non-abelian simple group is $G = \operatorname{GL}(3,2) \cong \operatorname{PSL(2,7)},$ which has order $168.$ (So elements of $G$ are $3 \times 3$ matrices with ...
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49 views

Using generators to write representations

Let $G=D_{12}=\{a,b\mid a^6=b^2=1, bab=a^{-1}\}$. Also let $A=\begin{pmatrix} e^{i\pi/3} & 0 \\ 0 & e^{-i\pi/3} \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. ...
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45 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...
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24 views

Prove that the inner product is the number of orbits of $G$ on $X \times Y$.

Let $X,Y$ be $G$-sets and $\mathbb{C}[X], \mathbb{C}[Y]$ the corresponding permutation representations. Prove that the inner product is the number of orbits of $G$ on $X \times Y$. Ive tried: ...
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hard lemma from a paper

I was reading a paper, there its mentioned as lemma that : Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $\phi_1 : G\to GL(V)$ is a irreducible representation of $G$ and $\phi_2 : H \to ...
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31 views

Auslander-Reiten Quiver [closed]

Can someone show me how to find A-R Quiver( example or class of examples). I know that we must find the indecomposable projective and the A-R sequences. Is there any methode for that?? For example ...
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Show that $\rho$ must be 2-dimensional

Let $G=D_8=\langle g,h |g^4=h^2=1, hgh=g^{-1} \rangle$. One can show that $G$ has $4$ $1$-dimensional representations. From first principles (no character theory). Suppose $\rho$ is an ...
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Jucys-murphy elements commute with each other.

In the group algebra $\mathbb{C}[S_n]$, for $1<i<j\le n$, $X_i=(1\ i)+(2\ i)+...+(i-1\ i)$ and $X_j=(1\ j)+(2\ j)+...+(j-1\ j)$ commute with each other. I have been trying to do it ...
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54 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
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How to understand the algebra $U_A(Lg)$?

Let $g$ be a complex simple Lie algebra and $Lg = g \otimes \mathbb{C}[t, t^{-1}]$. Let $q$ be a non-zero complex number and $U_q(Lg)$ the quantum loop algebra corresponding to $g$. Let $A = ...
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39 views

Showing representation of centralizer is irreducible.

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in $\mathbb{C}[G]$. How do I ...
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43 views

Finding a group that is not monomial

Definition. A group is called monomial if every representation of $G$ is induced from 1-dimensional representations of some subgroup of $G$. Question Give an example of a group that is not monomial. ...
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Proof of the Frobenius Schur indicator

I am trying to prove the Frobenius-Schur indicator for $\chi$ irreducible character. \begin{equation} i_{\chi} = \begin{cases} 0, & \text{if $\chi$ is not real valued} \\ \pm1, & ...
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Convolution product in Borel-Moore homology

I have a question about Exampla 2.7.10 from the book "Representation theory and complex geometry" by N. Chriss and V. Ginzburg. It concerns the convolution product. In the example we have $M_1 = M_2 ...
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Linear groups and isomorphisms

If two linear groups(subgroups of $\text {GL}(n,k)$ over some field $k$) $G(t)$ and $H(t)$ over $F(t)$, a transcendental extension of a field $F$, are isomorphic, then for each $f\in F$, are $G(f)$ ...
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1answer
50 views

Representations Isomorphic up to a Character

Suppose we have a finite group $G$ and with a normal subgroup $H$ such that the quotient is cyclic. Is it the case that two representations $\phi_1, \phi_2$ of $G$ are isomorphic when restricted to ...
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31 views

Finite representations of the Euclidean Group

What are the finite dimensional indecomposable representations of the special Euclidean group in three-dimensions, SE(3)? To clarify, I'm asking about the group $$SE(3) = \left\{ \begin{pmatrix} ...
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48 views

Product of class sums

Let $C_i$ be the conjugacy classes of a finite group $G$. Consider the class sums $z_i=\sum_{g\in C_i} g$. It is well known that ${z_i}$ form a basis of the center of the group algebra $\mathbb{C}G$. ...
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$GL_3(\mathbb{F}_2)$ is simple

Character table of $GL_3(\mathbb{F}_2)$. \begin{array}{|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 7A & 7B \\ \hline \chi_1 & 1 & 1 & 1 & 1 & 1 & 1\\ ...
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Cartan's Criterion for Solvability

I'm trying to understand the proof of Cartan's Criterion for Solvability given here, and have two questions: On page 15, about half way down, we assert the following: If $\mathfrak{g}=\mathfrak{g}_0 ...
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Middle bit of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Part of the proof needed to prove the above theorem is where you prove that: $\displaystyle \frac{\chi(C)}{\chi(e)}$ ...
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Number of orthonormal sets of vectors $\leq$ dim of the vector space

Theorem. If $V,W$ are arbitrary representations of $G$, say $$V=V_1^{a_1}\oplus \dots \oplus V_k^{a_k} $$ $$W=V_1^{b_1}\oplus \dots \oplus V_k^{b_k} $$ Then, $$\langle \chi_V,\chi_W \rangle ...
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Map to the submodule of invariants of a Lie algebra representation

If $G$ is a compact group and $V$ is a representation, the inclusion $V^G \to V$ has an easy-to-write-down retract: \begin{equation*} V \to V^G,\:\: v \mapsto \frac{1}{|G|} \int_G g\cdot v\;dg ...
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38 views

Auslander-Reiten theory

Suppose that every submodule of a projective $A-$module $M$ is a projective module. I want to prove that the functor $DTr$ is isomorphic to the functor $DExt^{1}$$_{A}(-,A)$. My solution(not ...
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Representations of a quiver and sheaves on P^1

We know from Beilinson that there's an equivalence of derived categories $D^b Rep(Q) \simeq D^b Coh(\mathbb{P}^1)$ where the lefthandside is the derived category of bounded complexes of ...
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Understanding restriction in $S_3$

Let $G=S_3$. \begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & 1 & -1 \\ \hline \chi_2 & 2 & -1 & 0 \\ ...
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1answer
48 views

Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V} $

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V} $$ is an algebraic integer. In the start of this proof we have: ...
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41 views

Isomorphism form $\mathbb{C}[G]$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.

What I want to ask is the proof of the Proposition 10. in "Linear Representations of Finite Groups" by Jean-Pierre Serre. Let $\rho_i : G \rightarrow GL(W_i)$ be the distinct irreducible ...
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48 views

2nd half of proof of $\dim V^G=\frac{1}{|G|}\sum_{g \in G}\chi(g) $

Lemma. Let $\rho: G \rightarrow GL(V)$ be a representation, character $\chi$. Then $$\dim V^G=\frac{1}{|G|}\sum_{g \in G}\chi(g) $$ Proof. RHS: $$\frac{1}{|G|}\sum_{g \in G}tr ...