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

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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 ...
1
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1answer
71 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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43 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 ...
1
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1answer
46 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 ...
2
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44 views

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 ...
0
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1answer
54 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 $B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. ...
2
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1answer
55 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 ...
2
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1answer
29 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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46 views

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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2answers
57 views

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 ...
2
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2answers
68 views

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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1answer
64 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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29 views

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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1answer
43 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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1answer
52 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. ...
3
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1answer
80 views

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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0answers
21 views

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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33 views

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 $G(f)$ ...
2
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1answer
54 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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1answer
36 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} ...
4
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2answers
51 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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1answer
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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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33 views

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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0answers
41 views

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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1answer
42 views

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 ...
0
votes
1answer
24 views

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 ...
2
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1answer
44 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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2answers
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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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41 views

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
52 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: ...
2
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1answer
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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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2answers
54 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 ...
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1answer
44 views

Spanning $\mathbb{I}$ in $V$

I have the following definition, Definition If $\rho$ : $G \rightarrow GL(V)$ is a representation we call $v \in V$ $G$-invariant if $$g \cdot v =v \ \ \forall g \in G $$ Then I have the statement ...
0
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1answer
30 views

Using $\displaystyle \mathbb{C}[G]\cong \bigoplus_{irreducible \ \rho}\rho^{\dim \rho}$ for $S_3$

Let $G=S_3$. $$\chi_{\mathbb{C}[G]}=(6,0,0)=1 (1,1,1)+1(1,1,-1)+2(2,-1,0)=1\chi_{\mathbb{I}}+1 \chi_{\xi}+2\chi_{\triangle} $$ since $\displaystyle \mathbb{C}[G]\cong ...
3
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0answers
35 views

Irreducibility of a certain polynomial associated to an irreducible representation of a finite group

Let $k$ be an agebraically closed field of characteristic 0. Let $G$ be a finite group of order $n$. A representation of $G$ is a homomorphism $\psi: G \rightarrow GL(V)$ where $GL(V)$ is the general ...
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8 views

Question on Discrete series representations of semisiple Lie groups

I am reading Knapp's book, representation theory of semisiple Lie groups. I am confused with the statements in the following. In page 310, Theorem 9.20: Let $\lambda \in (i\mathfrak b)'$ is ...
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57 views

Question about Poincare duality and homology of a cylinder.

I am reading the paper. I have some questions about Poincare duality and homology of a cylinder. On page 9, example 2.6. Let $X = \mathbb{R} \times S^1$ be a cylinder and $Y = X/(0 \times S^1 )$, ...
3
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1answer
104 views

Centralizer of $\mathbb{C}[G]$ in $\mathbb{C}[H]$

I found this result, but can't understand how to prove. Let $H$ be a subgroup of $G$. Then prove $Z(\mathbb{C}[G],\mathbb{C}[H])$ is commutative iff every irreducible $G$ module when restricted to ...
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restriction of irreducible representation to an ideal is irreducible

Let $A$ be a C*-algebra and $I$ a closed left ideal of $A$. Show that if $\{\pi,H\}$ is an irreducible representation of $A$, then the restriction of $\pi$ to $I$ is either zero representation or ...
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11 views

Prime ideals in Iwahori-Hecke algebras

Results on the ideals (especially the prime, completely prime ones) of Iwahori-Hecke algebras (espcially the ones with finite order) is needed. Thank you very much.
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1answer
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Dual of a matrix lie algebra

In fact I already calculate the dual space with a formula, but I did'd understand some steps of the formula. So, I want to calculate the dual space of The lie algebra of $SL(2,R)$. Knowing that ...
2
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1answer
50 views

Can two different characters of $S_n$ have the same _multiset_ of values?

As I was going through various representation-theory posts in the site, I stumbled upon this one: Characters of the symmetric group corresponding to partitions into two parts. Now, that question ...
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2answers
48 views

Semisimple implies complete reducibility

Why does a semisimple Lie algebra imply complete reducibility? I have that a semisimple Lie algebra is a Lie algebra with no non-zero solvable ideals. Complete reducibility means that every invariant ...
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26 views

References request: are there some references about simple modules of group algebras?

Are there some references about constructing the simples, determining the dimensions of simple modules and describing decompositions of tensor products of simple modules of group algebras? Thank you ...
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1answer
26 views

How to show that the ordinary quiver of a semisimple algebra is a quiver consisting of isolated points?

It is said that the ordinary quiver of a semisimple algebra is a quiver consisting of isolated points? How to prove this result? Thank you very much. Edit: the ordinary quiver is the quiver defined ...
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28 views

Irreducible representation of $G=\mathbb{R}$

How can one prove that the irreducible representation of $G=\mathbb{R}$ is $e^{kx}$? ($k\in\mathbb{C}$) Thank you.
2
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1answer
72 views

Bourbaki's proof of normal basis theorem Part 2

Let $K/k$ be a finite Galois extension of a field $k$, $G$ its Galois group. The normal basis theorem states as follows. There exists an element $\alpha$ of $K$ such that $\{\sigma(\alpha)\ |\ ...
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Projective representation of braid group

The representation theory of braid group $B_n$ is tough, not to mention the projective representation. But my problem is simpler: how to find out all the one-dimensional projective representations of ...
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2answers
53 views

Schurs Lemma for endomorphisms

Schur's lemma states that for an irrep $(\varphi,V)$ any endomorphism $\phi: V \mapsto V$ is given by a scalar mapping. Lets say we are in the complex case, then this would mean: $\phi = \lambda I$ ...
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1answer
36 views

Irreducible Representation of a $p$-group over field of characteristic $p$ is trivial (Dummit and Foote 18.1 #22)

I'm working on question 22 of Dummit and Foote 18.1. I know this question has been answered in other posts, but I'm confused about the method this text recommends using: Let $p$ be a prime, let $P$ ...