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

learn more… | top users | synonyms (1)

1
vote
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 ...
0
votes
1answer
49 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
votes
1answer
75 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, & ...
1
vote
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 ...
0
votes
0answers
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
votes
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 ...
0
votes
1answer
35 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
votes
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 $\mathbb{C}G$. ...
-1
votes
1answer
67 views

$GL_3(\mathbb{F}_2)$ 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\\ ...
0
votes
0answers
32 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 ...
0
votes
0answers
39 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: $\displaystyle \frac{\chi(C)}{\chi(e)}$ ...
1
vote
1answer
38 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
votes
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 ...
4
votes
2answers
64 views

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 ...
1
vote
0answers
39 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 \\ ...
1
vote
1answer
49 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
votes
1answer
42 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 ...
1
vote
2answers
51 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 ...
1
vote
1answer
41 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
votes
1answer
28 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
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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
votes
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 ...
0
votes
0answers
16 views

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 ...
0
votes
0answers
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.
0
votes
1answer
33 views

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
votes
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 ...
0
votes
2answers
47 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 ...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
0answers
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
votes
1answer
70 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)\ |\ ...
2
votes
0answers
41 views

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 ...
0
votes
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: $\phi = \lambda I$ ...
1
vote
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 $P$ ...
0
votes
0answers
14 views

Which representation of $ su(N) = A_l $ is $\Gamma(1,0,0,\dots,0,0,1)$?

I was wondering which representation of the Lie algebra $ su(N) $ is $\Gamma(1,0,0,\dots,0,0,1)$? where $ (1,0,\dots,0,1)$ are the Dynkin labels of the representation. My guess is that $ \Gamma $ is ...
2
votes
1answer
77 views

How to compute the pointwise stabilizer subgroup of a fixed-point subspace?

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
0
votes
0answers
72 views

A question about a proof in Lang's $SL_2(\mathbb{R})$

The following is a lemma in Lang's book $SL_2(\mathbb{R})$. It's the last line of the proof that I don't understand. Let $G=SL_2(\mathbb{R})$ , $E$ a Banach space, and let $\pi$ be an irreducible ...
0
votes
0answers
29 views

Basic Representation Theorey: Bijective Correspondence Between Representations (Dummit and Foote 18.1 #3)

I am working on the following question from Dummit and Foote: Prove that the degree 1 representations of $G$ are in bijective correspondence with the degree 1 representations of $G/G'$ (where $G'$ is ...
1
vote
2answers
37 views

Basic Representation Theory: One Dimensional Representation (Dummit and Foote 18.1 #2)

I'm working on question 2 in 18.1 of Dummit and Foote. The question states: Let $\phi : G \to GL_n(F)$ be a matrix representation. Prove that the map $g \to det(\phi(g))$ is a degree 1 ...
7
votes
2answers
102 views

Prove that $Q_8 \not < \text{GL}_2(\mathbb{R})$

Problem 18.1.10 in Dummit and Foote's Abstract Algebra, third edition: Prove that $\text{GL}_2(\mathbb{R})$ has no subgroup isomorphic to $Q_8$. [EA: The quaternion group]. [This may be done by ...
0
votes
0answers
17 views

Checking the definition of absolutely irreducible representations

The definition of an irreducible representation $(\rho, V) $ is one with no subrepresentations. Am I correct in saying that a absolutely irreducible means "it is irreducible over the algebraic ...
1
vote
1answer
32 views

Show that $ (φ^G )_K = (φ_{H∩K})^K $ with Mackey's theorem

Suppose H,K ≤ G e θ $ ϵ $ Char(H). Show that Z(θ)≤H. Suppose H,K ≤ G and HK = G. Se $ φ $ ϵ Char(H) show that $ (φ^G )_K = (φ_{H∩K})^K $. For the proof I have to use the Mackey's theorem. How do I ...
2
votes
1answer
33 views

Differences between primitive central idempotents and primitive orthogonal idempotents

I asked this question in mathoverflow. But it was closed. So I ask it here. If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, ...
1
vote
0answers
37 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...
1
vote
1answer
39 views

Functor from category of group representations to space of $G$ invariants

For a representation $(V,\rho)$ of a group $G$, define the subspace of $G$-invariants by $$ V^G=\{v\in V: \rho(g)v=v\quad \forall g\in G\} $$ and want to prove the following: $V\mapsto V^G$ ...
1
vote
1answer
40 views

Prove that $\mathrm{Ind}_{\mathbb{I}}^G \cong \mathbb{C}[G]$

Prove that $\mathrm{Ind}_{\mathbb{I}}^G \cong \mathbb{C}[G]$. Apparently: $$\langle \mathrm{Ind}_{\mathbb{I}}^G \mathbb{I}, \chi \rangle_G \overset{Frob.Rep.}= \langle \mathbb{I}, ...
1
vote
2answers
35 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...