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

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4
votes
2answers
128 views

Proof of Clifford's theorem for modules

http://en.wikipedia.org/wiki/Clifford_theory#Proof_of_Clifford.27s_theorem I've a very easy question that I just can't seem to find the answer to. I'm self-studying so I can't ask anyone else. ...
1
vote
1answer
15 views

Representation/Character theory of $S_3$: What is the Vector space $V$?

This is a basic question that I may have a misunderstanding on. When we study the character table of a group, say $S_3$, what vector space are we looking at? I understand that a linear ...
0
votes
1answer
15 views

Is there a direct sum decomposition of the tensor product of two representations of two group elements?

I know that I can decompose $\rho_a(g) \otimes \rho_b(g)$ into $U^\dagger \left[ \rho_c(g) \oplus \rho_d(g) \right] U$. Is there a similar way to decompose $\rho_a(g_1) \otimes \rho_b(g_2)$ into ...
2
votes
1answer
60 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
10
votes
1answer
247 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
1
vote
0answers
18 views

Composition series of a regular module.

Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$ e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0. $$ I am trying to find the composition series for ...
-1
votes
0answers
15 views

Simple modules of a finite dimensional k-algebra

Assume that $A$ is a finite-dimensional k-algebra, generated by some elements {${a_1, a_2, ... , a_n}$} . Is it true that the A-modules generated by $<a_i>$ are all simple A-modules, and ...
0
votes
0answers
26 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
45
votes
6answers
10k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
1
vote
0answers
15 views

Matrix representations of the generators of the full octahedral group

I want to find matrix representations of the generators of the full octahedral group which has the presentation $\{a,b,c|a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\} $ where a,b and c are the generators of the ...
3
votes
1answer
55 views

Prove that the sum of all simple roots is a root

Let $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = ...
2
votes
0answers
17 views

Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
0
votes
0answers
14 views

Irreducible representation restricted to index 2 subgroup

Suppose $G$ is a (not nec. finite) group with index 2 subgroup $H$ and $k$ is a field (possibly of positive characteristic). Suppose $$\rho:G\to\mathrm{GL}_2(k)$$ is an irreducible 2-dimensional ...
0
votes
0answers
22 views

About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if ...
1
vote
0answers
13 views

Every irreducible representation of $G_2$ appears in some tensor power of the standard representation

In the Book "Representation Theory" by Fulton and Harris, this fact ist stated on page 353 after looking at the weight diagrams of the complex Lie-Algebra $G_2$. The authors deduce that with ...
0
votes
1answer
21 views

Dual of a faithful representation

A representation $\sigma$ of a finite group G is said to be faithful if Ker$\sigma={1}$. Then is it true that dual of a faithful representation is also faithful?
0
votes
1answer
13 views

Modules generated by primitive idempotent elements

Assume that A is a finite dimensional k-algebra, and $e \in A$ is a primitive idempotent element. Is it true that the submodule of $A$ namely $<e>$ is simple $A$-module? If it is, how do we ...
2
votes
2answers
19 views

Finite dimensional representations of the Weyl algebra in characteristic $p>0$

I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra ...
0
votes
0answers
21 views

references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
0
votes
1answer
57 views

Every irreducible representation is either even or odd. [on hold]

Let $\rho:SL(2,\Bbb{Z}) \rightarrow GL(V)$ be a representation. It is even if $\rho(-I)=\Bbb{id}_V$ and odd if $\rho(-I)=\Bbb{-id}_V$. Then show that any irreducible representation is either even or ...
0
votes
1answer
44 views

A question about the representation theory of finite dimensional algebra

Let $A$ be a finite dimensional algebra, $M$ be a finite dimensional module of $A$.The socle of $M$, $\mathrm{soc}(M)$, is the maximal semisimple submodule of M. The top of $M$ is ...
5
votes
1answer
61 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
4
votes
1answer
56 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
1
vote
1answer
11 views

Composition Series of the regular A-module

Assume A is a finite-dimensional algebra over field K. How can we prove that any simple A-module occurs, as a composition factor (up to isomorphism) of an arbitrary composition series of A, as module ...
4
votes
1answer
51 views

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 ...
0
votes
0answers
17 views

How to represent the function of variables?

I have a function as $$E=\int_\Omega -\log\big( p_i(x)\big) dx$$ where $p_i(x)$ is density distribution which estimated by Parzen window method. $p_i(x)=\frac{1}{\Omega_i} ...
4
votes
0answers
39 views

Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
0
votes
0answers
25 views

Condition for appearance of singlet in product of two irreps.

By inspecting tables for tensor products of two finite-dimensional irreps of common Lie groups, I've noticed that a trivial subrepresentation only appears when the two irreps are conjugate of ...
1
vote
0answers
31 views

Linear represenation of a group(can be infinite also)

Let G be a group and let $\sigma :G \rightarrow GL(V) $ be a representation of G. Assume $\sigma$ is reducible. That is $\sigma=\sigma_1 \oplus \sigma_2\oplus .... \oplus \sigma_k $ or interms of G ...
1
vote
1answer
38 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
1
vote
0answers
14 views

Primitive Decomposition in a finite-dimensional algebra

Can you give me please, an explicit example of a primitve decomposition in a finite-dimensional algebra different than the usuals (for instance the decomposition in Mn(K))? Also, if e is an ...
3
votes
1answer
82 views

Compute the Jacobson radical of the group ring $\mathbb{F}_2S_3$.

Compute the Jacobson radical and the maximal semisimple quotient of the group ring $\mathbb{F}_2S_3$ of the symmetric group on three letters over the field with two elements, and compute the ...
0
votes
1answer
31 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
2
votes
0answers
18 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
0
votes
0answers
14 views

Bounds for the sum of the representations of the group $S_k \times S_j$

Let $S(n)$ be the sum of the degrees of the irreducible rational representations of the symmetric group on $n$ letters. I know that this number is the same as the number of involutions in $S_n$. ...
6
votes
1answer
71 views

Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
2
votes
1answer
53 views

Questions about Haar integral for the group $GL_2(\mathbb{R})$.

I have some questions about Haar integral for the group $GL_2(\mathbb{R})$. How to show that a Haar integral for the group $GL_2 (\mathbb{R})$ is given by \begin{align} I(f ) & = \int_{\mathbb{R}} ...
0
votes
0answers
22 views

Is $V \otimes V$ a $g \otimes g$-module?

Let $g$ be a Lie algebra and $V$ a $g$ module. Then $V \otimes V$ is a $g$ module under the action $X.(x \otimes y) = X.x \otimes y+x \otimes X.y$, $x, y \in V$, $X \in \mathfrak{g}$. Is $V \otimes V$ ...
4
votes
0answers
30 views

Structure of k[G]/J(k[G]) when char k divides |G|

I'm self-learning representation theory, so I'd like if possible someone kind to help me with the following. Given a field $k$ whose characteristic does not divide the order of a finite group $G$, we ...
2
votes
1answer
33 views

Showing that a very well-known representation is really a representation

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
1
vote
1answer
35 views

Equivalent conditions, reductive groups

I read the book Invariant Theory by T.A. Springer. There is the following definition of a reductive group: Definition. A linear algebraic group $G$ is called reductive if for any rational ...
0
votes
0answers
20 views

how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
0
votes
1answer
35 views

On the proof that one dimensional linear algebraic groups are either isomorphic to $\mathbb{G}_m$ or $\mathbb{G}_a$.

Let $G$ be a linear algebraic group of dimension one. The proof that I am looking at, in t.a springer's book (thm 3.4.9) proceeds by showing that $G$ must be either equal to its semisimple part ...
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0answers
17 views

Showing F-matrix representation is irreducible over $\mathbb{R}$

I have $G$ the cyclic group of order 4 and its $F$-(matrix) representation $T$ is $$\hat{T}(g) = \bigg[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \bigg]. $$ I am trying to show that ...
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vote
0answers
19 views

Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
1
vote
1answer
29 views

Product of Characters

Let $\rho:G\rightarrow\text{GL}_n(\mathbb{C})$ be a representation of a finite group and let $\chi_\rho$ be the corresponding character. If $\chi(e)>1$, then I want to show that ...
1
vote
0answers
26 views

All the Idempotent elements of a finite-dimensional algebra

Does there exist any way to determine whether or not, we have found all the idempotent elements of a finite-dimensional algebra A? In other words, if A is a finite-dimensional algebra with ...
0
votes
0answers
11 views

Finding the character table for Z_8

I am a bit confused about how to come up with the number of irreducible representations, as well to come up with the number of different conjugate classes. Starting me out would be highly appreciated ...
0
votes
0answers
15 views

Incidence algebra

Let $(I;\preceq)$ a finite poset, where $I=\{a_1,\ldots,a_n\}$ and $\preceq$ is a partial order on $I$. We define de incidence algebra $KI$ of the poset $(I;\preceq)$ with coefficients in $K$, where ...
0
votes
0answers
16 views

Positive definite functions coming from finite dimensional representations

Let $G$ be a topological group, let $\mathcal{H}$ be a complex Hilbert space, let $v\in\mathcal{H}$ be a nonzero vector, and let $\rho:G\rightarrow \mathcal{U}(\mathcal{H})$ be a unitary ...