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
0answers
19 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
2
votes
0answers
16 views

Irreducible components of tensor product representations.

Let $(\rho,V)$ be an irreducible representation of a finite group $G$, and let $W$ be a vector space. Then clearly $(\rho\otimes\text{Id}_{W},V\otimes W)$ is also a representation of $G$. I would like ...
0
votes
1answer
12 views

Lowering a non-zero weight vector gives a non-zero vector (representation of $\mathfrak{sl}(2)$)

In Lie algebras we study $\mathfrak{sl}(2)$ (the complex span of the usual matrices $X,Y,H$ where $X$ and $Y$ are the raising and lowering operators respectively). The defining commutator relations ...
1
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1answer
15 views

Simplify $\langle \operatorname{Ind}^G_1 1, \operatorname{Ind}^G_H\phi\rangle_G$

Let $G$ be a finite group and $H$ a subgroup. Let $\phi$ be an irreducible character of $H$ and $\mathbb 1$ the trivial character of the trivial subgroup $1$. Let $\langle,\rangle_G$ be the usual ...
0
votes
1answer
23 views

Torus action and multigrading.

Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? ...
2
votes
0answers
29 views

Moduli Spaces in Representation Theory of finite Groups

Recently I did work on Representation Theory of Finite Groups, in particular $p$-groups and recently I had a problem with something and I was wondering if I can put some geometry on that. So I thought ...
2
votes
2answers
29 views

Find all the homomorphisms from $D_8 \to \mathbb{C}^\times$

Find all of the homomorphisms from $D_8$ to $\mathbb{C}^\times$. So far I have: $\phi : D_8 \rightarrow \mathbb{C}^\times$ $\phi(a)^4 = 1$ so $\phi(a) = \pm 1, \pm i$ $\phi(b)^2$ = 1 so ...
-1
votes
1answer
32 views

Character of an $\mathbb{R}G$-module constructed from a $\mathbb{CG}$-module

I have been reading Representations and Characters of Groups by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module. ...
1
vote
1answer
17 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
1
vote
1answer
17 views

Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
2
votes
1answer
52 views

Center of a semisimple group and irreducible representations

Suppose that I am over an algebraically closed field of char $0$, and $G$ is a simply connected semisimple group. For a dominant weight $\lambda$, there is an irreducible representation ...
1
vote
1answer
25 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation ...
0
votes
1answer
33 views

Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ ...
0
votes
1answer
26 views

Constructing representation of $G$

Say we are given an arbitrary group $G$ and an arbitrary vector space $V$ over some field. How can we construct a representation of $G$ on some vector space from this data? Initially I wanted to ...
0
votes
2answers
20 views

Decompose the representation $V$ of $SO_2$ into irreducible representations

Let $V=\mathbb{C^2}$ be the standard representation of $SO_2$ Decompose $V$ into irreducible representations The standard unit vectors of $\mathbb{C^2}$ are $e_1$ and $e_2$ I am not sure how ...
0
votes
0answers
14 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
1
vote
1answer
33 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
3
votes
1answer
51 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 ...
0
votes
1answer
26 views

Sextonion Cayley Table

I've been reading up on the sextonions and was wondering if it would be possible to construct a Cayley table for the split sextonions the same way as one would do so for the split quaternions and ...
-1
votes
0answers
26 views

Character of subgroup of index 2

Let $\chi$ be an irreducible character of a finite group $G$; if $H\leq G$ is a subgroup of index 2, is $Res_H^G\chi$ irreducible? How do conjugacy classes change from $G$ to $H$?
2
votes
1answer
40 views

Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
2
votes
1answer
22 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 ...
2
votes
1answer
25 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 ...
2
votes
0answers
36 views
+50

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 ...
0
votes
1answer
59 views

When is this cyclic representation irreducible?

Let $G$ be a finite group, and let $(\rho, W)$ be a representation of $G$ on $W$. We assume that $W = \bigoplus_i W_i$ is a direct sum of equivalent irreducible representations $W_i$. There are many ...
0
votes
1answer
16 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
1answer
25 views

characters in semi-direct product.

The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 ...
0
votes
1answer
66 views

Every irreducible representation is either even or odd. [closed]

Let $V$ be any $n$-dimensional complex vector space and $SL(2,\Bbb{Z})$ is special linear group. Let $\rho:SL(2,\Bbb{Z}) \rightarrow GL(V)$ be a representation. It is even if $\rho(-I)=\Bbb{id}_V$ and ...
1
vote
0answers
15 views

Multiple reps if $g$ not conjugate to $g^{-1}$

If $g \in G$ is not conjugated to $g^{-1}$, how do I prove that $G$ has irreducible non-equivalent representations of the same order? I think the multiple representations are going to be in some way ...
4
votes
2answers
131 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
20 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
250 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 ...
0
votes
0answers
23 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
22 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
29 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 ...
3
votes
1answer
59 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 = ...
1
vote
0answers
28 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 ...
0
votes
1answer
22 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
18 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
20 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 ...
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votes
0answers
22 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
45 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
61 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
12 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
52 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
18 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} ...