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)

0
votes
3answers
41 views

What is the space $\operatorname{Sym}^2(V)$ and how does it act on the vector space $V$?

If $V$ is a vector space over $\mathbb{C}$ with basis vectors $e_i$, what is the space $\operatorname{Sym}^2(V)$? I am hoping someone can give me some insight into this space; perhaps by describing ...
1
vote
1answer
30 views

Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
2
votes
0answers
37 views

Find an irreducible extension of an irreducible representation

Let $G$ be a finite group and $C$ the center of $G$. Let$μ:C→F^×$ be a character of $C$. Prove that there is an irreducible representation $ρ : G → GL(V )$ such that $ρ(c)(v) = μ(c)v$ for all $c ∈ C$ ...
2
votes
1answer
46 views

Representation of Sylow which does not extend

Let $H$ be a subgroup of a finite group $G$ and $\rho$ a representation of $G$ such that the restriction of $\rho$ to $H$ is invariant under conjugation in $G$, in the sense that its character is ...
5
votes
1answer
104 views

Polynomials with $S_n \times \mathbb{Z}_2$ symmetry

Suppose that a polynomial $p(x_1\ldots x_n, y_1\ldots y_n)$ in $2n$ variables is invariant under the following operations: 1) $p(x_1\ldots x_n, y_1\ldots y_n)=p(y_1\ldots y_n, x_1\ldots x_n)$ 2) $\...
1
vote
0answers
19 views

Semidirect Products and Representations

Assume that $G$ and $H$ are two $p$-groups and $k$ a field of $char(k)=p > 0$. Also, assume that i denote by $G \rtimes H$ the semidirect product of the above groups. Do you know if there is any ...
3
votes
1answer
59 views

How does the sum of the absolute values of the diagonal entries of a matrix change when the matrix is written in a random basis?

The set-up is as follows: I have a complex, Hermitian matrix $H$ with $\mbox{Tr }H=0$, and such that the trace norm $\|H\|_1=1$ (i.e. the sum of the singular values $=1$). Let me define the functiona ...
1
vote
1answer
20 views

Witness for $2-$dimensional irreducible representation of $Q_8$ over an algebraically closed field

Show that the $2-$dimensional irreducible representation of $Q_8$ can be realized in the space $V$ of functions $f : Q_8 → F$ such that $f(gi)= \sqrt{−1}f(g)$ (the action of $G$ is by right ...
2
votes
0answers
52 views

The action of an element in the center of a group

Assume the base field is Algebraically closed. Let $C(g)$ be the conjugacy class of an element $g$ in a finite group $G$ of order n and $χ$ be the character of an irreducible representation $ρ$. Prove ...
-1
votes
2answers
16 views

Matrices for action wrt basis

Consider the permutation representation where $G=S_3$ on $\mathbb{C^3}$ with the action: $\pi(g)e_i=e_{g(i)}$ $W=\{ \lambda_i e_i ; \sum \lambda_i=0 \}$ is an invariant subsoace of vector space $V$ ...
1
vote
1answer
32 views

Conjugacy classes in PGL(2,n)

I'm working on a project that requires me to write out some character tables, and I know part of my computation is wrong for $PGL(2,n)$, would appreciate some input. So, restricting down from ...
1
vote
0answers
21 views

Expression of the Laplacian of the reduced Heisenberg group?

Let $\mathbb C^n$ be the n-dimensional complex field endowed with a positive definite hermitian form $H(z,w)$. The corresponding symplectic form is $E(z,w)= \Im (H(z,w))$, where $\Im $ denotes the ...
0
votes
0answers
29 views

Is G isomorphic to a Subgroup of $GL(2,\mathbb C)$

I'm stuck on a question on a past exam paper that asks if a group $G$ is isomorphic to a subgroup $GL(2,\mathbb C)$. We are given the character table for $G$ which I've attached below. It's the last ...
2
votes
0answers
19 views

Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, $...
2
votes
0answers
31 views

Holomorph of Cyclic $p$-groups

Assume that $G$ is a cyclic $p$-group (or up to isomorphism more explicitly $\mathbb{Z}_{p^r}$). Do you know if there does exist any explicit way to describe the holomorph of the above group? Moreover,...
1
vote
1answer
12 views

Infinite-Representation Type Group Algebras

Assume that $G$ is a $p$-group, $N$ a normal subroup and $k$ a field (infinite or finite) of $char(k)=p \gneq 0$. What can we say about the representations of $G$ if we already known about the ...
1
vote
0answers
27 views

Representations of $SL_2(\mathbb{F}_3)$

I am trying to determine all the irreducible representations of the group $SL_2(\mathbb{F}_3)$.I have determined its character table and I have seen that there is a unique $2$-dimensional ...
1
vote
1answer
116 views

An example of a discontinuous “$\ell$-adic Galois representation”

Let $\mathbb{F}_p$ be a finite filed with $p$ elements, and $G=\mathop{\mathrm{Gal}(\mathbb{F}_p^s/\mathbb{F}_p)}$ be its absolute Galois group. $G$ is a pro-finite group, with the Krull topology, see ...
0
votes
0answers
14 views

Decomposition irreducible representations $\mathbb{{SL}_2(R)}$ and $\mathbb{{SL}_2(C)}$

Let $V$ be the standard representation of $\mathbb{{SL}_2(C)}$ I decomposed $V \otimes Sym^2(V)$ into irreducible representations through analysis of highest weights and dimensions to get: $V \...
0
votes
0answers
17 views

Modular Representations of unitriangular matrices

Assume that $U_3(\mathbb{F}_p)$ is the group of unitriangular matrices with entries from the field $\mathbb{F}_p$ of $p$ elements. Do you have any idea how can we compute the indecomposable (modular, ...
0
votes
0answers
24 views

Solvability and Representation of a finite group

assume that $K$ is an algebraically closed field of characteristic $p$ and $G$ a solvable finite group (for instance a $p$-group). Can you tell me please, if there does exist any correlation among the ...
0
votes
0answers
10 views

Only two central extensions of $A_5$

Let $\bar A_5$ be central extension of $A_5$ with a faithful representation $\rho$ in $C^2$. Show that in fact there are exactly two such representations. Describe the decomposition of the third ...
2
votes
1answer
44 views

Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
3
votes
1answer
69 views

How to study for ring theory?

I want to study the theory of rings because it is used when I study representation theory. Here, a ring is not necessarily commutative and doesn't necessarily has unity. I know that there are a few ...
1
vote
1answer
56 views

Fulton and Harris exercise 5.7

I would like to know if I'm on the right track: We're trying to derive the character table for $PGL(2,q=p^k)$ from $GL(2,q)$. The table for $GL$ is given in the text. It gives the hint that the ...
1
vote
1answer
61 views

Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
1
vote
1answer
26 views

Applying the divided difference operator

This question is about divided difference operators. How do I perform $\partial _2$ or $\partial_3$ on $x_1^2x_2$? $\partial_i$ is defined as $\frac{p-r_i.p}{x_i-x_{i+1}}$, where $r_i$ is the ...
1
vote
0answers
63 views

How to show a group is infinite group by constructing epimorphism?

Consider a group G with representation $$\langle a,b|abab^{-1}a^{-1}b^{-1}\rangle$$ Prove that this group is an infinite group There is similar question here (Finding the kernel of an epimorphism onto ...
0
votes
1answer
47 views

Ordering on the weight lattice

When given a finite dimensional complex Lie algebra $\mathfrak{g}$ that is also semisimple and a choice of Cartan subalgebra $\mathfrak{h}$ we may talk about its weight lattice $\Lambda_{W} $ in $\...
2
votes
0answers
47 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
16 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 ...
2
votes
0answers
33 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
46 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 $\phi(b) = ...
1
vote
1answer
21 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
28 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? ...
9
votes
0answers
81 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 ...
1
vote
2answers
83 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
36 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
20 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 ...
1
vote
1answer
36 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
27 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 ...
2
votes
1answer
56 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 $W_{\lambda}...
5
votes
1answer
63 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 $\mathbb{R}$ $\mathbb{g}=\begin{...
6
votes
2answers
84 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 $SL_2(\...
0
votes
2answers
29 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
1answer
34 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
vote
1answer
38 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 ...
0
votes
1answer
30 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
65 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 ...
1
vote
0answers
17 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 ...