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

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5
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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) ...
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0answers
18 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
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1answer
54 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 ...
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1answer
19 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 ...
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50 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 ...
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2answers
14 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 $V$ ...
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1answer
31 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 ...
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20 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 ...
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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 ...
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14 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, ...
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28 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? ...
1
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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 ...
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0answers
25 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
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1answer
113 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 ...
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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 ...
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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, ...
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23 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 ...
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0answers
8 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
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1answer
39 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
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1answer
54 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 ...
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1answer
52 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 ...
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1answer
57 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 ...
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1answer
23 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 ...
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61 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 ...
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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 ...
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38 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
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1answer
15 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 ...
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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
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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 ...
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1answer
19 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
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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? ...
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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 ...
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2answers
81 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. ...
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1answer
32 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 ...
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1answer
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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 ...
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1answer
31 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
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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
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1answer
55 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 ...
5
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1answer
62 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 $\mathbb{R}$ ...
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2answers
81 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 ...
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2answers
28 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 ...
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1answer
31 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 ...
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1answer
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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 ...
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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 $H$?
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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 ...
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1answer
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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 ...
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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 ...
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1answer
21 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 ...
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1answer
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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 ...
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24 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 ...