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

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Representation theory of Lie groups and outer automorphisms

If $G$ is a simply connected Lie group (I have in mind $G=SL_n(\mathbb{C})$), then we have an isomorphism $Aut(G)/Inn(G)\rightarrow Aut(g)/Ad(G)$ induced by taking the differential at $1$; here $g$ is ...
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78 views

Uniqueness of the killing form

I would like to consider/prove the following problems: let $k$ be a field, $g$ a finite-dimensional simple Lie algebra over $k$ with Killing form $B$. If $\sigma:g\times g\rightarrow k$ is a ...
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59 views

Invariant ring for $S_5$ [closed]

For the standard representation of $S_5$, the ring of invariants is generated by the elementary symmetric polynomials and hence it is a polynomial ring. Now if we take the tensor product of standard ...
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32 views

Why do we have these values of the generalized character when evaluated with the scalar product?

Let $U \le G$ be a subgroup of odd order of the finite group $G$. Suppose $t \notin U$ is an involution with $u^t \in uU'$ for all $u \in U$, where $U'$ denotes the commutator subgroup of $U$. Set $T ...
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1answer
61 views

Calculating the basic algebra over a finite field in GAP

Assume $A$ is a (nonsemisimple) finite dimensional algebra over a finite field $K$ (for example a group algebra). I want to calculate the basic algebra $B$ of $A$ as a matrix algebra, constructed as ...
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23 views

twisted group ring: uniqueness of representation

$G$ is a multiplicative group, $K$ is a field. Let $\gamma$ and$\tilde{\gamma}$ be two twistings of $K^t[G]$ related by the equation $\tilde{\gamma}(x,y)=\delta(x)\delta(y)\delta(xy)^{-1}\gamma(x,y)$. ...
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34 views

Extending a linear character of $U$ to $TU$, where $T$ is generated by an involution normalising $U$

Let $U \le G$ be a subgroup of the finite group $G$ of odd order. And suppose $t \notin U$ is an involution normalising $U$, i.e. $U^t = U$ and $t^2 = 1$. Assume $t$ centralizes $U / U'$, i.e. $u^tU' ...
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$A_\mathfrak{q}(\lambda)$ module and Zuckerman's derived functor module

Does any reference book give the explicit definition for $A_\mathfrak{q}(\lambda)$ module? Or are $A_\mathfrak{q}(\lambda)$ and Zuckerman's derived functor module the same thing?
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80 views

A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
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26 views

$g_1,g_2 \in G$ such that for any complex character $\chi$ , $\chi (g_1)=\chi(g_2)$ ; does $g_1,g_2$ belong to same conjugacy class?

We know that any character on a finite group is a class function i.e. they each take a constant value on a given conjugacy class . Is the converse true ? that is let $G$ be a finite group , $g_1,g_2 ...
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Find missing values in character table (Representation Theory)

I am given part of a character table of a group G with conjugacy classes, $C_1, C_2, ..., C_5 $. Below each conjugacy classes in the table the size of the centraliser of one of its elements is given. ...
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48 views

How is an abelian $G$-operator group with $m1 = m$ a $\mathbb Z[G]$-module

Let $M$ and $G$ be groups. We call $M$ a $G$-group (or group with operators) if every $g \in G$ corresponds to an endomorphism of $M$, i.e. we have $$ (mn)^g = (n^g)(m^g). $$ (the application of ...
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2answers
31 views

socle(M) being simple gives an upper bound for the dimension of End(M)?

Suppose $k$ is a algebraically closed field of arbitrary characteristic. Let $A$ be a finite dimensional $k$-algebra and $M$ an $A$-module with finite dimension with respect to $k$. I have seen it ...
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1answer
25 views

Show permutation representation is reducible, by finding G-invariant subspace [duplicate]

$(\pi,V) $ is the permutation representation of the symmetric group $S_5 $, $ V=C^5$ and the action of standard basis vectors of $ V$ is given by $\pi(\sigma)e_i=e_{\sigma(i)} $ for $\sigma\in S_5 $ ...
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59 views

How are $G$-modules and linear group actions different

Let $M$ be an abelian group and let $G$ be a group acting on $M$ such that $M$ is a $G$-operator group, i.e. we have for $u, v \in M$ and $g,h \in G$ (1) $u\cdot 1_G = u$ (2) $(ug)h = u(gh)$ (3) ...
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33 views

Step in proof that a quotient is isomorphic to cohomology group

The following proof is about the cohomology group. But the fact I do not understand is something like, we have two groups $U \cong V$ and two normal subgroups $N \cong M$ in $U$ respectively $V$, what ...
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2answers
49 views

Modules over associative algebras are just special cases of “ordinary” modules over rings?

By module over a ring, I mean always a right-module. All rings are supposed to be unital, and the module fulfills $m\cdot 1 = m$. If $R$ is commutative and $M$ a right-module, we can define $rx := xr$ ...
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40 views

expressing a quadratic map as a complex map

Are there any known criterion when a real quadratic mapping $ Q:\mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n} $ can be expressed as a complex quadratic map $ Q:\mathbb{C}^n \rightarrow \mathbb{C}^n$? ...
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1answer
26 views

Does same character table imply isomorphic abelianizations?

We know two finite groups with the same character table might not be isomorphic (e.g. $D_4$ and $Q_8$), but the sizes of their abelianizations are equal (in fact equal to the number of linear ...
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31 views

An odd expression appearing in proof that kernel and image of certain map on group ring $\mathbb Z[G]$ are equal

Let $G = \langle g \rangle$ be a cyclic group of order $n$. Consider the free ring $\mathbb Z[G]$ of all formal sums of elements from $G$ with coefficients from $\mathbb Z$, with multiplication given ...
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36 views

Some questions about standard $K$-duality

Let $A$ be a finite dimensional $K$-algebra, where $K$ is an algebraically closed field. We define a funtor $D: mod A \to mod A^{op}$ called standard $K$-duality. Suppose that $M$ is an arbitrary ...
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1answer
29 views

Applications of $SO(3)$ irreps to spatial rotation

I've been on a kick learning about Lie Groups, with special emphasis on $SO(3)$ recently. I work in the field of spacecraft attitude determination and control, where is $SO(3)$ of interest in the ...
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56 views

Examples of Induced Representations of Lie Algebras

Given a (finite-dimensional) Lie algebra $\mathfrak{g}$, a subalgebra $\mathfrak{h}\subset\mathfrak{g}$, and a representation $\rho:\mathfrak{h}\rightarrow\mathfrak{gl}(V)$ of $\mathfrak{h}$, one can ...
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1answer
38 views

Character related to a maximal subgroup

I am trying to prove the following statement: If $H$ be a maximal subgroup of $G$ and $\xi=(1_H)^G$, where $1_H$ is principal character of $H$, and $\chi$ be a non-principal irreducible constituent ...
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27 views

Show the representation is an integral multiple of the regular representation

Let $(\pi_1, V_1), ... , (\pi_t,V_t)$ be all the irreducible representations of a finite group $G$ over the complex numbers with respective degrees $n_i$ and characters $\chi_i$. If $(\rho,V)$ is the ...
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51 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 ...
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34 views

Faithful irreducible representation of a finite $p$-group

I want to solve the following exercise in the representation theory field: A finite $p$-group $G$ has a faithful irreducible representation over an algebraically closed field whose characteristic is ...
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1answer
49 views

About an irreducible representation over an algebraically closed field

I want to prove the following statement that is an of the book "A course in the theory of groups" by D. Robinson: Let $n$ be the degree of an irreducible representation of a finite group $G$ over an ...
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37 views

$\mathbb{F}_p$ algebra with many $p$th roots of unity

An old qual problem reads Let $\mathbb{F}_p$ denote the finite field of $p$ elements. Consider the covariant functor $F$ from the category of commutative $\mathbb{F}_p$ algebras with ...
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33 views

The orbits of group representation

I will raise the question from an example. Let $G = \mathbb C^*$ be the multiplication group of nonzero complex numbers. Let $V$ be a $2$-dimensional vector space over $\mathbb C$. There is a ...
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20 views

How to recognize a monomial quiver algebra

Given a basic split finite dimensional algebra $A$ over a field K, A is isomorphic to $KQ/I_1$, for some quiver Q and a minimal(meaning it is generated by relations $x_i$, such that no relation is ...
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How does the weighted superposition of irreps make the Fourier transform of a finite group unitary?

This is supplementary to this question. In the lecture note of Andrew Childs on Nonabelian Fourier analysis, it is said that the Fourier transform of a finite group is the weighted superposition of ...
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30 views

Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?

I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note: My question: Why does it have to be weighted superposition and not equal ...
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33 views

Degree one irreducible representations

In section 2.5 of his Linear representations of finite groups (I have the french copy), Serre gives an example of determination of the character table of a group $G$. The group $G$ is taken to be ...
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Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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28 views

What is the use of a right-module?

It seems that only the left-module provides a representation of a group. So what is the use of a right-module?
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37 views

Exterior power of irreducible representation

I am new to representation theory. Suppose that $G$ is a finite group with an irreducible representation over a (real or complex) vector space $V$. In my application, $G$ is a symmetric group and the ...
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43 views

Degrees of irreducible characters of an extension of $A_5$ by an elementary abelian 5- group

I'm reading a recent paper of G. Navarro, The set of character degrees of a finite group does not determine its solvability, in which he construct two finite groups $H$ and $G$ with $cd(G)=cd(H)$ such ...
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116 views

What are some Group representation of the rubik's cube group?

The Rubik's cube corresponds to valid sequences of moves of the Rubik's cube. What are some group representations of this group (with respect to finite dimensional vector spaces on finite fields)? ...
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2answers
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Assume a homomorphism of groups gives a full and faithful functor on reps. Was it surjective?

Let $\phi: H \to G$ be a finite group homomorphism. Then there is a functor on representations $\operatorname{Rep}(\phi): \operatorname{Rep}(G) \to \operatorname{Rep}(H)$ given by precomposition with ...
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1answer
58 views

Degrees of Irreducible Characters of $GL(n,q)$

I know that in an old paper, R. Steinberg computed the irreducible (complex) characters of groups $GL_n(q)$ and $PGL_n(q)$ where $n \in$ {3, 4}. I want to know is there any known method for computing ...
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Is there an intuitive way of seeing why there are only finitely many irreducible representations?

Let $G$ be a finite group. A basic result in representation theory is that up to $\mathbb{C}[G]$-module isomorphism, there are only finitely many irreducible representations of $G$ over $\mathbb{C}$. ...
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Undecidability and the representation theory of $K<X,Y>$

The question comes from the problem here: http://mathoverflow.net/questions/73940/are-wild-problems-related-to-undecidable-ones It has already been proven that the representation theory of ...
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58 views

Definition of the category of group representations

One usually considers the category of complex linear group representations for a fixed group $G$. It is defined as the category whose objects are group morphisms $G \rightarrow GL(V)$ where $V$ is a ...
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Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$

I just computed the Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$. It would be nice if anyone could confirm it's correctness. Thanks.
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1answer
63 views

Does the supposed to exist functor considered in Langlands program bear a peculiar name?

I'm trying to figure out what a very rough sketch of the Langlands program could be. From what I (think I) understand, objects called reductive algebraic groups together with related so-called ...
4
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1answer
60 views

A Lie group that has an immersion in $\mathrm{GL}(n,\Bbb R)$ but no embedding?

Question: Is there a Lie group $G$ that admits a smooth immersion $$i:G\longrightarrow\mathrm{GL}(n,\Bbb R)$$ for some $n\in\Bbb N$, but no smooth embedding ...
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Is there a faithful linear representation of the additive group of integers?

In other words, does there (constructively) exist a faithful representation $\phi : \mathbb{Z} \rightarrow GL_{n}(\mathbb{C}) $?
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Why do we care about two subgroups being conjugate?

In classifications of the subgroups of a given group, results are often stated up to conjugacy. I would like to know why this is. More generally, I don't understand why "conjugacy" is an equivalence ...
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Flag varieties from the representation of a solveable Lie algebra

I've been reading Lie Algebras, and I've come across this problem: "Let $\mathfrak{g}$ be a solveable Lie Algebra over $\mathbb{R}$. $V$ a vector space over $\mathbb{R}$, and $\rho$ a representation ...