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

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Proving that an $FG$-homomorphism is surjective

Assume that $V$ is an $FG$-module.Prove that the subset $$V_0 = \{v \in V : vg = v \space \forall \space g \in G \}$$ is an $FG$- submodule of $V$. Also show that the function $$\phi: v \to ...
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Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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Why do roots span dual space of maximal toral subalgebra?

Suppose $\Phi$ is the root system of a semi simple Lie algebra with maximal toral subalgebra $H$. I read that $\Phi$ spans $H^\ast$. The Killing form on $H$ is nondegenerate, so $H\cong H^\ast$ by ...
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The definition of Fell topology

Let $G$ be a Lie group, $\pi$ is a representation, then with some conditions, we have the following branching law $\pi|_N=\int^{\oplus}m_\pi(\mu)\mu\mathrm{d}\mu$ where $m$ is the multiplicity ...
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35 views

Character table $S_3 \times Z_2$

I need get of character table of $S_3 \ \times \ Z_2$ how make this character table? The representation is a $\psi (g,h) = \rho_1 (g) \rho_2 (h)$ with $\deg (\rho _2) = 1$ and $\rho _1 $ irreducible. ...
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36 views

Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
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$\mathfrak g = [\mathfrak g,e]\oplus {\rm Ker}({\rm ad}f)$ for an $\mathfrak {sl}_2$-triple $\{e,h,f\}$.

Let $\mathfrak g$ be a finite dimensional semisimple lie algebra over $\mathbb C$. Let $\{e,h,f\}$ be an $\mathfrak{sl}_2$-triple in $\mathfrak g$ (i. e. with relations $[h,e] = 2e$, $[h,f]=-2f$ and ...
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36 views

Permutation representations of finite abelian groups [closed]

What is a good source to study from about permutation representations of finite abelian groups, specifically $\mathbb{Z}_{p}$? If reference for the specific topic is not available, I would like to ...
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28 views

A Question about Quantization and Partition Function

I have a question about quantization and partition function, which sound a little bit inappropriate. But I still want to ask for help. I think that quantization is a unitary representation of the ...
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Formula for $\theta:\mathfrak{P}(L)^{G}\to \mathfrak{P}(H)^{W}$ for $\mathfrak{sl}_2$; exercise in Humphrey

Let $L=\mathfrak{sl}(2,\mathbb{F})$ with standard basis $(x, y, h)$ and dual basis $(x^{*}, y^{*}, h^{*})$, $H$ a CSA, $W$ the Weyl group and $G=\operatorname{Int}L$. Let $\mathfrak{P}(L)^{G}$ be ...
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49 views

How to obtain real irreducible representation matrices for finite point groups?

I would like to generate the irreducible representation matrices in real (not complex) form for any finite point group, in order to use them in a projection operator. At least I require the diagonal ...
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injectivity of natural map $Hom_G(V,W)\otimes V\to W$ (Kraft Procesi Exercise 3.2.4)

Exercise 4a on page 27 of Kraft and Procesi's Primer on Invariant theory http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf asks to show the following: $V$ is an irreducible finite dimensional ...
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42 views

Is there a theory of induced representations for semigroups?

Given a semigroup $G$, a subgroup $H\subseteq G$ (not merely a subsemigroup) and a representation $\rho: H\rightarrow GL(V)$ for some vector space $V$, is there a canonical definition of an induced ...
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35 views

Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
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51 views

Induction of an irreducible group representation

I'm having some trouble finding the answer to the following question. Any ideas on how to get started? Let $H$ be a subgroup of a group $G$ and let $U_{1}$, ...,$U_{k}$ be the irreducible ...
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23 views

Criterion for semisimplicity for $\mathfrak{so}_6(\Bbb C)$

I'm trying to prove that $\mathfrak{so}_6(\Bbb C)$ is semisimple. There exists a criterion which says that, given a Lie algebra $L\le\mathfrak{gl}(V)$, where $V$ is an irreducible $L$-module, then ...
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Hopf algebra associated to $GL_V$ for $V$ infinite rank?

Let $V$ be a free $\mathbb{Z}$-module of rank $n$, and let $F$ be the functor associating to a ring $R$ the group $Aut_R(V \otimes R)$. If $V$ was finite, say $Z^n$, this functor is representable by ...
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Are the elements of the adjoint represetnation normal operators

Given a Lie group $G$ with Lie algebra $\mathfrak{g}$ on has the adjoint action of each $g\in G$ given by $Ad_g(\mathfrak{g})$. Is $Ad_g: \mathfrak{g} \rightarrow \mathfrak{g}$ a normal operator ...
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32 views

How to understand that the left regular representation of a division algebra is irreducible?

In Weyl's book The classical groups, it is said the regular representations of a division algebra are faithful and irreducible. The key step is to show the ideal of the division algebra is $\{0\}$ ...
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Finitely generated $\mathbb{C}$-algebra, uncountably many $\lambda \in \mathbb{C}$. [closed]

Let $A$ be a finitely generated $\mathbb{C}$-algebra and let $a \in A$ be a nonalgebraic element. My question is, are there uncountably many $\lambda \in \mathbb{C}$ such that the element $a - ...
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A question about tensor products of representations

Let $V$, $W$ be real finite diemnsional vector spaces. What is the relationship between the subgroup of $GL(V\otimes W)$ whose elements can be written as tensor products (I think we can write this ...
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33 views

Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
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Inverses of elements in group algebras

If $G$ is a finite group whose elements are $g_1,\ldots,g_n$ and let $F$ be the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. We define a vector space over $F$ with ...
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Matrix-like Representation of any linear map using Hamel Basis

Let $X$ and $Y$ be two arbitrary linear spaces over a field $\mathbb{K}$. Let $B=\{x_\alpha:\alpha \in S\}$ and $C=\{y_\beta:\beta \in T\}$ be Hamel basis for $X$ and $Y$ respectively. Denote the ...
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Nonlinear Lie group from Fulton & Harris

On page 138 of my copy of the celebrated Representation Theory by Fulton & Harris, a proof is outlined to show that the real group of $3\times 3$ upper-triangular unipotent matrices modulo a ...
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Explain this proof in more details

The following is the proposition 3.3 of folland "A Course in Abstract Harmonic Analysis" book. please Explain its proof in more details: I do not know the cause of contradiction. that is, how ...
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Question concerning an isomorphism between a module of $\operatorname{add}(M)$ and a hom space

Let $M$ be a $\Lambda$-module of an artin algebra $\Lambda$. Let $N$ be in $\text{add}(M)$. Let $\Gamma:=\text{End}_\Lambda(M)$. Assume further that $\Lambda\cong \text{End}_{\Gamma}(_\Gamma M)$. ...
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How to construct an explicit isomorphism between two special endomorphism rings

Let $\Lambda$ be an artin algebra and $M$ a $\Lambda$-module. Let $\Gamma:=\text{End}_\Lambda(M)$ and let $D$ be the standard duality. How can you give an explicit isomorphism ...
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Representing real numbers from matrices of non-negative-reals.

Consider $I = \left(\begin{array}{cc} 1&0\\0&1\end{array}\right)$ and $N = \left(\begin{array}{cc} 1&1\\1&1\end{array}\right) - I = \left(\begin{array}{cc} ...
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Proving irreducibility of representations using matrix representation

For the quaternion group $Q_8$ we have the presentation $$Q_8 = \big<a,b : a^4 = 1, a^2=b^2, b^{-1}ab=a^{-1} \big>$$. Now knowing that for matrix $$ A = \begin{bmatrix} i & 0 \\ 0 & -i ...
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PSL(2,q) has no nontrivial irreducible representation of small dimensions

I am a bit new to this site, and wondering how to show that PSL(2,q) has no nontrivial irreducible representation of small dimensions. Thanks.
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60 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
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Generating function of symmetric power representation

Let $\rho:G\rightarrow GL(V)$ be a complex representation. For each $n$, let $\chi_{\text{Sym}^n}$ be the character of the n-th symmetric power of $V$. Prove for each $g\in G$, $$\sum_{i=0}^\infty ...
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If $A$ is an associative algebra Show that $End_{A}(A)=A^{op}$ the algebra with opposite multiplication [duplicate]

Let $A$ be an associative algebra, And let $V$ be a representation of $A$. By $End_{S}(V)$ one denotes the algebra of homomorphisms of representations $V \to V$ Show that $End_{A}(A)=A^{op}$ the ...
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Order of group elements from a character table

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I'm trying to answer a question which goes the ...
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64 views

Using character table to find normal subgroups

I am working through a question on an old character theory exam. I've answered the first two parts ok, but am now struggling on the third part. Here is the part that I can't do: I've computed ...
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45 views

Tensor product of simple finite-dimensional modules

I am trying to understand the following one-liner that appears in http://arxiv.org/abs/0901.0827v5, Theorem 2.26 (and I'm afraid I must be missing something really basic): Let $V$ and $W$ be ...
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If two non-equivalent representations are irreducible then sum is 0

We have a finite group and two representations $D_1:G\to GL_n(\mathbb{C}),D_2:G\to GL_m(\mathbb{C})$ for some positive integers $m,n$. We define $$T=\sum_{g\in G}D_1(g)BD_2(g^{-1})$$ where $B\in ...
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Computing plethysms of the adjoint representation using the Littlewood Richardson rule

Let $N$ be an integer (let's imagine very large), and let $G$ be the group $\mathrm{GL}_N(\mathbb{C})$. I would like to compute various plethysms of irreducible representations which are not ...
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In a vector space over a finite field, can the orbit of a point under matrix multiplication have dependent subsets?

Let $\mathbb{F}_q^n$ be the vector space of dimension n over the finite field of order q, $\vec{v}$ a vector in the space, and $M$ an invertible $n \times n$ matrix over $\mathbb{F}_q$. We know that ...
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Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
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Showing given representation is summand by averaging

This problem is from Artin 1st edition 9.2.2. Problem : Let $\rho$ be the standard three-dimensional representation of T, and let $\rho'$ be the permutation representaion obtained from the action of ...
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Cohomology of permutation representation

Consider the action of $S_n$ over $\{1,...,n\}$ consider the associated representation with integral coefficients $X_n$. What are $H^r(S_n,X_n)$? More in general is there a nice way to predict the ...
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Matrix representations of $\mathfrak{g}^*$.

Let $g = sl_2$. Then there is a matrix representation of $g$ as follows. The Lie algebra $g$ is a three dimensional vector space with a basis $E, F, H$ such that $[E,F]=H$,$[H,E]=2E$,$[H,F]=-2F$. The ...
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What is the determinant of Ad(g)?

In more generality, if a matrix acts on a group of matrices by conjugation, what is the determinant of this action (if such a notion exists)? Is it simply the determinant of the matrix being used to ...
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1answer
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Why is $B^2 = \mathbb{I}_2$ where $B = T^{-1}AT$

I am currently studying group representations. And I know that two If $\phi:G \to GL(n,F)$ is a representation of $G$ over $F$ and if $\gamma:G \to GL(m,f)$ is also a representation of $G$ over $F$we ...
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Differing conventions regarding “modulus character” of $k$-points of smooth affine $k$-group, $k$ non-Archimedean

Let $\mathbf{G}$ be a smooth connected affine $k$-group, where $k$ is a non-Archimedean local field, $G=\mathbf{G}(k)$ the group of $k$-rational points, a locally $k$-analytic group. Since $G$ is in ...
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Converse of Maschke theorem

Let F be a field and G be a group and FG the group ring. Let H:=$\{\sum_{g\in G}\lambda_g g\in FG : \sum_{g\in G}\lambda_g=0\}$. Then H is codimension 1 subspace of FG and is an FG submodule. ...
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Weightspace decomposition of a semisimple Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a (finite dimensional) semisimple Lie algebra. Let $H$ be a maximal toral subalgebra of $L$. Consider a representation $\pi: L \to \mathfrak{gl}(V)$. It is ...
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Inducing representation for groups of order $p^3$

For groups $G$ such that $|G|=p^3$ one can show that $(1)$ $Z=Z(G)\cong C_p$ $(2)$ $G'=Z$ $(3)$ $G/Z \cong C_p \times C_p$ Take any $x \in G/Z$. Then $N=\langle x,Z \rangle$ is an abelian normal ...