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

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*$G$-invariant* symmetric bilinear form

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
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14 views

Tensor product of representations of a product group?

Given some group $G$ that can be written as product of two other groups $$G = G_1 \times G_2 $$ and some representation of this group written in terms of representations of $G_1$ and $G_2$ $$R = ...
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30 views

Solving the Character table for $A_4$ and derived algebra characteristics.

Say I am given a group with 4 conjugacy classes $C_1, C_2, C_3, C_4$ with orders 1,3,4,4 respectively. I will call the conjugacy class characters $\chi_1,\chi_2,\chi_3,\chi_4$ respectively. I am ...
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Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & ...
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36 views

Fulton and Harris: Exercise 1.3 in section 1.1

This is exercise 1.3 on page 5 of Fulton and Harris Representation Theory: A First Course. Exercise: Let $G$ be a finite group, let $V$ be an $n$-dimensional $\mathbb C$-vector space and let $\rho: ...
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Stone representation theorem and right(or left-) one-sided ideals in a ring

Consider Marshall Stone's representation theorem: https://en.wikipedia.org/wiki/Stone's_representation_theorem_for_Boolean_algebras I would like to know in whichspecific way, if any, it is connected ...
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30 views

Find character table for symmetric group $S_3$

This group contains all permutations of 3 elements, so it has order 3!=6. Its three congruency classes are {1}, {(1,2),(1,3),(2,3)}, {(123),(132)}. As we know that the number of congruency classes ...
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17 views

Every base of a root system arises as indecomposable positive roots of a regular element?

I'm confused about a line in the Theorem p48 in Humphrey's's book on Lie Algebras. He's proving that every base $\Delta$ of a root system $\Phi$ arises as the set of $\Delta(\gamma)$ of ...
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19 views

Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
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75 views

Irreducible representations of $\operatorname{GL}_3(\mathbb{F}_q)$

I am trying to find all irreducible representations of $G = \operatorname{GL}_3(\mathbb{F}_q)$. I know that the order of $G$ is $(q^3-1)(q^3 - q)(q^3 - q^2)$ and the number of conjugacy classes is ...
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26 views

Existence of groups corresponding to dimensions of irreducible representations

If there are $r$ irreducible representations of a group $G$, we know that $|G| = \sum_{i=1}^{r}d_{i}^2$ and $d_i$ divides $|G|$. Suppose we have a decomposition of $N$ such that $N = ...
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38 views

Algebraic Peter-Weyl theorem in the case of $G=SL_2$.

The algebraic Peter-Weyl theorem says that for a linear reductive group $G$ we have $\mathbb{C}[G] = \oplus_{V} V \otimes V^* $, where $V$ runs over the set of all non-isomorphic irreducible ...
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19 views

References about an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$.

Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times ...
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What would be the “action” in functional analysis?

I am reading Simmons' "Topology and Modern Analysis". He keeps bringing up the idea of studying the set of all structure preserving mappings to obtain info regarding the structure of a certain normed ...
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48 views

Equality on pg. 40 of Humphrey's Lie Algebras, $\kappa(t_\lambda,t_\mu)=\sum_{\alpha\in\Phi}\alpha(t_\lambda)\alpha(t_\mu)$?

I don't understand part of an equality on page 40 of Humphrey's book on Lie Algebras. Suppose $L$ is a semi-simple Lie algebra over an algebraically closed field of characteristic $0$, and $H$ a ...
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95 views

All simple modules are projective $\Rightarrow$ semisimple [duplicate]

Let $A$ be a finite dimensional algebra over a field $K$. It is clear that if $A$ is semisimple, then every simple module is projective. Does the converse hold ? It seems false, but I can't find a ...
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42 views

Nonlinear representation of SU(2) and SU(4)

Consider a nonlinear representation of a group acting on the space of real vector $\phi_i$ in the form: \begin{equation} \phi_i\rightarrow \sum_jM_{ij}\phi_j+\delta\phi_i \end{equation} where $M$ is ...
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51 views

Right modules Vs Left modules.

I have been reading Frobenius Algebras, Volume 1 By Andrzej Skowroński, Kunio Yamagata. On page 18 I came across the following paragraph, and I founded interesting, I will quote it and then ask my ...
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38 views

Questions about fundamental representations of $SL_3/U$.

Consider the group $SL_3$. Let $U$ be the subgroup of $SL_3$ consisting of all upper triangular unipotent matrices. Then the algebra $\mathbb{C}[SL_3/U]$ is generated by $a_{11}, a_{21}, a_{31}, ...
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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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18 views

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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29 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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27 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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1answer
27 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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26 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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42 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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23 views

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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32 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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31 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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48 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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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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85 views

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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32 views

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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25 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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38 views

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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48 views

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.