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

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CG-modules: what does this notation mean?

I am trying to solve a question, but I do not know what the notation used means. If anyone could help me out that'd be great! I don't need help doing the proof, just what the notation means would be ...
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15 views

Questions on the proof of Beilinson-Bernstein localization theorem

I am trying to understand the Beilinson-Bernstein localization theorem (following the book by Hotta, Takeuchi and Tanisaki). I got stuck at the following two steps. Any help will be greatly ...
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Lifting the projective property through the affine centre

Let $\mathbb{k}$ be an algebraically closed field. There are many interesting examples of $\mathbb{k}$-algebras $R$ which admit a large central subalgebra $Z_0$ such that $R$ is a free $Z_0$-module ...
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24 views

How to compute $df(e)$ explicitly?

I am reading the book. On page 244, the formula (9.2.3.4). I would like to compute the bracket on g^* induced from the Poisson bracket on C[G] explicitly in the example of $G=SL_2$. The formula is: ...
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21 views

Evaluate the Projection Operator for this Irreducible Representation of Dihedral Group

I am trying to compute the projector for the Dihedral group of order 12 ($D_{12}=D_{2n}$) for a certain Irreducible Representation. The representation is two dimensional and so I need to caculate ...
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How to construct all Indecomposable representations of a quiver (up to isomorphism)?

Can you give me a sketch of how to Construct all (up to isomorphism) indecomposable representations of a quiver of type $E_{6}$ and $D_{6}$. Perhaps using Gabriel theorem. Thank you
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Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
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49 views

How to wite a Statement of Purpose for a Summer Program in Representation Theory. [on hold]

I want to attend a summer research program in Representation theory,$\;$for that I need to write a statement of purpose or simply a write up, so I want to know prerequisites for this course, and what ...
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14 views

Multiplicity free representation contain irreducible representation (for type I representation)?

While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central ...
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1answer
26 views

Characterization of simple representations

I am trying to solve exercise 10 from chapter 2 of Peter Webb's book on representation theory: Prove the following theorem of Burnside: let $G$ be a finite group and let $k$ be a algebraically ...
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9 views

Normalizer problem for finite Metabelian Groups with abelian sylow 2-subgroup

I am studying Normalizer problem (which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\Bbb{Z}G))$ and I came ...
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Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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1answer
24 views

Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace

The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of ...
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20 views

Show that we have a ring isomorphism $\varphi : D^{op} \rightarrow {End_{M_n {(D)}}}(D^n) $. [on hold]

I am trying to solve the following Representation Theory question: Suppose that $d \in D$ and define the map $$ \varphi_d \colon D^n \rightarrow D^n $$ by $$ \varphi_d((v_1, \ldots, v_n)) ...
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1answer
39 views

In proof of $|G| = n_1^2 + n_2^2 + \ldots + n_h^2$ how could equality of dimensions concluded from ring isomorphism

In Derek Robinson, A Course in the Theory of Groups on page 224 he proves: Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of ...
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1answer
47 views

Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of ...
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29 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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Show that all simple modules of ${M_n}{(D)}$ are isomorphic to $D^n$, where $D$ is a division ring. [duplicate]

I am trying to solve part (c) of the following Representation Theory question: Let $D$ be a division ring and let $n$ be a positive integer. For $ 1 \leq l \leq n $ let $$C_l= \{A = (a_{ij}) \in ...
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1answer
14 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
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1answer
22 views

Show ring isomorphisms $End_R{({S_1}^{n_1} \oplus \ldots \oplus {S_r}^{n_r} )} \cong End_R{({S_1}^{n_1})} \times \ldots \times End_R{({S_r}^{n_r})}$

I have been struggling with this Representation Theory question for the past week: Let $R$ be a ring and $S_1, \ldots, S_r$ simple $R$-modules with $S_i$ not isomorphic to $S_j$ whenever $i \neq j$ ...
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34 views

Is this a linear transformation? in the context of group representations

Let $G$ be a group. A regular representation is given as $V=\mathbb{C}[G]$, a vector space, where $l: G \to GL(V)$ be the action is given by $l(g)(\alpha)(h) = \alpha (g^{-1}h)$ for all $g,h\in G, ...
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1answer
28 views

The set $W^{⊥⊥}$ in a Hermitian space

Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$ I am trying to figure out a good strategy for this proof. I know that: $W$ is a subspace of $V$ ...
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1answer
24 views

Representations on an infinite dimensional graded vector space

I have two questions. First, given a finite dimensional complex vector space $V$ and a finite group representation $\rho:G \to GL(V)$, Maschke's theorem tells us that we may decompose $V$ into a ...
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21 views

Proving this representation is irreducible (converse of Schur's Lemma?)

I have a projective, unitary representation $V:G\rightarrow GL_n(\mathbb{C})$ of a (for now) finite group $G$. I have shown that the following is true: for every n x n matrix $X$, $$ \sum_{g\in G} ...
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48 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} ...
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Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$

Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K ...
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35 views

Action of an algebraic group induce a representation of its Lie algebra

Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation : $$\Gamma : Lie(G) \to gl(V)$$ If yes, how ? ...
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Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some ...
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1answer
15 views

Why is the Young symmetrizer non-zero?

Suppose $\lambda$ is a partition of the natural number $n$ and $T$ is a standard Young Tableaux of shape $\lambda$. Let $$P_{\lambda}:=\lbrace g\in S_n:g\text{ preserves the rows of }T\rbrace$$ and ...
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If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$

Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism $$ \varphi(1) = (1,0), ...
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About a representation of D4

Could you please give ma an example of a linear representation of the dihedral group D4? all references give this matrix representation. How could I express it linearly? Thanks.
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38 views

Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong \mathbb{R}/2\pi\mathbb{R}$ .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong ...
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Find the subrepresentation of a cyclic group

This is a spin-off of this question: Show that representation $\rho$ can be divided I came across the problem of dividing representation $\rho$ of a cyclic group given as below: $$ g \longmapsto ...
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Why is $(A \otimes_\mathbb{Z} V_p )\bigcap R(G) = V_p$

I am reading the proof of brauers theorem in Serres book Linear Representations of finite groups and I have trouble understanding Lemma 5(page 75). Let $G$ be a group of order $g$. First let $g=p^nl$ ...
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Show that representation $\rho$ can be divided

let $\rho$ be a representation of a cyclic group of order 6 defined by the following relationship: $$ 1 \longmapsto \begin{pmatrix} 1 & -1 \\ 1 & 0 \\ \end{pmatrix} $$ ...
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How can we prove that the group order is an upper bound both for the number and the dimensionalities of the irreducible representations?

For example, in $S_{3}$ there is 6 number of members for the group while there are 3 different irreducible representation with dimensinalities of 1, 1 and 2 in which gives: $$\sum_{\mu}{n_{\mu}^2} = ...
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35 views

example of infinite group that maschke's theorem is not hold [closed]

Show by giving an an example that Maschke's theorem does not hold for all infinite groups.
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25 views

About character table and normal subgroups

I have read that: All normal subgroups of a group $G$ can be recognized from its character table. The kernel of a character $\chi$ is the set of elements $g$ in $G$ for which $\chi(g) = \chi(1)$; ...
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Computing character for the partition $(2, 2, 1, 1)$ using Murnaghan–Nakayama rule

I am trying to understand an example from Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. The group ...
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Understanding the Murnaghan–Nakayama rule

I am trying to understand the Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. Here is the ...
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highest weight of adjoint represesentation

Let $\mathfrak{g} = \mathfrak{gl}(3,\mathbb{C})$ and let $\mathfrak{h}$ be the subalgebra of $\mathfrak{g}$ consisting of diagonal matricies. Then for $1 \leq i \leq n$, let $\epsilon_i \in ...
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Action on Flag manifold

When $G$ is of type A,D,E and $B_4$ then the group of Dynkin diagram automorphisms is non-trivial. If $B$ is a Borel subgroup of $G$, then is there a nice action of the Dynkin diagram automorphism ...
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Metaplectic representation carried on the space of generalized analytic functions

Background to my question The defining representation of the symplectic group Sp(2m,R) are $2m\times 2m$ matrices in block form, \begin{equation} D(g)= \left[ \begin{array}{cc} B & -A ...
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1answer
35 views

Trying to show $End_R(S^n) \cong M_n(End_R(S))$

Trying to show $End_R(S^n) \cong M_n(End_R(S))$ where S is simple R-module. I'm using Schur's Lemma so I know the ring of endomorphisms is a division ring, I'm not sure how to manipulate this to get ...
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1answer
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Trying to show $End_R(U_1 \oplus U_2) \cong End_R(U_1) \times End_R(U_2)$

I'm trying to prove $End_R(U_1 \oplus U_2) \cong End_R(U_1) \times End_R(U_2)$ Where $U_1, U_2$ are simple R-modules s.t $U_1 \not\simeq U_2$ So far I have constructed a mapping $\phi \mapsto (\pi_1 ...
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1answer
28 views

Connection between quivers and representations of Lie algebras

Can anyone recommend a reference to study the connection between quiver theory and representation theory of Lie algebras? Supposedly those two things have something to do with each other, with the ...
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39 views

Matrix representation of a 6-dimensional Lie algebra

The question is about the matrix representation of the following 6-dimensional Lie algebra, with 6 generators $t_1,t_2,t_3,t_4,t_5,t_6$. This Lie algebra is nilpotent, non-abelian, non-reductive and ...
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1answer
38 views

Is the dihedral group Dn linearly primitive for n>2?

Let $D_n$ be the Dihedral group (of order $2n$). For $p>2$ a prime number, $\mathbb{Z}/2$ is a core-free maximal subgroup of $D_p$, then $D_p$ is a primitive permutation group, and so linearly ...
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1answer
34 views

I don't know Maschke's theorem in the group representation.

I have a question about the Maschke's theorem in the group representation. I know that Maschke's theorem says that "Every representation of a finite group having positive degree is completely ...
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1answer
25 views

An inequality for the minimal number of generators of a finite group II

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating exactly the left regular representation (with ...