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

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Decomposition of standard Borel subgroup.

I am reading the lecture notes. On page 3, formulas (1.7), (1.8), (1.9), let $P$ be the standard Borel subgroup, we have $P=MN$. Why $M, N$ must be (1.8), (1.9)? I know that $P$ should be a upper ...
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32 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
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Representation theory and point groups

Hello everyone :) I have a doubt. I have the point group $C_{3v}$, which is the group $$C_{3v}= \lbrace e, C_{3}, C_{3}^{2}, \sigma_{v_{1}}, \sigma_{v_{2}}, \sigma_{v_{3}} \rbrace$$ $C_{3}$ and ...
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What values can a character $\psi$ take on an element of order $2$?

If $\psi$ is the character of a degree $2$ complex representation $\varphi\colon G\to GL_2(\mathbb{C})$, and $x\in G$ has order $2$, then $\psi(x)=0,\pm 2$. I noticed this by seeing $\varphi(x)$ ...
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Product of characters in representation theory

Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation? I think this is true, say for instance if $R_1$ is a matrix representation with ...
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Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $ \chi(g^{-1})=\overline{\chi (g)}$ ...
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Real regular representation of cyclic group

I am looking for help to answer the following questions: What are the irreducible real representions $ρ: C_n → GL(V ) $ of a cyclic group of order n? How does the real regular representation $RC_n$ ...
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Occurrences of trivial representation is equal to dimension of $\{v\in V:\varphi(g)v=v\}$.

Suppose $\varphi\colon G\to GL(V)$ is a complex representation with character $\psi$. If $W=\{v\in V:\varphi(g)v=v,\ \forall g\in G\}$, why is $\dim W=(\psi,\chi_1)$, where $\chi_1$ is the principal ...
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Infinitely many direct sum decompositions of $M$ into direct sum of irreducible $\mathbb{C}G$-modules?

I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4. Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, ...
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Number field attached to a finite group.

Let $G$ be a finite group. I know that the set of irreducible representations of $G$ over the complex numbers (up to isomorphism) is finite. Let us fix our attention on some irreducible ...
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Proof that $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective

I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb ...
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Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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Irreducible unitary representation of a solvable lie group

Determine the equivalence classes of irreducible unitary representations of a solvable lie group. $$\begin{bmatrix}ae^t & 0\\ 0 & be^{-t}\end{bmatrix}$$ for $a,b\in \mathbb{R}, t\in ...
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What is the representation induced by $Z(G)$?

If $G$ is a finite group, and we have a representation of the center $\varphi\colon Z(G)\to GL(V)$, what is the induced representation on $G$? I've seen it mentioned in places, but not its definition. ...
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1answer
21 views

Character of action of permutations on a subset

I have a problem involving characters of a certain $\mathbb C$$G$-module, where $G = S_n$. The module is the vector-space $V$ with basis $\{v(I) | I\subset\{1,...,n\},|I| = k\}$, where $k\le n$. With ...
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scalar multiple of Young symmetriser

The following is a lemma on Fulton and Harris' book -Representation theory,a first course (page 53): Lemma: For all $x\in \mathbb{C}\mathfrak{S}_r$, $c_{\lambda}\cdot x\cdot c_{\lambda}= scalar ...
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37 views

What's the meaning of $d^{\times } a$?

In the lecture notes, the last line of on page 51, what is the meaning of $d^{\times } a$ in the integral? Thank you very much.
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How does GAP understand $SL_2(\mathbb{F}_3)$?

When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats the field on which the representations are. There GAP ...
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Relation between Jordan Normal Form and Irreducible Matrix Representations.

Ok so I have learned some very basic things about groups and matrix representations of groups. I have learned that it can be possible to find a "minimal basis" or "irreducible basis" for which a ...
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Representation of complex Clifford algebra on exterior algebras when quadratic form has odd index

Overview This problem entails the explicit construction of representation of Clifford algebra upon the exterior algebra, using orthogonal complex structure or polarization, namely, given a ...
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1answer
27 views

Projection matrix (C* algebra.. but linear algebra question) [on hold]

The subject is $C^*$-algebra, but I think my question might be linear algebra related type. I have a question from the book Operator Algebras Theory of C*-Algebra by Blackadar. On page 351, in the ...
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2answers
44 views

Existence of projection in proof of Maschke's theorem

In a proof of Maschke's theorem, my lecturer writes "If $N$ is a $\mathbb C G$-module and $N\leq M $ is a submodule, let $\pi: M \to M$ be a projection (i.e. a map with $\pi^2=\pi$) with image $N$." I ...
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Proving $Ind_H^G1_H=\pi_X$

Lemma Let $\psi=1_H$ the principal character of $H$, then $Ind_H^G1_H=\pi_X$, the permutation character of $G$ on the set $X$ of left cosets of $H$ in $G$. Proof Let $\{t_i \}$ form a tranversal. ...
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30 views

Does there exist a non-quasi-split torus?

In a homework, I was asked to prove that any torus is isomorphic to a quotient of a finitely many product of Weil restrictions $Res_{L/k}\mathbb{G}_m$. While solving this, I got an impression that ...
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1answer
35 views

Prove that if G has a faithful complex irreducible representation

I am struggling with the proof that if G has a faithful complex irreducible representation then $Z(G)$ is cyclic: Let $\rho:G \rightarrow GL(V)$ be a faithful complex irreducible representation. ...
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Definition of a matrix by using coaction.

Let $V$ be a vector space with a basis $e_1, \ldots, e_n$. Let $C$ be a coalgebra and $V$ a $C$-comodule. Consider the coaction $\delta: V \to C \otimes V$ given by $\delta(e_i) = \sum_{j=1}^n c_{ij} ...
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What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?

It seems that there is a action by which $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ permute the $p^2$ ordered tuples in $\mathbb{F}_p^2$. What is the map from the $2 \times 2$ matrices over ...
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How does $\mathrm{Res}_{\{I\}}\chi = \mathbb{I} \oplus \dots \oplus \mathbb{I}$?

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
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How to read GAP's output on “IrreducibleRepresentations”?

For example for the group $SL_2(\mathbb{F}_3)$ I get the following, ...
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Is the tensor product of two semi-regular representations again semi-regular?

Given a finite group $G$ and two semi-regular representation $D_1$ on $V_1$ and $D_2$ on $V_2$, i.e. representations containing all irreducible representations of that group, is $D_1 \otimes D_2$ ...
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1answer
51 views

Find the conjugacy classes of $A_5$

I was trying to find the conjugacy classes of $A_5$. So I started by writing out all the conjugacy classes of $S_5$ in the hope that I could just restrict the set of them. The conjugacy class ...
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Permutation representation of $S_4$ acting on $\{1,2,3,4\}$

I am considering the permutation representation of $S_4$ acting on $\{1,2,3,4\}$ with character $\chi_{\pi}$. The answer is $\chi_{\pi}=(4,2,1,0,0)$ Now I know that for the permutation representation ...
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1answer
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Any 1-dimensional character $\otimes$ irreducible character is irreducible

I would like to prove that any 1-dimensional character $\otimes$ irreducible character is again irreducible. (I think this is character but there is chance I have misinterpreted my notes and it may be ...
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Definition of Schur Functors on morphisms

I've been learning about Schur functors on nLab: http://ncatlab.org/nlab/show/Schur+functor A definition is given, for $R$ some finite dimensional representation of $S_n$, by the formula $S_R(-)=R ...
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Does every Young diagram have a unique minimal major index?

Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ ...
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Recipe to compute dimension and decompose product of $SO(N)$ group representations

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
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1answer
42 views

Lifting representations, kernels and invariant subspaces

Let $G$ be a group, $N \triangleleft G$, $G/N$ the corresponding quotient group. Suppose $\rho : G/N \longrightarrow GL(\mathbb{C})$ is a representation of $G/N$. Then the composition ...
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+50

At what position do we insert the new coefficient in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
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Homomorphisms in $Q_8$ [duplicate]

Prove directly that the 2-dimensional irreducible representation $\rho$ of $Q_8$ is not realisable over $\mathbb{R}$. Suppose $\rho: Q_8 \rightarrow GL_2(\mathbb{R})$ is a representation with ...
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1answer
20 views

Adjoint representation and tangent vectors

Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra, $\text{Ad}:G\rightarrow GL(\mathfrak{g})$ the adjoint representation of $G$. Then, for $X,Y\in \mathfrak{g}$, \begin{align*} ...
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Are derivatives of generic representations generic?

I am trying to learn about the Bernstein-Zelevinsky derivatives. If $\pi$ is a generic representation of $GL_{n}$, then will the $k$-th derivative $\pi^{(k)}$ be generic? Thanks.
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Considering transitive $G$-set

Question. Suppose that $X$ is a transitive $G$-set of size greater than $1$ and let $\pi$ be the associated permutation representation with the character $\chi$. Show that some element $g \in G$ ...
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Matrix representations and idempotent/nilpotent elements

A conceptual question: Let's say there's a linear matrix representation of a particular algebra. I'm wondering just how much this matrix representation can tell us about the 'outstanding' elements ...
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Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
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Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always hold?

Let $G$ be a group and $H < G$. Let $\Psi$ be a character. Let $\mathbb{I}$ be the trivial representation Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always ...
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Inducing $A_4$ from $\langle (123) \rangle$

Let $G=A_4$ and $H=\langle (123) \rangle < G$. Compute $Ind_{H}^G \chi$ for every irreducible $\chi$ of $H$. Choose the right transversal of $H$ in $G$ as $V_4=\{ 1, (12)(34),(13)(24),(14)(23) ...
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Choices of decompositions of a representation into irreducible components (Serre, Ex. 2.8)

The following is exercise 2.8 in Serre's Linear Representations of Finite Groups. If $V$ is a representation of a group $G$, recall it has a canonical decomposition $V=\oplus_1^n V_i$, where each ...
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1-loop quiver and the classification of quivers

Gabriel's theorem states that finite type quivers are exactly the ones whose underlying graphs are ADE type Dynkin diagrams. Furthermore, the quivers whose underlying diagrams are ADE type affine ...
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179 views

Symmetric kernel of tensor product

Let $V,W$ be two real vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with distinct kernels $K_i$ of dimension $1$. Consider the tensor product of these maps ...
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Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...