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

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Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
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A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
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Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
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Question about the equivalence of two linear representations.

I would like to know if this approach is correct. I have two distinct permutation representations and I have to prove that the associated linear representations are equivalent. In order to do this I ...
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Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module.

Exercise: Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module. I'm not sure about all different kind of modules, but this is a question of a book about ...
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$\overline{\phi}: G/H \to GL(V)$ irreducible representation then $\phi= \overline{\phi}\circ \pi :G\to GL(V)$ it's irreducible

Let $H\trianglelefteq G$ be a normal subgroup of $G$ and let $\pi: G\to G/H$ be the canonical projection. Suppose that $\overline{\phi}: G/H \to GL(V)$ it's an irreducible representation. Define the ...
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Infinite Cyclic group representation

I am trying to learn Group representation and have a basic question regarding infinite cyclic groups. I am trying to find a representation of infinite cyclic group in $GL_n(\mathbb{C})$ and ...
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Restriction of a Specht module to the alternating group

Let $n\in\mathbf{N}$ and denote by $S_n$ the symmetric group on $n$ letters. For $\lambda\vdash n$ a partition of $n$ the Specht module $S^\lambda$ defines an irreducible representation. What ...
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Tensors furnish representations of the group

I'm bad at english, so what exactly does it mean in simple english that Tensors furnish representations of the group?
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Let $v\in V-0$, then $\varphi _{v}: k[x]\rightarrow V : f \mapsto f.v$ is a surjective $A$-module homomorphism.

Proposition. Let $A=k[x]$ and let $(V,\rho )$ be a finite dimensional irreducible $A$-module. Let $v\in V-0$, then $\varphi _{v}: k[x]\rightarrow V : f \mapsto f.v$ is a surjective $A$-module ...
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multiplicities of irreducible representations

Let $G$ be a finite group and $G'$ be a subgroup. Let $\rho$ be a one-dimensional group of $G'$. Then define $\psi$ to be the induced action of $\rho$ - $\psi:= Ind_{G'}^G \rho$ Is there any general ...
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Describing all $\rho$-invariant inner products

Let $z$ satisfying the equation $z^3=1$ be a generator of the cyclic group $\mathbb{Z}_3= \{ 1 , z,z^2 \}$. You are given that $\rho : \mathbb{Z}_3 \to GL(\mathbb{C}^2)$ defined by $$\rho(z) = ...
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Is there anywhere some explicit Bruhat decompositions are written down?

Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of ...
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Orbits of $Sp(n,R)$ under action of $Gl(2n,R)$ by conjugation

These questions arose from a question related to K-theory, I am hoping for (big) results from the theory of linear algebraic groups to be helpful. Maybe somebody with a better background there can ...
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37 views

Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
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Spin Representations and Galois correspondence?

I have a vague question regarding the Spin representations. Is there a "quick" way of seeing that $Spin(2n)$ has exactly two irreducible representations which do not factor through $SO(2n)$, and one ...
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What are the non-linear representations of $SO(3,1)$?

The classification of the representations of the Lorentz group $SO(3,1)$ is well known, but the representations are usually expressed in linear form. My question is whether there is a framework to ...
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Iwasawa decomposition of $GL_n\times GL_m$

One knows that any reductive group, in particular GL$_n$, has an Iwasawa decomposition $G=NAK$. Is the Iwasawa decomposition of $GL_n\times GL_m$ simply the diagonal decomposition, $$GL_n\times ...
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When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
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Every representation of a finite group is reducible?

I somehow "proved" that every representation of a finite group is reducible. While I'm fairly sure the error is something silly, I can't seem to place it. Could someone please help me figure out what ...
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Matrix representation and permutation matrices

In order to find the matrix representation associated to a permutation representation I identify each permutation with the corrisponding matrix representation. How can I prove that these matrices ...
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How does Fulton and Harris establish that the differential of a group hom respects ad?

Fulton and Harris, Representation Theory, Section 8.1 (pages 104 - 107 in my copy) is concerned with showing that group homomorphisms $\rho : G \to H$, where $G$ is connected, are completely ...
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limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals: $$\int_{G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & ...
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A question about positive forms on involutive algebras.

A linear form $f$ on an involutive algebra $A$ is said to be positive if $f(x^\ast x)\geq 0$ for every $x$ in $A$. To be useful, this definition requires that is not always possible to write ...
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left regular representation of SU(2)

in Sepanski's book Compact Lie groups, he describes the representation theory of SU(2) as being isomorphic to $\mathbb{N}$ (SU(2) acts irreducibly on the (n+1)-dimensional space of homogeneous ...
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Show that if $V$ is isomorphic to $A/I$ for some left ideal $I$, then $V$ is a cyclic representation of $A$ over $k$

Suppose we have a representation $V$ of an algebra $A$ over a field $k$. Now assume that there exists a left ideal $I$ in $A$ such that $V$ is isomorphic to $A/I$. Now I have to show that $V$ is a ...
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Proving an Irreducible Representation

Consider the representation $$\pi\colon \mathbb R \to GL(\mathbb R^2)$$ by $$\theta \mapsto \text{rotation by }\theta.$$ I want to show that it is irreducible. I start with a non-zero invariant ...
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235 views

Cyclic representation

Suppose $V\neq0$ is a representation of an algebra A. Definition: $v\in V$ is cyclic if and only if it generates $V$, thus $Av=V$. If a representation has a cyclic vector we call the representation ...
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Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe ...
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Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
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Number of prime divisors of element orders from character table.

From wikipedia: It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group ...
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Is every unitary irreducible representation an induced reperesentation?

I have recently read about induced representations and I have the following perhaps naive question about them. Let $G$ be a finite or infinite (Lie) group. Can we construct all irreducible unitary ...
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Induced Representation on $S_4$

(This is my first question so please let me know when something is wrong) I'm a bit confused about the induced representations in Fulton and Harris. I tried exercise 3.23(i) but I'm not sure if I've ...
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Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
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Do all Groups have a representation?

I know that many kind of groups can be represented by matrices; for example: rotation groups can be represented by matrices. Especially all elements of rotation groups can be represented by ...
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Finding inequivalent representations in a given group

I am studying characters of representations and how number of conjugacy classes is same as the number of inequivalent representations in a group. However, my question is, how do we actually find all ...
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Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
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dimension of the intertwiner (homomorphism) between equivalent irreducible representation is 1

How do I show this using the Schur's lemma? (Schur's lemma). Let $\phi, \rho$ be irreducible representations of $G$, and $T \in Hom_G(\phi,\rho)$. Then either $T$ is invertible or $T = 0$. ...
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Every representation of a finite group is completely reducible

Is this equivalent to saying that a representation is diagonalizable matrix in matrix form?
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Maximal Kostka Numbers

Let $\lambda\vdash n$ be a partition of $n$ and assume that $\lambda$ has $k$ parts. Then let $\mu$ run through all the other partitions of $n$ and consider the Kostka-number $K_{\lambda,\mu}$. Can ...
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Dynkin diagram automorphisms and weights

Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
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Indecomposable representations of Lie algebra

Let $\mathfrak{g}$ be the nonabelian $2$-dimensional complex Lie algebra. It can be generated by two independent vectors $e_1,e_2$ such that $[e_1,e_2]=e_1$. Thus, $\mathfrak{g}$ is solvable and it ...
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Computing quotient representations and Hom set fort wo representations

Consider the representation $M$ defined by We want to find all subrepresentations quotient representations of $M$, and $\mathrm{Hom}(M,N)$, where $N$ is a representation with $N \cong M$. I put B ...
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Finding a basis for $sp(4,\mathbb{C})$ and related basis.

Let $$L = so_4(\mathbb{C})= \{x \in End(\mathbb{C}^4)|^txS + Sx = 0 \} \text{ where }S = \left(\begin{array}{cc} 0 & I_2 \\ -I_2 & 0 \end{array}\right)$$ Letting $x = \left(\begin{array}{cc} ...
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Reduced norms of matrix algebras

I'm trying to understand a few basic notions on the reduced norm of division algebras, and more specifically the relation between the norm of an algebra and the norm of algebras similar to it. ...
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Properties of Group representations, duality and the derived subgroup

I am trying to understand why 1) all finite-dimensional complex representations $V$ of $G$ are self dual, and 2) How the derived subgroup $[G,G]$ is a union of particular conjugacy classes. My ...
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Question on GL(n,F) representation

Let A be the group of all invertible n x n matrices over F, A+/- the subgroups of all upper/lower matrices. F^n as an A-module is irreducible? Is this because F^n has only one orbit under A? Why is ...
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The indecomposable projective A-modules

Let Q be the quiver bound by $αβ = 0$, $γδ = 0$. The indecomposable projective A-modules are given by where $A=KQ/I$. This an example in Assem-Simson-Skowronski book (Elements of the ...
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quasi-split algebraic group

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...