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

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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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Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
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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 ...
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Finite dimensional representations of semi-simple Lie algebras

I've been trying to understand the proof of the following statement: An injective map of $\mathfrak{g}$-representations of a semisimple Lie algebra splits. I'm supposed to show this considering the ...
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Why can we write the weights of a representation in terms of the simple roots?

I'm currently trying to get my head around the fact that we can write the weights of any representation in terms of the simple roots of the algebra. Is there any, not too-technical, explanation? I ...
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centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
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Complex irreducible representation of solvable lie algebra

How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1? What I know is that one ...
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What the's contradiction in showing the regular representation is indecomposable in characteristic $p$?

Suppose $G$ is a nontrivial $p$ group, and $F$ is a field of characteristic $p$. The group ring $FG$ is a module over itself affording the regular representation $g\cdot g_i=gg_i$. Why is $FG$ ...
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Irreducible representation of $C^*(D_\infty)$

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$ Ultimately, I'm interested in finding a ...
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Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
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Is the unreduced Burau representation completely reducible?

To be specific, my question is about specializations $\beta \colon B_n \to GL_\text{n}\left( \mathbb C \right)$ of the unreduced Burau representation given by \begin{array}{cr} \beta \left( \sigma_{i} ...
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How can I find the Weights of a Subalgebra

I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group. For example, if we start with the 16-dimensional representation of ...
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Representations of group $G=\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} $

I need to find all in-equivalent irreducible representation of a group $G \equiv \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} $ I know that $\mathbb{Z}/p\mathbb{Z}$ is a cyclic finite group. ...
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One-dimensional submodules of $\mathbb{C}^4$; direct sum of submodules.

I'm having some trouble understanding my lecture notes. I need to find a one-dimensional submodule $U$ of $\mathbb{C}^4$. Is $u=1+x+x^2+x^3$ valid? My reasoning: because $ux = x + x^2 + x^3 + 1 = u ...
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Completing the character table

Let $G=\{a,b|a^6=1,a^3=b^2,b^{-1}ab=a^{-1}\}$ be a group of order 12. $G$ has 6 conjugacy classes $$\{1\},\{a^3\},\{a,a^{-1}\},\{a^2,a^{-2}\},\{b,a^2b,a^4b\},\{ab,a^3b,a^5b\}.$$ Name them ...
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Degenerate Hecke algebra

Problem: Define $H_2$, the degenerate Hecke algebra generated by $X_1, X_2, s$. Let $a,b\in \mathbb{Z}$. Consider the following matrices defining an action of $X_1, X_2$ on a 2-dimensional space ...
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How to find subgroup centralizer?

Having found that a group G has a normal Sylow 2-subgroup P, how do I find $C_P(g_i)$, where $g_i$ is a conjugacy class representative? I have the character table, and have previously found ...
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What is the group $C_2^4$?

I'm trying to do a problem which asks me to show that a certain group is isomorphic to $C_2^4$. What is this group?
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Natural representation of GL(V)

Let $V$ be a vector space over some field. Is the natural representation $V$ of the group $GL(V)$ irreducible? Is it absolutely irreducible? Is the span of $GL(V)$ inside $End(V)$ all of $End(V)$? I ...
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Complete the character table

Let $G$ be a group of order 10 having 4 conjugacy classes and the above character table. Complete the table. It's easy to get the degrees of the remaining 2 irreducible characters to ...
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Why is it called the category of representations?

Let $A$ be a (Hopf) algebra. Let $C_A$ be a category whose objects are $A$-modules and whose morphisms are $A$-linear maps. This category is called "the category of representations". My question is: ...
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Show that every finite simple group G has a faithful irreducible representation

A representation $ \rho $ : G $ \rightarrow $ GL(V) is faithful if ker($ \rho $)={$ e $}. A representation is irreducible if it contains no proper invariant subspaces G is a simple group its ...
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Conjugacy classes of the representatives

I have just started reading about chracter theory. I am confused on few things like how can we find the order of a group G and Centralizer of element $g_i$ i.e $C_G(g_i)$ of a group G of the ...
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Extension of the duality of the space of distributions over $X$ locally profinite space

I have two questions on a definition that appears in the book "Répresentations des groupes réductifs $p$-adiques" by David Renard (http://www.math.polytechnique.fr/~renard/Padic.pdf). Let $X$ be a ...
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Decomposition of the Regular Q8 Module

For a worksheet we were asked to find the decomposition of the regular $Q_8$ module into a direct sum of simple modules. This isn't me asking for help on homework though, the problem is that I already ...
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Parametrization of the split orthogonal group O(n,n)

I would like to find or construct an explicit parametrization of the $2m$-by-$2m$ matrix representation of the real indefinite orthogonal group $O(m,m)$ associated to the bilinear form with matrix ...
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Is decomposition of a semisimple Lie algebra unique?

A semisimple Lie algebra is defined to be the sum of simple Lie algebras. But is this decomposition to simple Lie algebras unique? If not can you give an example?
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Pure state on a C*-algebra

Let $\tau$ be a pure state on a C*-algebra $A$, $(\pi_\tau, H_\tau, \eta_\tau)=(\pi,H,\eta)$ be the corresponding cyclic representation of $\tau$, and $\xi$ a unit vector in $H_\tau$ such that ...
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Sum of characters formula

Let $G$ be a finite group and $g\in G$. Let $\{\rho_{i}\}$ be the set of irreducible representations of $G$, where $\rho_{i}: G \rightarrow GL(V_{i})$ and $\chi_{i}$ be the character of $\rho_{i}$. ...
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Supercuspidal representations

If $(\pi,V)$ is a representation of $G=GL_n(F)$ where $F$ is a nonarchimedean local field, and $0 \subset V_2 \subset V_1 \subset V$ is a filtration of $V$ into $G$-invariant subspaces, with $V/V_1$ ...
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On Irreducible Representations of $A_n$

I am seeking some literatures on Representation Theory of $A_n$, the alternating group of $n$ elements. Is there any article discussing the all possible Irreducible representations of $A_n$?
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Why is the trace submodule of $FS_n$ the unique $1$-dimensional submodule?

Suppose $V$ is an $n$-dimensional vector space over a field $F$, with basis $e_1,\dots,e_n$. Then $S_n$ acts on $V$ by the action $\sigma\cdot e_i=e_{\sigma(i)}$, and $V$ is a $FS_n$-module, where ...
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Weights in $\mathfrak{sl}(3,\mathbb{C})$

Let $\mathfrak{h} \subset \mathfrak{sl}(3,\mathbb{C})$ be the set of diagonal matrices. Then for $A = \begin{pmatrix} a_1 & 0 & 0 \\ 0 & a_2 & 0 \\ 0 & 0 & a_3 \end{pmatrix} ...
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A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...
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question on lemma in Bushnell and Henniart, irreducible components of a particular induced representation

I have a question on a lemma that appears in the book "The Local Langlands Conjecture for GL(2)" by Bushnell and Henniart. The setting is as follows: we let $G = GL_2(k)$ where $k$ denotes a finite ...
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Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
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Can the adjoint representation on a discrete centered group merge connected components?

Let $\mathfrak{G}$ be a Lie group with algebra $\mathfrak{g}$ and with in general many connected components but which is either centerless or has at most a discrete center. Then we have an ...
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Show that the action is transitive

$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in ...
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Representation Theory group element as a vector

This is a very introductory level question. I am reading "The Symmetric Group" by Bruce Sagan and I am stuck on a concept. In Example 1.3.4 he talks about the regular representation of the cyclic ...
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Killing form of a reductive symmetric Lie algebra

suppose $(g; , k ,p)$ is a reductive symmetric Lie algebra. i.e. $k$ is a sub-algebra of $g$, $[k,p] \subset p$ , $[p,p] \subset k$ and $g= k \oplus p$. this is actually from Lepowsky and McCllum's ...
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Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...