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

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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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33 views

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 ...
3
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1answer
44 views

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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30 views

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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22 views

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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1answer
30 views

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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26 views

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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1answer
47 views

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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1answer
35 views

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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27 views

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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1answer
38 views

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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35 views

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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15 views

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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28 views

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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1answer
22 views

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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1answer
23 views

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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27 views

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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19 views

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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2answers
54 views

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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1answer
53 views

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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30 views

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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28 views

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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23 views

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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1answer
22 views

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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1answer
29 views

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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1answer
34 views

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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1answer
21 views

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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35 views

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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1answer
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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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1answer
38 views

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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1answer
48 views

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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1answer
34 views

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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1answer
64 views

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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33 views

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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61 views

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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1answer
22 views

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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1answer
19 views

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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39 views

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 $|G| = 8n$, ...
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35 views

How to compute perverse sheaves?

In the video, from 49:00 to the end of the video, there is an example of computing $IC(S, L)$ and equivariant local systems. I don't understand some parts of the computations. Let $X$ be the variety ...
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1answer
28 views

Why Demazure operator is an endomorphism of $\mathbb{Z}[P]$?

Let $P$ be the weight lattice of some Lie algebra. Let $$ \Delta_{\alpha}(u) = \frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}, $$ where $\alpha$ is a root, $u \in P$. In the article, it is said that ...
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1answer
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Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
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Corollary to Maschke's Theorem.

If in Maschke's Theorem, for group ring KG where K is any field s.t. char $K \nmid |G|$, I take G to be finite , then I know Maschke implies that KG will be semisimple so it is isomorphic to a direct ...
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65 views

Expression of character of coset representation

Let $G$ be a finite group with subgroup $H$. Consider the set of all cosets $G/H$ and a corresponding transversal $\{x_1,...,x_m\}$. We now have a representation $\rho$ of $G$ on $\mathbb{C}[G/H]$: ...
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Is trace of regular representation in Lie group a delta function? [duplicate]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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1answer
67 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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1answer
70 views

direct sum and tensor product of representation of lie algebra

Let $(p_1,V_1)$ , $(p_2,V_2)$ representation of a lie algebra $g$ on $V_1,V_2$. I have to prove that: $ i) $ the direct sum $p_1 \oplus p_2$ is a representation of $g$ in $V_1 \oplus V_2$ $ ii) $ ...
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42 views

Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$

Let $G=S_3$ and $H=\langle (12) \rangle < G$. Compute the character $Ind_{H}^G \mathbb{I}$ by computing the character of $\mathbb{C}[G/H]$. I have computed the character using the induction ...
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1answer
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Book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups?

Does anybody have book recommendations for reading A. Okounkov and A. Vershik's approach for complex irreducible representations of symmetric groups? Preferably, I am looking for a book that is ...