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

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Algebra Homomorphism

This is a follow-up to a question I asked here yesterday. It's coming from a (non-examinable) exercise sheet and I really can't get my heard around how this question is posed and to be approached. ...
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49 views

Group algebras, Maschke's lemma and direct sums of matrix algebras

Let $G=\{g_1,g_2,\dots,g_n\}$ be an arbitrary finite group. We consider its representations over $\mathbb{C}$. There is Maschke's theorem which states that each representation of $G$ is a direct sum ...
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50 views

Finite simple nonabelian groups with the same character table

Conjecture: If $G_1$ and $G_2$ are finite simple nonabelian groups, and if $G_1$ and $G_2$ have the same character table, then $G_1\cong G_2$. I am looking for a proof, or at least some intuition. Or ...
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47 views

Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...
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56 views

Direct Sums of Matrix Algebra

This is the first half of the question introduced in Representations of direct sums of matrix algebras Let $A_1, A_2....A_n$ be n algebras with units $u_1, u_2,...u_n$ respectively. Let $A = A_1 ...
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Does every Lie algebra come from commutator of some associative product operation?

Suppose $\mathfrak{g}$ is an Lie algebra. Is it possible to define an associative product operation $\star$ on $\mathfrak{g}$ such that $[A,B]=A\star B - B \star A$ ? If it is not possible to do so ...
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41 views

Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}

Let $k$ be a field and let $A$ be a $k$-algebra. Denote by $End_A(A)$ the set of all $A$-homomorphisms of the regular $A$-module $A$ into itself. Fix $a ∈ A$, and define the $A$-module homomorphism $r_a ...
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367 views

the converse of Schur lemma

I am interested in the converse of the following form of Schur's lemma: Lemma. (Schur) A group G, a $\mathbb{C}$-vector space V and a homomorphism D : G $\rightarrow$ GL(V) is given. Suppose that D ...
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43 views

Square-integrable representations of noncompact groups

In Marc Rieffel's paper "Square-integrable Representations of Hilbert Algebras," he establishes (Corollary 5.12) that a nonfinite discrete group has no square-integrable, irreducible representations. ...
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321 views

Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
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106 views

Faithfulness of adjoint representation of Lie algberas

Are there any simple or useful conditions (necessary & sufficient) under which the adjoint representation lie algebra is faithful ? One sufficient condition is semisimplicity, but perhaps this is ...
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92 views

Consequence of the branching rule of S_n representations

Let $V_\lambda$ be the irreducible $S_n$-representation (a left $kS_n$-module) over a field $k$ of characteristic $0$ associated to the partition $\lambda\vdash n$. By abuse of notation let $S_a$ and ...
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150 views

Existence of an irreducible $L$-submodule

Suppose $L$ is a finite dimensional Lie algebra. Let $V$ be an $L$-module (i.e. $V$ is a vector space which $L$ acts upon). We are assuming that $V$ has a finite dimension. My question is the ...
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80 views

$\text{Hom}$ of irreducible modules and restrictions

This question is in reference to this paper. More specifically it is in reference to the proof of proposition 1.4 on page 8. First a defintion: Let $A$ be a semisimple finite dimensional ...
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57 views

Representations of the Nil-Coxeter algebra

For $i=1,\ldots,n$, let $u_i$ belong to the Nil-Coxeter algebra $\mathcal{N}_n$ which is defined through: \begin{align} u_i^2&=0\\ u_iu_j&=u_ju_i, \ \ \ \ \ \ \ |i-j|\geq 2\\ ...
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20 views

Irreducible unitary representations of a fondamental group.

Let $C$ be a compact Riemann surface with genus $2$. It is well-known that $\pi_1(C) \simeq F/N$, where $F$ is the free group with $4$ elements (say $a_1,b_1,a_2,b_2$) and $N$ is a normal subgroup ...
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Representation in $\mathbb{C}[S_3]$

By $A$ we denote the algebra over $\mathbb{C}$, generated by $y_1,y_2,s$ such that $y_1y_2-y_2y_1=0, s^2-1=0$ and $sy_1-y_2s-1=0$. Could you help me to build a homomorphism $A\rightarrow ...
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28 views

Partial converse to fact about isomorphic finite groups and their representations

If two finite groups are isomorphic then, they have the same irreducible characters (if $G_1\cong G_2$, we must send elements in a conjugacy class of $G_1$ to elements of the corresponding class in ...
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65 views

Nondegenerate representation vs faithful representation

There are two kinds of injection in the representation theory, nondegenerate representation and faithful representation. Does any relation between them? I want to get some good examples to ...
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Group of order 24 with no element of order 6 is isomorphic to $S_4$

Proposition: Given a group $G$ with $|G|=24$ such that $\nexists g\in G$ with $|g|=6$, then $G\cong S_4$. I understand methods you can employ to deduce the number of Sylow $p$-groups in $G$ by ...
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Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
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129 views

Towards a Quantum Peter Weyl Theorem

This is taken from Timmermann's Invitation to Quantum Groups and Duality. Let $(A,\Delta)$ be a *-Hopf algebra and let $\chi:V\rightarrow V\otimes A$ be a corepresentation of $(A,\Delta)$ on a vector ...
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is there a polynomial-form minimal representation for SO(3)?

Is there a minimal local representation for $SO(3)$ such that if $(x_1,x_2,x_3)$ is the representation for some $R\in SO(3)$ then I can write the entries of the 3x3 rotation matrix for $R$ as a ...
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206 views

Finding Idempotents?

I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive ...
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41 views

Finite representations of $\mathbb{C}[x]$

Let $\mathbb{C}[x]$ be the ring of polynomials with complex coefficients, how can I find any finite representation of $\mathbb{C}[x]$?
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170 views

Absolutely irreducible representations of the absolute Galois group of $\mathbb{Q}_p$

Let $p$ be a prime number. Denote by $G$ the absolute Galois group of (a finite extension of) $\mathbb{Q}_p$. Let $\ell$ be a prime number. For $\ell= p$, I guess it is well known that the ...
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41 views

Bounding the inner product in root systems.

Let $R$ be a root system (irreducible if that makes this easier) in the real vectorspace $E$. Let $\lambda$ and $\mu$ in $E$ with $w_0(\lambda)\leq \mu \leq \lambda$ where $w_0$ is the longest ...
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160 views

Schur's lemma and Invariant subspaces of direct sums of irreducible representations

There is a corollary to Schur's lemma which says that : If $V$ is a finite dimensional irreducible complex representation of a group G or Lie algebra and $\phi :V \rightarrow V$ is an intertwining ...
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27 views

Embedding of $GL_n(F)$ inside another matrix groups

We can embed $GL_n$ inside $SL_{n+1}$ easily. is there any other such embedding of $GL_n$ or its subgroups inside any other group of invertible matrices? Thanks in Advance.
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Exterior square of multiset in representation theory

General Setting: In a paper I'm working on, the author uses multisets to describe the representation theory of the cyclic group $G = C_n = <\sigma>$ of ...
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25 views

Proving that representation is differentiable

Let $V$ be a real or complex finite-dimensional vector space, and let $\pi$ be a continuous representation of $(\mathbb{R}, +)$ on V with: $$\pi(t + s) = \pi(t) \pi(s), t, s \in \mathbb{R} \: \: ...
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a exercise in Berkovich‘ book

in Berkovich' book characters of finite groups there is a exercise in page 59. exercise 15, a group G is a Q-group if and only if for any cyclic subgroup Z of G who can tell me how to prove it ? ...
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Indecomposable vector space

Let us define $V$ as a quotient space $\mathbb{K}[t]/(p^m)$, where $p$ is an irreducible polynomial. Condsider the linear operator $\phi\in Hom_{\mathbb{K}}(V,V)$, which sends each $q+(p^m)$ to ...
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what does a “rational structure” mean in algebraic group theory

For an algebraic group $G$ and a finite field $\mathbb F_q$, what does an "$\mathbb F_q$-rational structure of $G$" mean? Is it always related to a Frobenius map? I encountered this while reading ...
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231 views

Reference request for studying Lie group & Lie algebra representations

I am learning representation theory of Lie groups & Lie algebras from the book by Brian Hall. Unfortunately, this does not discuss infinite dimensional representations. Which books should I study ...
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169 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
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Questions about Lusztig's $\mathbf a$-function

In chapter 13 of Lusztig's Hecke Algebra with Unequal Parameters, the function $\mathbf a$ is defined to be $$\mathbf a(z) = \max_{x,y} \deg h_{x,y,z},$$ for any $z$ in the Coxter group, where the ...
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179 views

Wedderburn decomposition of $D_{5}$

I'm wondering how to solve this question. Find the Wedderburn decomposition of $D_{5},$ the dihedral group of order 10, over the field $\mathbb{F}_{3}.$ I have shown that the irreducible ...
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1answer
57 views

Four term exact sequence for Artin algebras

Suppose $\Lambda$ is an Artin algebra, i.e. an algebra finitely generated as $R$-module for a commutative Artin ring $R$, $A$ is a finitely generated left $\Lambda$-module, and $X$ a finitely ...
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Homomorphisms from the additive groups of virtual characters into certain idele groups

This is a question from Frohlich's book 'Galois Module Structure of Algebraic Integers', Ch.1. Let $K$ be a number field and $\Omega_K=\text{Gal}(K^c/K)$ where $K^c$ is the separable closure of $K$. ...
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153 views

Interesting problems using group/representation theory

I've been going through this representation theory lecture notes, and I've found the ''hungry knights'' problem very interesting. Do you know any interesting problems similar to that one? To define ...
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96 views

How to find the representation of Lie algebra

I read a book about the Lie algebra, but I really don't understand the calculation of $ad(X)$. For example, we have a Lie algebra of bases: $$e_1=\left[\begin{array}{cc}1 & 0\\0 & ...
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39 views

Does the induced representation always contain a non-trivial representation

Let $H$ be a proper subgroup of a finite group $G$ - not normal. Does $Ind_H^G 1$ contain a non-trivial representation? The Frobenius character formula was my original approach, but I can't rule out ...
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136 views

Invariant Subrepresentation of Induced Representation

In what follows we let $G$ be a compact Lie group and $H\lneqq G$ a closed subgroup. We denote by $\mu_{G},\mu_{H}$ the respective Haar measures. Let ...
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How can these be the weights of the adjoint representation?

This is perhaps a stupid question. We consider $G =\text{SU}(3)$ and $\pi : G \to \textrm{GL}(\mathfrak{g})$ the adjoint representation that sends $g \in G$ to $Ad_g$ that acts on the Lie algebra ...
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26 views

The character of $\mathbb{C}[S_n]b_\lambda$

Let $b_\lambda = \sum_{g \in Q_\lambda} (-1)^{sgn(g)}g$, where $Q_\lambda$ is the subgroup of elements that permute the numbers in columns, $\lambda$ a Young diagram corresponding to partition ...
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Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
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self-adjoint subalgebras of matrix algebra

Is there any classification theorem for the self-adjoint matrix subalgebras of $M_n(\mathbb{C})$ the algebra of $n \times n$ matrices over $\mathbb{C}$ ?
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Action of a lie group on its lie algebra via the adjoint representation

I am a physics undergrad. The adjoint action of a group on itself is $\operatorname{Ad}: G \times G \to G$ is defined to be $\operatorname{Ad}:(g,h) \to g^{-1}hg$. The adjoint representation of the ...
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Reference request for Lorentz group and unitary representations

More precisely, I often read or listen that Lorentz group has not (non trivial) unitary finite dimensional irreducible representations because it is not compact. Now, I know that the "converse" part ...