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

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A family of commuting endomorphisms is semisimple if each element is semisimple

If $\phi : V \rightarrow V$ is an endomorphism of a finite-dimensional (say real) vector space, $\phi$ is called "semisimple" if any $\phi$-invariant subspace of $V$ has a complimentary ...
4
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1answer
57 views

Doubt: “A group representation is exactly like a module over the group ring”

It is traditional to say that a representation of a group $G$ over a field $F$ is "exactly like" a module over the group ring $F[G]$. I think it is inaccurate. I think a module over $F[G]$ encodes ...
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1answer
50 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
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30 views

Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
3
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1answer
39 views

Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
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2answers
60 views

Application of representation theory

I often read that one can use representation theory in the field of quantum physics or for the analysis of symmetries in physics or chemistry. Unfortunately I coundn't find a concrete example for ...
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98 views

Definition of Verma modules

I have a question regarding different (but equivalent!?) definitions of Verma modules of semisimple Lie algebras: Let F be a field and denote the following: $ \mathfrak{g}$ , a semisimple Lie ...
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2answers
90 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
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1answer
73 views

Exterior power respects $G$-action

Basic setting: Let $V$ be a $k$-vector space of finite dimension and $V^*$ its dual space. Let $\bigwedge^n V$ denote the $n$-th exterior power of $V$. Now the canonical pairing $$V \times V^{*} ...
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1answer
136 views

Direct sum and tensor product of two representations of a group

Our lecturer gave us a hard exercice to go further in group theory (we stopped at group actions) : Let G be a group, V and W complex vector spaces and $\rho_1 : G \mapsto GL(V) $ be a group ...
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18 views

Finding an orthonormal basis for a gl(3) module

I'm trying to find an orthonormal basis for gl(3)-module V(ε1-ε3), where ε1-ε3 is the weight (1,0,-1) of the highest-weight vector. Using Gelfand-Tsetlin (/Zetlin/Zeitlin) patterns, I'm at the point ...
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1answer
28 views

Regular module endomorphisms into itself

Let $k$ be a field and let $A$ be an algebra over $k$. Denote by $End_A (A)$ the set of all endomorphisms of the regular $A$-module $A$ into itself. Fix $a \in A$, and define the A-module homomorphism ...
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4answers
162 views

Representation Theory book other than Fulton's

Fulton/Harris's book on representation theory seems to be the "definitive" introductory text on the subject. But is there perhaps a lower level introduction to the subject? Most of the very first ...
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79 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = ...
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0answers
30 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
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52 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
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39 views

Volume of a ball for SO(n).

Let us equip the special orthogonal group $SO(n)$ with a normalized Haar measure $\theta_n$ and let $G_r$ be the subset of rotations $\Omega$ which differ from the identity by (sufficiently small) $r$ ...
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65 views

Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
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22 views

A question about Cayley-Chow forms

I'm reading some papers about $k$-stable theory and I have a question about Cayley-Chow forms. Maybe this question looks silly. Let X be a variety of $\mathbb{P}^N$ with dimension n and degree d. ...
2
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1answer
139 views

Haar measure of $SO(3)$ obtained from $SU(2)$

I am reading 'Analysis on Lie groups, an introduction' by Faraud and don't understand the following statement … the image by the map Ad of the Haar measure $\mu$ of $SU(2)$ is equal to the ...
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1answer
221 views

Representation Theory of the Dihedral Group $D_{2n}$

So I'm pretty new into Representation Theory having so far covered only a couple of example sheets. I'm thinking about the following question: Suppose we have the group $D_{2n}$ (for clarity this is ...
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1answer
49 views

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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1answer
44 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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42 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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1answer
41 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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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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1answer
299 views

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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1answer
35 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 ...
3
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1answer
217 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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21 views

Representation of a G on $L^2(\mu)$

Let $G$ be a group acting on a locally compact Hausdorff space $S$, then $G$ also acts on functions of $S$ by $$[\pi(x)f](s)=f(x^{-1}s)$$I'm trying to show that $\tilde\pi:G\to L^2(\mu)$ defines as ...
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26 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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231 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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1answer
61 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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43 views

Multiplicity free module and centralizer of subalgebra

This is related to a previous question I posted here. Let $A$ be a semisimple algebra and $B\subset A$ be a subalgebra. Define $$C_A(B) = \{a\in A | ab=ba \quad \forall b\in B\}$$ Let $V$ and ...
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24 views

Characters and ''twisted dimensions''

Can someone shed some light at this part of wiki article about character theory: http://en.wikipedia.org/wiki/Character_theory#.22Twisted.22_dimension ? It kind of just stands there without any ...
3
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1answer
74 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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1answer
111 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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$\text{Hom}$ 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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1answer
31 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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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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1answer
25 views

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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26 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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56 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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1answer
195 views

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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137 views

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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1answer
114 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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32 views

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 ...
2
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1answer
149 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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1answer
39 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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1answer
134 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 ...