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

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Nice parameterization of linear complex structures on the real plane?

A linear complex structure on a real vector space $V$ is an endomorphism $J$ such that $J \circ J=-\mathrm{id}$. What do all the linear complex structures on $\mathbb{R}^2$ look like? If we let $$ ...
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227 views

Schur -Weyl duality for $sl_2$ and $S_n$

$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts ...
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184 views

Irreducible Representations and Maschke's Theorem

$\mathscr L(V,W)^G = \mathscr L(V_1,W)^G \oplus...\oplus\mathscr L(V_k,W)^G$ and for each irreducible represtation of G on a space W, the number of $j\in (1,...,k)$ for which $V_j \cong W$ is ...
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118 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
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350 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
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121 views

What theorems/examples will make me really understand representation theory?

Okay, so I've been through some basic results on representation theory. I've gone over the proof of Burnside's $pq$ theorem using characters. I've also read though the basics of Lie groups and ...
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67 views

elementwise conjugate but not conjugate homomorphisms

Does there exist a finite group $G$ and two group homomorphisms $\rho_1,\rho_2:G\to PGL(2,\mathbb{C})$ such that (i) For all $g\in G$ there exists $M=M(g)\in PGL(2,\mathbb{C})$ such that ...
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74 views

If $f$ is an irreducible representation, what can we say about $g:x\mapsto f(x^{-1})$?

Let $G$ be a finite group, $K$ a field which characteristic does not divide the group order and $V$ a $K$ vector space. Suppose there is an irreducible representation $f: G \rightarrow GL(V)$, $x ...
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64 views

Locally finite dimensional representations of a Lie algebra

A representation $V$ of a Lie algebra $\mathfrak{g}$ is "locally finite-dimensional" if $\dim U(\mathfrak{g}) v < \infty$ for every $v \in V$. I want to show that this condition holds if and only ...
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57 views

Representation of non-Abelian, dimension 2 Lie algebra

Let $k$ be a field and $\mathfrak{g}=kx\oplus ky$ with $[x,y]=y$. Show that $\rho(x)=t\,\frac{d}{dt}$ and $\rho(y)=t\cdot$ (mult. by $t$) define a representation $\rho:\mathfrak{g}\to ...
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171 views

Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
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44 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
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1answer
63 views

Matrix of a representation given a decomposition

I am having a course about group representations and I saw this today: If $T : V \rightarrow V$ is a linear transformation and $B$ is a basis for $V$ , then we shall use $[T]_B$ to denote the ...
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149 views

Are “FG-module characters” sometimes used, too?

I am only beginning my study of group representations and characters. So far I have already encountered the regular group algebra $FG$. Although in an FG-module the multiplication is only defined for ...
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140 views

Hermitian conjugation and representations of the Lorentzian Clifford algebras

The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
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139 views

Kernel of induced representation

Let $G$ be a group and $H\leq G$ be a subgroup of $G$ with index $n$ and $\mathbb{k}$ be a field with characteristic 0. Let $V$ be a $m$-dimensional $\mathbb{k}$-vector space. For a representation ...
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118 views

Existence of a finite group having a certain kind of 2-dimensional representation.

Is there a finite group $G$, an element $c$ of order 2 in $G$, and an irreducible 2-dimensional complex representation $\rho$ of $G$ such that all the following are true: 1) $\rho(c)$ has trace zero ...
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45 views

Prime divisors of irreducible degrees

I would like to ask that : assume that $N$ is a normal subgroup of a finite group $G$, if $p$ is a prime divisor of $\chi(1)$ for some $\chi\in Irr(N)$, does it imply that $p$ divides $\varphi(1)$ for ...
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225 views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
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54 views

check subgroups using representation theory

I am not sure if this is the right place to ask such a question? But I will give it a try here. Given two groups $G,H$, there is a problem we encounter very often: whether $H$ is isomorphic to a ...
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168 views

Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
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69 views

Existence of a Lie subgroup

Let $G=SU(k)\times T^1$, $S$ a subgroup of the center of $SU(k)$ ($Z(SU(k)\cong \mathbb{Z}_k$) and $\eta$ a homomorphism from $S$ into $T^1$. Suppose $(S, \eta)$ denotes the subgroup of $G$ contains ...
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289 views

cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
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1answer
72 views

the index in representation theory

I'm studying quantum field theory. In my text book, generators of compact groups are normalized by $\rm{Tr}$$(T^aT^b)=\frac{1}{2}\delta^{ab}$. However, the index $T(R)$ is defined by $\mathrm{Tr} ...
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296 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
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201 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
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181 views

Product with primitive central idempotent in a complex group ring

Let $G=S_n$ be the permutation group on $n$ letters and $e\in\mathbb C[G]$ a central, primitive central idempotent. Let also $f\in\mathbb C[G]$ be central. I am reading this paper where the author ...
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128 views

irreducible induced representation hom(V,W)

If $ \rho: G \to GL_{\mathbb{C}}(V)$ and $ \sigma: G \to GL_{\mathbb{C}}(W)$ are irreducible representations, is it necessarily true that the induced representation $G \to GL_{\mathbb{C}}(Hom(V,W))$ ...
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79 views

Simple faithful $KA$-modules over finite fields

Let $V$ be a vector space of dimension $n$ over finite field $K=\mathbb{F}_q$ and let $A$ be an abelian group such that $V$ is simple, faithful $KA$-module. Then $A$ is cyclic. Moreover, for every ...
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172 views

Isomorphism types of simple $KG$-modules with $G=S_3$

I am trying an exercises on determining all isomorphism types of simple $KG$-modules with $G=S_3$, the symmetric group of degree $3$. If $K$ is algebraically closed then we can use the following ...
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258 views

Uniqueness of Hermitian inner product

Let V be an irreducible representation of a finite group G.How to show that up to scalars,there is a unique Hermitian inner product on V preserved by G. i know of how to get an inner product. but i ...
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59 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
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52 views

Representation and the central of a algebra

Suppose $A$ is an algebra over a field $k$. I look to the following exercise: Show that if $V$ is an irreducible finite dimensional representation of A then any element $z\in Z(A)$ (with $Z(A)$ the ...
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202 views

Orthonormal basis of Cartan subalgebra relative to Killing form

I'm trying to understand a step in a proof: Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let ...
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1answer
242 views

Projective indecomposables versus general indecomposables

Given a finite dimensional algebra, what is the exact relation between the indecomposable projective modules, and a general indecomposable module? In the case of an oriented quiver without cycles for ...
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79 views

Characters of subrepresentation

Given an algebra $A$ with finite dimensional representation $V$ with action $\rho$, I want to prove the following statement: If $W\subset V$ are finite dimensional representations of A, then ...
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59 views

Problem about decomposition of modular space into eigenspace

It is said we can use the operator $$ \pi_\chi=\frac{1}{\phi(N)}\sum_{d\in\mathbb{Z}_N^*}\chi(d)^{-1}\langle d\rangle $$ to project function in $\mathcal{M}_k(\Gamma_1(N))$ into the $\chi-$eigenspace ...
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92 views

Connection between $GL(\mathbb{Z}_p)$ and $GL(\mathbb{F}_p)$

Suppose there is a finite group $G$. Is there a connection between indecomposable representations over $\mathbb{Z}_p$ and $\mathbb{Z}/p \mathbb{Z}$. I know what to do if $G$ is cyclic. But if not? If ...
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135 views

When are all irreps left ideals in the group algebra and generated by idempotents?

Let $A$ be an associative algebra. I am wondering under what conditions we can get all irreducible representations of $A$ as left ideals $A\cdot e$ with $e\in A$ an idempotent. This is certainly the ...
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112 views

If $K[Q_8]\cong K[D_8]$, char $K=p$ odd, $p=?$

Denote $Q_8$ to be the quaternion group, and $D_8$ to be the dihedral group with order 8, then we know that the group algebra $\mathbb{C}[Q_8]\cong \mathbb{C}[D_8]$ since $Q_8$ and $D_8$ have the same ...
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263 views

Distinguishing the two irreducible representations of odd-dimensional complex Clifford-Algebras

The complex Clifford algebra $A$ of a complex, non-degenerate quadratic space $(V,q)$ of odd dimension $2k+1$ admits up to isomorphism exactly two non-trivial, irreducible and finite-dimensional ...
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69 views

A group of linear isomorphisms of $\mathbb C^n$ must have an invariant subspace

Let $G$ be a finite group acting linearly on $\mathbb C^n$, and suppose that $|G| < n^2$. I am trying to show that there is a nonzero invariant subspace $W\subset\mathbb C^n$, i.e. $g(w) \in W$ ...
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113 views

Do field automorphisms of a character imply outer automorphisms of the group?

Apologies for the imprecise wording of the title. In studying the basic representation theory of finite groups, I've been struck by a pair of phenomena present in every example I've worked with but ...
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124 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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Young tableaux: evaluate action of permutation

Consider the irreducible representation $V$ in the symmetric group $S_5$ corresponding to the Young diagram (these are meant to be boxes): $$[\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;$$ (a) List ...
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132 views

Invariant inner products on infite-dimensional representations

Let $G$ be a compact group and let $V$ be it's continuous representation. It is well known that if $V$ is finite-dimensional, then there is an $G$-invariant inner product on $V$. I haven't found a ...
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238 views

Is there formula name and proof for this theorem ? ( guess it's called Burnside character formula)

The formula answers: how many tuples $(\sigma_1,\sigma_2,\dots,\sigma_n)$ of elements of a given group $G$ such that (1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class. (2) ...
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327 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
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Understanding the proof of Schur-Weyl Duality

I am teaching myself representation theory on $GL(V)$ and $S_n$ using my friend's lecture notes, and have reached a proof of the Schur-Weyl Duality theorem; on reading through I'm struggling to make ...