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

learn more… | top users | synonyms (1)

0
votes
0answers
16 views

Combining infinitesimal generators of diferent dimensions

I am reading a paper about ways in which you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting ...
0
votes
0answers
17 views

$U(1)$ generators of $SU(2)$

I wanna get $U(1)$ out of $SU(2)$. I know for example that this can be done using the diagonal Pauli matrix, but I wonder if there are more $U(1)$-s in $SU(2)$. So, which are the all the ways in ...
0
votes
0answers
11 views

How to transform the following direct product of the group representations?

Let's have 4-vector $A_{\mu}$ which transforms as $\left(\frac{1}{2}, \frac{1}{2}\right)$ representation of the Lorentz group. So the product $A_{\mu}B_{\nu}$ refers to the direct product $$ \tag 1 ...
0
votes
0answers
11 views

Properties of characters that remain true for infinite compact groups

Which properties of irreducible characters for finite groups still hold for infinite (compact) groups? In particular, is it still true that the irreducible characters form a basis for the space of ...
0
votes
0answers
36 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
0
votes
0answers
19 views

Finite-dimensional commutant of unitary representation

Suppose that we are given a unitary representation $\rho\colon G\to\mathcal{U}(\mathcal{H})$ of some group $G$, that moreover satisfies $$\dim\rho(G)'=n<\infty,$$ i.e., the space of intertwining ...
0
votes
0answers
11 views

A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$

Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to ...
0
votes
0answers
44 views

Definition of a splitting field of a finite group

This is a basic question from the journal 'Mathematische Zeitschrift' 208 (1991) page 243. Let $K/F$ be a finite Galois extension of number fields and $G={\rm Gal}(K/F)$. Also let $L$ be any number ...
0
votes
0answers
30 views

How do non-semisimple modules over $\mathbb C\mathrm{GL}_n(\mathbb C)$ look like?

[Separated from another question] Can you give an example of a non-semisimple module over $\mathbb C\mathrm{GL}_n(\mathbb C)$? (Preferably one without direct summands, i.e. an indecomposable module) ...
0
votes
0answers
6 views

Augmented Positive Definite G-Invartiant Hermitain Form

We know that we can create a G-invariant positive definite Hermitian form on V by picking a arbitrary positive definite Hermitian form $\{ , \}$ and applying the averaging process: $$<v,w> = ...
0
votes
0answers
29 views

Suppose that V is a 2 D FG module and that there exists g,h in G, v in V such that (gh).v is not equal to (hg).v.

Show that V is irreducible. I think I have possibly proved this by contradiction, but I just wanted to make sure my answer is thorough enough. The question gives the hint to use Maschke's theorem: Let ...
0
votes
0answers
42 views

Obtaining representations of the symmetric group

Consider the following permutation representations three elements in $S_3$: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = ...
0
votes
0answers
42 views

graded k-algebras

Suppose we are given a positively graded $k$-algebra $A$ such that $A_i=0$, for $i\neq 0,1$ (i.e $A=A_0\oplus A_1$). Suppose furthermore all $A_i$ are finite dimensional as $k$ vector spaces and that ...
0
votes
0answers
32 views

Suppose f:V->W is an FG homomorphism. Show that ker(f) is a submodule of V.

I think I have done this I just want to check that this is enough to show that it is true, given it is worth 12 marks. First we recall from Linear Algebra that her(f) is a subgroup of V. Suppose that ...
0
votes
0answers
35 views

Show that $U=\langle v_1-v_2,v_2-v_3,…,v_{n-1}-v_n\rangle$ is a sub module of $V$.

$G=S_n$ and $V$ is a vector space over a field $F$, with basis $\{v_1,....v_n\}$, then $V$ is an $FG$ module with action defined by setting $g · v_i = v_{g(i)}$ for all $g\in G$ and $1 < i < ...
0
votes
0answers
11 views

Closed form for 3j-symbol ratio

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch Gordan coefficients. ...
0
votes
0answers
20 views

Why $\widehat{G^{\mathbb{C}}}$ can be identified with the space of highest weights

Let $G$ be a compact connected Lie group and $G^{\mathbb{C}}$ be the complexification of Lie group $G$ and we denote $\widehat{G^{\mathbb{C}}}$ the set of isomorphism classes of irreducible rational ...
0
votes
0answers
18 views

How to decompose a representation of $so(n)$ into representations of a subalgebra

In some cases, it is possible. For instance the representation $16$ of $so(9)$ decomposes as $8_c+8_s$ of $so(8)$. Now I would like to do the same with representations of $so(8)$ into a sum of ...
0
votes
0answers
80 views

Prove that every 2 dimensional FG-Module with gh not equal to hg is irreducible.

Basically the questions is as follows. Suppose that $V$ is a 2-Dimensional $FG$-Module where F=The complex numbers and that there exists $g,h$ elements of $G$ and $v$ an element in $V$ such that ...
0
votes
0answers
27 views

Weyl Character Formula to find $M_\lambda(\mu)$

In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the ...
0
votes
0answers
33 views

Equivalent representations of $\mathfrak{sl}_2$

Hello I have a question about the equivalence of two representations of the Lie algebra $\mathfrak{sl}_2$. The first representation is $(ad,\mathfrak{sl}_2)$ the adjoint representation with map ...
0
votes
0answers
17 views

derived equivalence of coalgebras

let $C$ and $D$ be two coalgebras over a field, $C$ and $D$ are called derived equivalent if the derived categories $D(C-comod)$ and $D(D-comod)$ are equivalent as triangle categories. if $C$ and ...
0
votes
0answers
43 views

Representation of $\mathbb C$

Let $M_2(\mathbb R)$ be the ring of $2\times 2$ matrices with real entries. Its group of multiplicative units is $GL_2(\mathbb R)$, consisting of the invertible matrices in $M_2(\mathbb R)$. (a) ...
0
votes
0answers
16 views

Multiplicity of the tensor product of $A_n$-modules

In Coutinho, A Primer of Algebraic D-Modules, Theorem 13.4.1 (p. 128), I read: Theorem. Let $M$ be a finitely generated $A_m$-module, $N$ a finitely-generated $A_n$-module. $d(M ...
0
votes
0answers
25 views

How to show that $Ind_K^G \mathbb{C}_\chi$ is natrally isomorphic to $\mathbb{C}[G]e_{\chi}$?

Let $K \subset G$ be finite groups and $\chi: K \to \mathbb{C}^*$ be a homomorphism. Let $\mathbb{C}_{\chi}$ be the corresponding 1-dimensional representation of $K$. Let $$ e_{\chi} = \frac{1}{|K|} ...
0
votes
0answers
27 views

Holomorphic representation of $G$ on vector space $V$.

Let $G$ be a Lie group. What is the definition of holomorphic representation of $G$ on vector space $V$.
0
votes
0answers
36 views

Weights of group in terms of it fundamental weights

how can I found the weight of a group G ( I find the fundamental weights, but I don't know how found the linear combination of fundamental weights that give me the weights ). so we found the ...
0
votes
0answers
36 views

Show $\mathbb F_{p}[x]/(x^{p}-1)$ is indecomposable as a representation of $\mathbb Z/ p\mathbb Z$

Let $R=\mathbb F_{p}[x]/(x^{p}-1)$. $R$ has both ring and vector space structure. I am trying to show that, given a representation $\rho : \mathbb Z/ p\mathbb Z\rightarrow GL(R)$, any invariant ...
0
votes
0answers
20 views

subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
0
votes
0answers
14 views

Invariants of exterior power of Lie algebras

Let $\mathfrak{g}$ a simple finite dimensional Lie algebra, and consider $$\bigwedge(\mathfrak{g}\oplus\mathfrak{g}).$$ Let $\{e_i\}$ and $\{f_i\}$ be dual basis of $\mathfrak{g}$ with respect to the ...
0
votes
0answers
19 views

Invariants of representation of simple Lie algebras.

Let $\mathfrak{g}$ a finite dimensional simple Lie algebras and let $V$ a representation of $\mathfrak{g}$ such that $$V=\bigoplus_{i,j\in I}(L(\mu_i)\otimes L(\mu_j)).$$ Where $L(\mu_i)$ is the ...
0
votes
0answers
22 views

Clifford Theory Proof Representation Theory

I have a few questions about the proof of clifford theory and basic applications. Note: I am using that for class functions $\chi_1,\chi_2$ we have that $\langle \chi_1,\chi_2^x\rangle =\langle ...
0
votes
0answers
32 views

Irrep dimensions of non semisimple Lie algebra

I'm mostly interested in Lie algebra "numerology". The book "Birdtracks" and the website http://www-math.univ-poitiers.fr/~maavl/LiE/form.html answered me everything on irrep dimensions for semisimple ...
0
votes
0answers
17 views

What is the relationship between the spaces $\mathscr K (G)$ and $L^2(G)$?

The context is that $G$ is a locally compact Hausdorff group, $\mathscr K (G)$ is the space of continuous compactly-supported functions $G \to \mathbb C$ equipped with the inner product $(f|g) = \int ...
0
votes
0answers
52 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
0
votes
0answers
58 views

The annihilator of finitely generated modules over PID

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Suppose there exists prime ideal $p$ and integer $i$ such that $p^i=\operatorname{Ann}(M)$. Then prove: (1) there ...
0
votes
0answers
75 views

Irreducible representation of SO(3)

When we look at irreducible representations of $SO(3)$, we find the denoted by physicists $D_j$ of dimension $2j+1$, where $j \in \mathbb{N}$. We can see the vector space where the $D_j$ act as the ...
0
votes
0answers
53 views

Irreducible representations of groups of order $pq$: induction from normal subgroups

Consider a group $G$ of order $pq$ ($p$ and $q$ are distinct primes and also $p<q$). It is easy to show that the dimension of each irreducible representation of $G$ is $1$ or $p$. Also, it can be ...
0
votes
0answers
47 views

SU(2) representations and differential equations in physics.

I studied that $SU(2)$ has a spin $j$ representation $U_j$on a homogeneous space of 2 variables with dimension $2j+1$. Now I am trying to understand the following sentences. Suppose $\phi: ...
0
votes
0answers
54 views

Irreducible and regular representation

I was working on representations of $S_3$ and thought about the following problems: The symmetric group $S_4$ acts on $\mathbb{C}^{4}$ by permuting coordinates. Decompose this representation into ...
0
votes
0answers
38 views

Character tables of finite groups in positive characteristic - not the modular case

Let $G$ be a finite group and let $F$ be an algebraically closed field of characteristic $p$ with $p \nmid |G|$. So, the group algebra $F[G]$ is semisimple. What are the techniques to compute the ...
0
votes
0answers
18 views

Understanding analytic construction of induced representation

I'd like to get some intuition for analytic construction of induced representations as described on Wikipedia. Algebraic construction also described there is much more intuitive and clear to me, but ...
0
votes
0answers
67 views

Representation theory

I am trying to study representation theory from the book Algebra by Artin. I came across the following problem which seemed interesting: Prove that the linear operator $T=\sum_{g\in C} \rho_{g}$ is G ...
0
votes
0answers
47 views

Is there any groups $G$ with the property $(*_d)$?

Let $G$ be a finite group of even order has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$): $(*_d)$: There ...
0
votes
0answers
48 views

Show that we have an algebra homomrohpsim

I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field. I suspect it's really easy but I don't know what to do. Is ...
0
votes
0answers
50 views

A corollary to the Wedderburn-Artin theorem.

Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field $F$, then $$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k), $$ where ...
0
votes
0answers
37 views

how to see that Group Orthogonality Theorem is correct

For the group representation theory, how did people discover the Group Orthogonality Theorem? ...
0
votes
0answers
12 views

Integration of Representative Function of a compact Lie group

Let $G$ be a compact Lie group. Let $f$ be a representative function with representation $V$ such that $V$ has no trivial representation summand in its decomposition. Why is it that $\int_G f = 0$?
0
votes
0answers
28 views

Does locally defined $L(H)=u(1)\oplus su(n)$ necessarily imply global $H=U(1)\otimes SU(n)$?

Consider local gauge transformation groups (defined on spacetime manifold). Does local condition $L(H)=u(1)\oplus su(n)$ necessarily imply global $H=U(1)\otimes SU(n)$? Why? Sorry, I'm not sure ...
0
votes
0answers
41 views

How to understand the direct product of group representations (on example)?

The algebra of the Lorentz group $SO(3, 1)$ can be represented as direct product of $SU(2)$ or $SO(3)$ algebras. How to understand this statement?