3
votes
0answers
47 views

Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
3
votes
1answer
60 views

Transitivity of representation induction

Let $K\subset H\subset G$ be some triple of finite groups and $T: K\longrightarrow \mathrm{GL}(V)$ - some representation f $K$. We are to prove the transitivity of induction: $Ind_K^G(V)\simeq ...
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
1answer
39 views

A Tensor Product Identification

Hi folks just looking at Timmermann's "Introduction to Quantum Groups and Duality" and looking at the algebra of representative functions on a compact topological group $G$. I take off in Example ...
1
vote
1answer
87 views

Equivalence of tensor reps & tensor products of reps

Let a finite-dimensional vector space $V$ over $\mathbb R$ or $\mathbb C$ with dual $V^*$ and a group $G$ be given. Let $\rho:G\to\mathrm{GL}(V)$ be a representation, and let $T_kV$ and $V^{\otimes ...
1
vote
1answer
30 views

Show the $\mathbb{C} S_3$-module of dimension 2 has $S(V \otimes V)$ is not irreducible

Consider the $\mathbb{C} S_3$-module of dimension 2, call it $V$. I want to concretely show that $S(V \otimes V)$ is not irreducible. I found a representation for $S_3$ over $\mathbb{C}$ of degree ...
2
votes
1answer
98 views

properties of Sym^2 vector subspace/properties of tensor products

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
1
vote
1answer
127 views

Representations - Tensor Product prove properties of tensor product

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
1
vote
2answers
77 views

Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$

Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...
8
votes
2answers
828 views

The physical meaning of the tensor product

I have come across tensor products many times in physics, namely for matrices, vector-space elements, Hilbert-space elements (quantum states), and representations of groups and algebras. However, the ...
4
votes
2answers
184 views

Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
2
votes
1answer
305 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
1
vote
1answer
71 views

$\mathbb{Q}$-dimension of f. g. $\mathbb{Q}[\mathbb{Z}/p^l]$-modules.

My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} ...
0
votes
1answer
45 views

If $\rho: G \to GL(V)$ is a representation with sub-representation $\tau$, is $\tau^{\otimes n}$ a subrepresenation of $\rho^{\otimes n}$?

I'm working over an algebraically closed field of characteristic $p>0$ so I'm not assuming that $\tau$ is a direct summand of $\rho$. I think I can prove this by looking at the Kronecker product of ...
6
votes
0answers
173 views

Invariant element in the tensor product of rectangular Specht modules?

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
6
votes
1answer
71 views

r-th transvectants and $\mathbb{C}G$-module maps

Suppose $V=\mathbb{C}^2$ and $G=SL(V)=SL_2(\mathbb{C})$. We define $C_n = H_{\mathbb{C},n}(V,\mathbb{C}) \cong S^n(V^*)$, the n-th symmetric power of the dual of $V$, i.e. the homogeneous polynomials ...
7
votes
1answer
454 views

Irreducible representations of a tensor product

Let $A, B$ be finitely generated (noncommutative) algebras over a field $k$ (say, algebraically closed). Can we get all irreducible representations of $A \otimes_k B$ from tensoring representations of ...
2
votes
0answers
105 views

induced representation of tensors of irreducibles

Let $V_{\lambda}$ and $V_\mu$ be representations of the symmetric groups $\mathfrak{S}_d$ and $\mathfrak{S}_m$ respectively where $\lambda$ is a partition of $d$ and $\mu$ is a partition of $m$. It is ...
1
vote
1answer
179 views

What is meant by “direct summand in a tensor product”?

I am currently working on the topic of Lie - Algebras and I have stumbled a few times over the expression "direct summand in a tensor product". The text says that $\ V(\lambda) $ as an ...
2
votes
1answer
92 views

fixed space of tensor representation

Let $V$ be a finite-dimensional irreducible representation of a finite group G over an algebraically closed field. If $V \otimes V$ has a fixed subspace acted by $G$, why should it be 1-dimensional? I ...