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)

2
votes
1answer
39 views

How to show that Yang-Baxter equation is the same as braid equation?

The quantum Yang-Baxter equation is $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$. The braid equation is $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. It is said that these two equations are equivalent. How to ...
1
vote
1answer
55 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...
0
votes
1answer
46 views

character of a representation of the group $S_n$

Let $\Phi$ be a representation of the group $S_n$ in the space with basis $(e_1, \ldots, e_n)$ such that $\Phi(\sigma)e_i=e_{\sigma(i)}.$ Find character of $\Phi$ This is looks like a regular ...
7
votes
1answer
52 views

Non-polynomial representations of $GL_n$

Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the ...
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 ...
1
vote
0answers
26 views

Skew polynomial algebra and deformation

Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x ...
1
vote
1answer
63 views

Isaacs exercise 5.4 (Character Theory of Finite groups)

Any advice/hints how to prove the following statement? If $G$ is a finite group and $b(G)=\max\{\chi(1); \chi\in \mathrm{Irr}G\}$ is the maximal degree of irreducible characters and $H\leq G$, then ...
0
votes
0answers
31 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
1answer
42 views

Show that $W = \langle v_1 + \dots + v_n\rangle $ is a submodule of $V$ .

Let $G = S_n$. Let $V$ be a vector space over a field $F$ with basis $\{v_1,\dots,v_n\}$. Then $V$ is an $FG$-module with action defined by setting $g · v_i = v_{g(i)}$ for all $g \in G$ and $1\leq i ...
0
votes
1answer
160 views

Heisenberg XXX spin model

Let $\pi$ be the standard representation of $sl_2(\mathbb{C})$ on $\mathbb{C}^2$. Let $p_1,p_2,p_3$ the three Pauli matrices. Define $S^a:=\frac{1}{2}\pi(p_a)$. What does such matrices looks like?
3
votes
1answer
35 views

1-dimensional FG-Modules

Suppose $V$ is a two-dimensional FG-Module and there exists $g,h \in G$ $v \in V$ such that $(gh).v \neq (hg).v$. Show that $V$ is irreducible. I can understand the idea of this is to use Maschke's ...
1
vote
0answers
29 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
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. ...
3
votes
1answer
95 views

Modules and submodules

Let $G=S_n = Sym_n$ be the symmetric group and $V$ a vector space with basis $\{v_1,...,v_n\}$, then $V$ is a module with action defined by $g$. $v_i$=$v_{g(i)}$ for 1$\leq$i $\leq$ n and extending ...
1
vote
2answers
84 views

Let G be a finite group and V be the regular CG-Module. Find a submodule of V which is isomorphic to the trivial CG-Module

As in the question. I have read through different books and articles and they seem to set W=<$\sum_{g \in G}$ g> as a submodule. I can understand that this is unique, but I fail to see how this is ...
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 ...
4
votes
1answer
40 views

Reference Request: Characters of Finite General Linear Groups

I've been looking at J.A. Green's article The Characters of Finite General Linear Groups and it seems that Green in this article comes up with a way of calculating all irreducible characters of a ...
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 ...
1
vote
1answer
39 views

Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Question: Is $dim(V^H) \le ...
1
vote
0answers
24 views

Does the representation ring functor preserve limits?

If I have a diagram of groups $\{H_J\}$ and let $G$ be the limit of that diagram, how well does the representation ring functor "preserve the limit", IE: If I have $\lim_{J} H_J = G$ is it true that ...
1
vote
1answer
26 views

Maximal subgroup and representations (dual part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Let $g \in G$, $K = H \cap ...
2
votes
1answer
29 views

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
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 ...
8
votes
1answer
119 views

$\chi(g)$ group character $\Rightarrow$ $\chi(g^m)$ group character

Let $G$ be a group of order $n$ and and $\gcd(m,n)=1$. Let $\chi:G\rightarrow\mathbb{C}$ be a class function and define $\chi^m\!: g\mapsto\chi(g^m)$. How can one show that $\chi^m$ is a character iff ...
1
vote
1answer
35 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
3
votes
1answer
63 views

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

In Introductory 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 ...
1
vote
1answer
38 views

Irreducible representations of finite lamplighter group

Let $G = \mathbb{Z}_2 \wr \mathbb{Z}_n$ be the finite lamplighter group. What are the irreducible representations of $G$ - can anyone provide a clear reference? Austin, Naor and Valette list ...
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 ...
3
votes
1answer
41 views

Modular function of the unipotent radical of a parabolic subgroup of a reductive group

Let $G = \text{GL}_n(\mathbb{R})$. For a partition $\underline{n} = (n_1,\ldots,n_t)$ of $n$, let $P = P_{\underline{n}}$ denote the standard, block-upper-triangular parabolic subgroup of $G$ ...
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
1answer
58 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
3
votes
1answer
37 views

What am I missing about Schur functions?

Let's say I only know the following about Schur functions: you give me a partition $\lambda$ of $d$ such that $\lambda$ has $n$ parts $\lambda_1,\ldots,\lambda_n$, and I can compute the Schur function ...
1
vote
0answers
29 views

Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
2
votes
0answers
27 views

A question in the proof that the weight of a finite dimensional module is W-invariant

Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which ...
1
vote
1answer
34 views

Finding irreducible representations

This might be a very elementary question in representation theory, but I dare to ask Suppose I am asked to complete the character table of $S_5$, I know it has 7 conjugacy classes as follows : ...
4
votes
2answers
55 views

Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
1
vote
0answers
20 views

representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
3
votes
1answer
73 views

Representation theory over $\mathbb{Q}$

I am looking for books or papers which tell me something about representation theory of finite groups over $\mathbb{Q}$ (or finite extensions thereof which are not splitting fields of the group ...
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) ...
2
votes
2answers
132 views

Representation theory of infinite groups?

I am familiar with the representation theory of finite groups (at least of the symmetric groups over the field of complexes) And I know that the group algebra of an infinite group is not semisimpe ...
1
vote
1answer
26 views

Questions about cuspidal representations of $GL_2(\mathbb{F}_q)$.

All representations of $GL_2(\mathbb{F}_q)$ are classified in the book. They are principal series representations, complementary series representations, 1-dimensional representations. They form all ...
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 ...
2
votes
0answers
42 views

question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
5
votes
0answers
48 views

Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
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|} ...
2
votes
1answer
24 views

Questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$.

I have some questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in GL_2(\mathbb{F}_q)$ has eigenvalues which ...
2
votes
2answers
82 views

About the converse of Maschke's theorem

The Maschke's theorem say that\ Let $G$ be a finite group and $F$ a field whose characteristic does not divide $\mid G \mid$. Then every $FG$-module is completely reducible (I'm using the notation of ...
0
votes
1answer
78 views

representation of abelian which is noncyclic

Let $G$ be a noncyclic abelian group acting by conjugation on an elementary abelian $p$-group $V$, where $p$ is a prime not dividing the order of $G$. (a) Prove that if $W$ is an irreducible ...