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)

3
votes
1answer
38 views

How to construct an explicit isomorphism between two special endomorphism rings

Let $\Lambda$ be an artin algebra and $M$ a $\Lambda$-module. Let $\Gamma:=\text{End}_\Lambda(M)$ and let $D$ be the standard duality. How can you give an explicit isomorphism ...
3
votes
1answer
30 views

Representing real numbers from matrices of non-negative-reals.

Consider $I = \left(\begin{array}{cc} 1&0\\0&1\end{array}\right)$ and $N = \left(\begin{array}{cc} 1&1\\1&1\end{array}\right) - I = \left(\begin{array}{cc} ...
3
votes
1answer
48 views

Proving irreducibility of representations using matrix representation

For the quaternion group $Q_8$ we have the presentation $$Q_8 = \big<a,b : a^4 = 1, a^2=b^2, b^{-1}ab=a^{-1} \big>$$. Now knowing that for matrix $$ A = \begin{bmatrix} i & 0 \\ 0 & -i ...
3
votes
0answers
22 views

PSL(2,q) has no nontrivial irreducible representation of small dimensions

I am a bit new to this site, and wondering how to show that PSL(2,q) has no nontrivial irreducible representation of small dimensions. Thanks.
0
votes
0answers
46 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
-1
votes
0answers
41 views

Socle Friendly Groups

Are dihedral groups socle friendly? Can anyone give me any link where I can get the proof or a details Explanation? Thanks in advance. These are the two links where you will get introduced to socle. ...
4
votes
1answer
23 views

Generating function of symmetric power representation

Let $\rho:G\rightarrow GL(V)$ be a complex representation. For each $n$, let $\chi_{\text{Sym}^n}$ be the character of the n-th symmetric power of $V$. Prove for each $g\in G$, $$\sum_{i=0}^\infty ...
0
votes
1answer
14 views

If $A$ is an associative algebra Show that $End_{A}(A)=A^{op}$ the algebra with opposite multiplication [duplicate]

Let $A$ be an associative algebra, And let $V$ be a representation of $A$. By $End_{S}(V)$ one denotes the algebra of homomorphisms of representations $V \to V$ Show that $End_{A}(A)=A^{op}$ the ...
3
votes
1answer
88 views

Order of group elements from a character table

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I'm trying to answer a question which goes the ...
2
votes
1answer
60 views

Using character table to find normal subgroups

I am working through a question on an old character theory exam. I've answered the first two parts ok, but am now struggling on the third part. Here is the part that I can't do: I've computed ...
1
vote
1answer
43 views

Tensor product of simple finite-dimensional modules

I am trying to understand the following one-liner that appears in http://arxiv.org/abs/0901.0827v5, Theorem 2.26 (and I'm afraid I must be missing something really basic): Let $V$ and $W$ be ...
1
vote
0answers
27 views

If two non-equivalent representations are irreducible then sum is 0

We have a finite group and two representations $D_1:G\to GL_n(\mathbb{C}),D_2:G\to GL_m(\mathbb{C})$ for some positive integers $m,n$. We define $$T=\sum_{g\in G}D_1(g)BD_2(g^{-1})$$ where $B\in ...
1
vote
1answer
24 views

Computing plethysms of the adjoint representation using the Littlewood Richardson rule

Let $N$ be an integer (let's imagine very large), and let $G$ be the group $\mathrm{GL}_N(\mathbb{C})$. I would like to compute various plethysms of irreducible representations which are not ...
2
votes
1answer
55 views

In a vector space over a finite field, can the orbit of a point under matrix multiplication have dependent subsets?

Let $\mathbb{F}_q^n$ be the vector space of dimension n over the finite field of order q, $\vec{v}$ a vector in the space, and $M$ an invertible $n \times n$ matrix over $\mathbb{F}_q$. We know that ...
4
votes
1answer
33 views

Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
3
votes
0answers
41 views

Showing given representation is summand by averaging

This problem is from Artin 1st edition 9.2.2. Problem : Let $\rho$ be the standard three-dimensional representation of T, and let $\rho'$ be the permutation representaion obtained from the action of ...
1
vote
0answers
37 views

Cohomology of permutation representation

Consider the action of $S_n$ over $\{1,...,n\}$ consider the associated representation with integral coefficients $X_n$. What are $H^r(S_n,X_n)$? More in general is there a nice way to predict the ...
0
votes
2answers
39 views

Matrix representations of $\mathfrak{g}^*$.

Let $g = sl_2$. Then there is a matrix representation of $g$ as follows. The Lie algebra $g$ is a three dimensional vector space with a basis $E, F, H$ such that $[E,F]=H$,$[H,E]=2E$,$[H,F]=-2F$. The ...
2
votes
1answer
56 views

What is the determinant of Ad(g)?

In more generality, if a matrix acts on a group of matrices by conjugation, what is the determinant of this action (if such a notion exists)? Is it simply the determinant of the matrix being used to ...
1
vote
1answer
20 views

Why is $B^2 = \mathbb{I}_2$ where $B = T^{-1}AT$

I am currently studying group representations. And I know that two If $\phi:G \to GL(n,F)$ is a representation of $G$ over $F$ and if $\gamma:G \to GL(m,f)$ is also a representation of $G$ over $F$we ...
0
votes
0answers
14 views

Differing conventions regarding “modulus character” of $k$-points of smooth affine $k$-group, $k$ non-Archimedean

Let $\mathbf{G}$ be a smooth connected affine $k$-group, where $k$ is a non-Archimedean local field, $G=\mathbf{G}(k)$ the group of $k$-rational points, a locally $k$-analytic group. Since $G$ is in ...
1
vote
2answers
40 views

Converse of Maschke theorem

Let F be a field and G be a group and FG the group ring. Let H:=$\{\sum_{g\in G}\lambda_g g\in FG : \sum_{g\in G}\lambda_g=0\}$. Then H is codimension 1 subspace of FG and is an FG submodule. ...
1
vote
1answer
23 views

Weightspace decomposition of a semisimple Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a (finite dimensional) semisimple Lie algebra. Let $H$ be a maximal toral subalgebra of $L$. Consider a representation $\pi: L \to \mathfrak{gl}(V)$. It is ...
3
votes
1answer
93 views

Inducing representation for groups of order $p^3$

For groups $G$ such that $|G|=p^3$ one can show that $(1)$ $Z=Z(G)\cong C_p$ $(2)$ $G'=Z$ $(3)$ $G/Z \cong C_p \times C_p$ Take any $x \in G/Z$. Then $N=\langle x,Z \rangle$ is an abelian normal ...
2
votes
1answer
32 views

submodules of $V^n$

I'm stuck with Exercise 3 on page 25 of Kraft and Procesi's Primer on Invariant theory http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf. It goes as follows: let $\rho:G\to GL(V)$ be an ...
1
vote
0answers
22 views

Permutation module $M^\lambda$ as induced module

If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module: $M^\lambda := ...
0
votes
2answers
37 views

Why does complex conjugation permute the rows (columns) of a character table

If $\chi$ is the character of $\rho$, then $\overline{\chi}$ is the character of $\rho^*$ (dual) and $\chi_{irreducible} \iff \overline{\chi_{irreducible}}$. This implies complex conjugation ...
0
votes
0answers
38 views

Axial vector and vector representions of C$_{4v}$ group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way $$ \vec{V} ...
0
votes
0answers
26 views

Subrepresentations of $\mathbb{I} \oplus \xi$

$G=C_2=\{e,h \}$. $\mathbb{I}$ is the trivial representation and $\xi$ is the sign representation. Let us consider $\mathbb{I} \oplus \xi$ where $e \mapsto \begin{pmatrix} 1 & 0\\ 0 & 1 ...
2
votes
0answers
32 views

Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
2
votes
1answer
71 views

Definitions of $\mathrm{Hom}(V,W)$

I have the definition of a homomorphism as map such that $\varphi(g_1g_2)=\varphi(g_1)\varphi(g_2)$ I have the definition of $\mathrm{Hom}(V,W)$ as $$\begin{align}\mathrm{Hom}(V,W) &= ...
1
vote
1answer
19 views

Etingof problem 2.16.2: Irreps of Two-dimensional Lie algebra over a field of positive characteristic

This is problem 2.16.2 in Etingof's introduction to representation theory. Note that problem 2.16.1 is a proof of Lie's theorem. I'm having trouble with the second case, where the base field has ...
1
vote
0answers
20 views

Dimension of the restricted representation

My definition of restriction is: Let $H < G$, $\rho: G \rightarrow GL(V)$. The restriction of $\rho$ to $H$, $\rho: H \rightarrow GL(V)$. Its character is $$Res_H\chi(h)=\chi(h) \ \ \forall h \in ...
1
vote
1answer
25 views

Invertibility of character table

Corollary. The character table of a group is an invertible square matrix. The theorem that is a corollary to states that the character table is a square matrix and the explanation for invertibility ...
1
vote
0answers
52 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
0
votes
1answer
29 views

Show that a representation of a finite group is isomorphic to its dual if its character takes only real values

This appeared as a part of showing that a representation of a finite group is isomorphic to its dual if and only if its character takes only real values. The "only if" part was easy to show. For the ...
1
vote
1answer
36 views

Verify regular representation?

Let $G$ be a finite group and let $V$ be the vector space of functions from $G$ to $\mathbb{C}$. For $g \in G$ and $f \in V$, let $R(g)(f)$ be the function $$(R(g)f)(x) = f(xg^{-1}).$$ How can I show ...
1
vote
1answer
35 views

A Question About Group Representation

I’m studying the representations of finite groups.We all know that the group representations are very important tool in the study of finite groups by allowing many group theoretic problems to be ...
1
vote
0answers
7 views

How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
1
vote
0answers
19 views

A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
1
vote
1answer
20 views

Identification of the Lie algebra of an isotropy group with the tangent space - stuck with a statement

I think I am stuck with the following statement that I read on the Encyclopedia of Mathematics website regarding Isotropy representations: "If $G$ is a Lie group acting smoothly and transitively on ...
1
vote
1answer
49 views

Etingof problem 2.15.1 Representations of sl(2)

I'm studying from Etingof's Introduction to Representation Theory. This is problem 2.15.1, part a. I feel I'm close to the solution. Here's what I have. Problem: A representation of sl(2) is a vector ...
2
votes
1answer
46 views

Characters of transitive finite permutation group

I know that Frobenius reciprocity helps us to solve this problem, but I don't know why: Let $ G $ be a transitive finite permutation group with permutation character $ \pi $. If $\chi $ is an ...
1
vote
1answer
55 views

Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $ H $ be a subgroup with index $ m $ in the finite group $ G $. Let $ F $ be an algebraic closed field of characteristic $ 0 $. ...
1
vote
0answers
24 views

A question and a conjecture on $USp(N)$ group

$USp(N)$ with $N$ an even integer is defined as the group of unitary matrices $M$ that satisfy $M^TJM=J$, where $M^T$ is the transpose of $M$ and $J$ is the anti-symmetric $N$-by-$N$ matrix ...
0
votes
0answers
47 views

transitivity of induction

I want to prove the "transitivity of induction" property: Let $ H\leq K\leq G $ where $ G $ is finite. Let M be an $ FH $-module, where $ F $ is any field. Then $ (M^K)^G\simeq^{FG} M^G $. Would you ...
3
votes
0answers
58 views

Relation between the characters of subgroups of a finite group

Let $ H $ and $ K $ be subgroups of a finite group $ G $. Let $ \chi_1(H) $ and $ \chi_1(K) $ denote the trivial characters of $ H $ and $ K $ over an algebraically closed field of characteristic $ 0 ...
3
votes
0answers
23 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
1
vote
0answers
12 views

Relation between induced and coinduced spaces

Let $G$ be a compact Lie group and $H$ a closed subgroup of it. Let $X$ be a $G-$space. The induced $G-$space is defined to be $$G\times_H X$$ with the equivalence $(gh, x)=(g, hx)$, for any $g\in G, ...
0
votes
1answer
26 views

Properties of generalized characters

I search for generalized characters which are not characters.Also I want to know that why every generalized character is a difference of characters.