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)

-1
votes
0answers
24 views

C*-algebras: States?

I'd like to better understand states on C*-algebras. What properties should I investigate and in which order? (Positive functionals, extremal states, Schwarz's inequality, Kadison's inequality, what ...
2
votes
1answer
78 views

Is my understanding for group algebra correct?

Let $G$ be a group, $k$ be a commutative unital ring. Consider $\mathbf{Alg}_k^{inv}$ the category of unital $k$-algebras whose multiplicative semigroup is a group. Then there is a forgetful functor ...
2
votes
1answer
35 views

Constructing an irreducible representation for a finite group

This is not a homework. Recently, I have been starting to study the representation and character theory and I am doing some exercises in this field, but I have some problem with some of them, like: ...
1
vote
2answers
52 views

A question in finite group theory with proof using representation theory

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises in this field, but I have a problem with an exercise: I want to prove ...
2
votes
0answers
21 views

Fundamental chiral representations of SU(2)

During class we arrived to the statement that the $(1/2,0)$ or right-handed representation of SU(2) is realized by: $$ R_R=\exp\left(i\theta^a\frac{\sigma^a}{2}-\eta^a\frac{\sigma^a}{2}\right) $$ ...
15
votes
1answer
124 views

Decomposing $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles, formula for all $n$?

$``$Let $G = \text{SU}(2)$, and let $V_n$ be the space of homogeneous degree $n$ polynomials in $\mathbb{C}[x, y]$. Decompose $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles.$"$ For ...
14
votes
0answers
34 views

Determinant of Character table as a matrix

I'm studying for finals and came across this problem in a book. Suppose $G$ is a finite group with conjugacy class representatives $g_1,...,g_k$ and character table $Z$. Consider $Z$ as a matrix. ...
4
votes
1answer
55 views

Is there a “unifying framework” for harmonic analysis?

Recently, I was exposed to a basic harmonic analysis course. Although the course is almost over, I still can't put my finger on what harmonic analysis is about. I have a vague idea that it is ...
2
votes
1answer
38 views

A question in character theory of finite groups

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises from "A Course in the Theory of Groups by Robinson" and "character ...
0
votes
1answer
63 views

Fixed Spaces for Group Elements

what is the GAP code for finding the fixed space? A list of row vectors that form a base of the vector space $V$ such that $v M = v$ for all $v$ in $V$ and all matrices $M$ in the list $mats$.
5
votes
1answer
42 views

representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition). Define a representation $\pi_{m,\,n}: \text{SU}(n) \to ...
4
votes
1answer
29 views

How to check if a representation of su(2) is irreducible

I have found a representation $\rho$ of the group $G=Su(2)$. I want to show that this representation is irreducible but I don't know how. Finding all invariant subspaces seems very difficult. I ...
3
votes
1answer
27 views

Character of a tensor product of $\mathfrak{sl}_2$-modules

Let $V$ be a finite-dimensional $\mathfrak{sl}_2$-module. There is a standard base $\{e,f,h\}$ in $\mathfrak{sl}_2$, I use standard notation ($h$, for instance, is the diagonal matrix with $1$ and ...
10
votes
2answers
58 views

$SU(2)$ acting by conjugation, decomposition into irreducibles

I am attempting past exam questions of the Cambridge Math Tripos. I know how to do the first few parts, which involves giving the irreducible representations of $U(1)$ and $SU(2)$. But I am not sure ...
7
votes
0answers
120 views
+50

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
1
vote
1answer
268 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?
-1
votes
0answers
27 views

Module of representation matrix

Can someone please show me why the module of any representation matrix in a one-dimensional representation of a finite group is equal to 1? and please define module of a representation as well. ...
3
votes
1answer
35 views

“Converse” Schur's lemma [duplicate]

For representations over an algebraically closed field one can formulate Schur's lemma in the following form: Every endomorphism of irreducible representation is of the form $\lambda\cdot id$ I ...
2
votes
0answers
28 views

Decomposiom of the representation of $SU(N)$

Let $T$ be the "fundamental" representation (I mean the one in which the matrices representing the group elements are simply themselves) of $SU(N)$ group. I have \begin{pmatrix} SU(N-1)& 0\\ ...
4
votes
2answers
52 views

sub-$G$-representations

"So let $G$ be a finite group, $H$ a proper, nontrivial normal subgroup of $G$. For any representation $\rho: G \to \text{GL}(V)$ define the $H$-invariants of $V$ as $$V^H := \{v \in V \text{ }|\text{ ...
6
votes
1answer
49 views

Exercise on Induced Representations of 1 dimensional complex representation

I'm having a hard time trying to solve the following problem coming from Eingof's book "Introduction to representation theory" (page 55 of the book in PDF format ...
0
votes
1answer
20 views

Representatives of simple $\mathbb C[\mathbb D_3]$-modules (left modules)

Problem Let $\mathbb D_3$ be the symmetry group of the equilateral triangle. Give a complete list of the representatives of the simple left $\mathbb C[\mathbb D_3]$-modules. My attempt at a solution ...
0
votes
1answer
21 views

Is the restriction of the regular representation of a finite group always a multiple of the subgroup?

For an inclusion of groups $H \hookrightarrow G$, define the restriction $\operatorname{Res}^G_H$ of representations as precomposition with the inclusion map. Also, define the complex regular ...
0
votes
0answers
18 views

Similar transformation matrix restricting determinant to be 1.

How do you prove that if restricting the determinant of a similar transformation matrix between two equivalent irreducible unitary representation of a finite group to be 1, then this transformation ...
4
votes
1answer
61 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
6
votes
2answers
80 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
4
votes
2answers
29 views

No invariant complement?

How do I show that the representation $\rho: \mathbb{Z} \to \text{GL}_2(\mathbb{C})$ with $$\rho(1) = \begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$$ has an invariant subspace with no invariant ...
3
votes
1answer
42 views

No character theory, any representation ${\bf{GL}}_N(\mathbb{C})$ is reducible, upper bound

Let $G$ be any finite group. How do I show without character theory that there is a number $N = N(G)$ so that any representation $\rho: G \to {\bf{GL}}_N(\mathbb{C})$ is reducible (and finding an ...
6
votes
1answer
98 views

$G$-representations, $W \otimes V^* \to \text{Hom}(V,W)$

Let $V$ and $W$ be finite-dimensional vector spaces. I know how to construct an explicit isomorphism of vector spaces $W \otimes V^* \to \text{Hom}(V,W)$ and show that it's an isomorphism. But if I ...
2
votes
1answer
91 views

Failure of the Krull-Schmidt Theorem?

Theorem 1.19 of Representation Theory of Finite Groups: Algebra and Arithmetic is the Krull-Schmidt theorem, which I screenshotted and uploaded it here, I don't have any problem with this theorem and ...
5
votes
0answers
192 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
2
votes
1answer
21 views

Subalgebra condition in Engel's theorem

An equivalent version of Engel's theorem says that Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V\ne 0$, then there exists ...
3
votes
1answer
26 views

Lie subalgebra in $Der(\mathbb{C}[z])$ isomorphic to $\mathfrak{sl}_2$

I am to prove that $\{(az^2+bz+c)\frac{\partial}{\partial z}:a,b,c\in\mathbb{C}\}$ regarded as a Lie algebra is isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. I guess it is possible to build a basis ...
1
vote
0answers
25 views

Extensions of representations

I'm again confronted with an exercise from Etingof's book "Introduction to representation theory" (page 30 of http://math.mit.edu/~etingof/replect.pdf) Problem 2.22. Let ...
-2
votes
0answers
59 views

Show that G is a Lie group and find a adjoint representation for G

$$G = \{ A \in GL(2,R): AA^t = p^2I, p>0, \det A >0\}$$ Show that G is a Lie group and find the explicity expression for their elements. And find a adjoint representation for G. Hi, I tried to ...
14
votes
1answer
59 views

random walk on finite cyclic group

Suppose that I have a random walk on the finite cyclic group of order $d > 2$, where the initial probability distribution $Q$ assigns the values $p, q, r$ to $-1, 0, 1$, respectively, where $p + q ...
6
votes
0answers
38 views

Is SL(2, 3) a subgroup of SL(2, p) for p>3?

As the title says, I was wondering whether SL(2,3) is a subgroup of SL(2,p) for p>3. I know that it is for p=5 (it can be found explicitly using the quaternionic representation), and I have some ...
5
votes
4answers
333 views

Ring of polynomials as a module over symmetric polynomials

Consider the ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a module over the ring of symmetric polynomials $\Lambda_{\mathbb{k}}$. Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free ...
3
votes
1answer
49 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
0
votes
0answers
35 views

matrix representation of free group with metric requirement

Look at this Cayley diagram of the free group generated by 2 elements, $F_2 = \langle a, b \rangle$: The 2 elements marked by green and pink are "unrelated" in the sense that they are far apart in ...
1
vote
0answers
33 views

Matrix representations of free groups?

What is the general form of faithful matrix representations of free groups? How about for the simple case of $F_2$?
0
votes
0answers
15 views

Representation groups over Dedekind domains

I am interested on groups defined over $O_K$ the ring of integers of a number field $K$. Given a linear representation $T:Gl_N(O_K)\rightarrow Gl(W)$ with $W$ a free $O_K$-module, What are the main ...
2
votes
0answers
21 views

Closure relations of the cells in the Bruhat decomposition of the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$. How do we prove the closure relations between the cells, which ...
3
votes
0answers
24 views

Is a semidirect product of linear groups a linear group?

It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product? Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, ...
0
votes
2answers
35 views

Why representations become functions?

I am trying to answer Problem 5 below, but why irreducible representations become functions(i.e. $f_{m}$), aren't representations homomorphisms from $G$ to $GL(V)$? (It would also be helpful if ...
1
vote
1answer
32 views

Questions for compact lie group representations.

I have two questions about representations of compact Lie group. If all irreducible representations of a compact Lie group are one-dimensinal, then G is abelian. An infinite compact Lie group has an ...
0
votes
0answers
18 views

matrix coefficients are representative functions.

If matrix coefficients and representative functions are stated below, then why matrix coefficients are representative functions if $V$ is finite dimensinal?
3
votes
0answers
29 views

12 in the definition of Virasoro algebra and Regge symmetry

In the definition of Virasoro algebra, there is a following condition on the generators: $[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$ Now, Regge symmetry is the following ...
0
votes
1answer
21 views

How to determine $Hom (M,M)$ for an irreducible $R$-module $M$?

More exactly, I'm considering $R$ to be a finite dimensional $\mathbb C$-algebra. For any $R$-module $M$, the $\mathbb C$ vector space $Hom_R(M,M)$ contains scalar multiplication and hence contains ...
0
votes
1answer
19 views

Can we define unitary representations on semigroups

A representation on a semigroup $S$ is a pair $(\pi,H_\pi)$ where $\pi$ is a homomorphism from $S$ into $B(H_\pi)$ and $H_\pi$ is a Hilbert space. In the group case, a representation $\pi$ of a group ...