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

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7
votes
1answer
90 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
2
votes
0answers
10 views

Intuition behind the construction of Young Symmetrizer

I've been studying representation theory of group on Tung's "Group Theory in Physics". I understood Young Symmetrizers of different Young diagrams are essentially primitive idempotents of group ...
0
votes
1answer
52 views

character theory question [on hold]

Can you offer me some useful books in representation and character theory of finite groups? I want to see the applications of group representation theory and character theory in other sciences too.Can ...
5
votes
1answer
50 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 ...
5
votes
0answers
22 views

Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
3
votes
1answer
37 views

Proving a relation for representations of gauge groups

Let ${\cal G}$ be a Lie group - possibly disconnected. Let ${\mathfrak g}$ denote the corresponding Lie algebra. Let $R_k$ be a general unitary representation of ${\cal G}$ and $R$ be the adjoint ...
20
votes
2answers
77 views

any $2$-dimensional rep of a finite, non-abelian simple group is trivial

Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering ...
0
votes
0answers
21 views

Why are certain representations called supercuspidal and generic?

For instance, is the condition that all the Jacquet modules are zero somehow analogous to the first Fourier coefficient of a certain modular form being 0? I also have no idea why generic ...
0
votes
0answers
6 views

Restriction of Irreducible Admissible representations

Let $F$ be a $p$-adic field. It is a theorem of Bernstein that if $(\pi_1, V)$ is an irreducible admissible representation of $GL_{n}(F)$ and $(\pi_2, W)$ is an irreducible admissible representation ...
3
votes
1answer
47 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: ...
0
votes
0answers
33 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 ...
3
votes
1answer
85 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 ...
1
vote
2answers
55 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
23 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
126 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
39 views

Determinant of Character table as a matrix [duplicate]

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
59 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
40 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
64 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 $mats$.
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
28 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
59 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
146 views

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
36 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
30 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
22 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 ...
7
votes
2answers
82 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
30 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
44 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
99 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
92 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
193 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
60 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
39 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
334 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 ...