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

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22 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
76 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
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1answer
33 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: ...
14
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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. ...
1
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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
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 ...
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: $$ R_R=\exp\left(i\theta^a\frac{\sigma^a}{2}-\eta^a\frac{\sigma^a}{2}\right) $$ ...
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 ...
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
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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 ...
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 ...
7
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0answers
114 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 ...
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\\ ...
0
votes
1answer
19 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 ...
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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. ...
0
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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 ...
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 ...
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 ...
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 ...
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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 ...
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 ...
-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 ...
0
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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 ...
0
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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 ...
1
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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 $F_2$?
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 ...
2
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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
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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
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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
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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 ...
4
votes
4answers
437 views

Can someone tell me in dummy terms what Left and Right regular representations are.

In my book it just says that the left regular representation is the map f in the Cayley's Theorem proof, but I just don't understand what are they? Why we need to find them?
0
votes
1answer
24 views

Should Ext-quiver be a full sub-quiver of its AR-quiver for a basic hereditary algebra A over algebraic closed field K?

For a basic hereditary algebra A over algebraic closed field K, prove its Ext-quiver $\Gamma_{A}$ is a full sub-quiver of its AR-quiver $\Delta_{A}$. I have no clue for this.
1
vote
1answer
33 views

Lie algebra representations and tensor product decompositions.

Find the weights for $V_{L_1 - 2L_3}$, where $L_1, L_2, L_3$ are the weights for the standard representation of $\mathfrak{sl}_3 \Bbb{C}$ on $V \cong \Bbb{C}^3$. In order to find these weights, ...
3
votes
1answer
77 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
15
votes
1answer
123 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 ...
7
votes
4answers
89 views

A trigonometric integral identity

How can we prove the following identity? $$ \int_{0}^{2\pi}\cos^{n}\left(\,\theta\,\right) \sin\left(\,\left[\,n + 1\,\right]\theta\,\right)\sin\left(\,\theta\,\right) \,{\rm d}\theta ={\pi \over ...
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 ...
10
votes
2answers
57 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 ...
0
votes
0answers
38 views

Jacobson radical of a particular group algebra

I am studying for my Algebra test and I got stuck in this question: Determine the Jacobson radical of the group algebra $F[G]$, where $F$ is a field of characteristic $p>0$ and $G$ is the cyclic ...
0
votes
1answer
49 views

Decomposition into irreducibles of representations of semisimple Lie groups.

Let $G$ be a connected semisimple Lie group and $\mathfrak{g}$ it's Lie algebra. Then $\mathfrak{g}$ is semisimple. Let $V$ be a finite dimensional representation of $G$. Viewing $V$ as a ...
1
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1answer
22 views

Irreducible representation

I know that correspondence every pure state on a C*-algebra $A$, there is an irreducible representation of $A$. Also we have the following theorem: Let $A$ be a C*-algebras and $(\pi,H)$ be an ...