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)

0
votes
0answers
48 views

Who are the mathematicians in US who are working on expander graphs right now?

I am familiar with only the "big" names doing this research like Gharan, Nikhil Srivastava, Dan Spielman, Jean Bourgain, Luca Trevisan, Elina Fuchs, Peter Sarnak , Amin Saberi and Terence Tao. I would ...
0
votes
1answer
37 views

Proving that this function is an Endomorphism?

Given that $\mathbb{C}G = W_1 \oplus W_2$ and $1 = e_1 + e_2$ where $e_1 \in W_1$ and $e_2 \in W_2$ Also Knowing that $$w_1e_1 = w_1, w_2e_1 = 0$$ $$w_1e_2 = 0 , w_2e_2 = w_2$$ Let $x \in G$ Now it ...
5
votes
1answer
53 views

What shall I learn in order to understand Auslander-Reiten theory and tilting theory?

I work on cluster algebras and quivers and hence I need to understand Auslander-Reiten theory and tilting theory as soon as possible. I have read some noncommutative algebra and homological algebra ...
3
votes
0answers
29 views

Minimal polynomial of endomorphism of permutation module

Let $G$ be a transitive permutation group on a set $\Omega$. If $n$ is the degree and $M\in\mathbb{Z}^{n\times n}$ is a symmetric matrix that is also contained in ...
2
votes
3answers
73 views

How to show that $\int_G f(t) dt = \int_G f(t^{-1}) dt$?

I am reading the lecture notes. On page 34, line 13, it is said that $\int_G f(t) dt = \int_G f(t^{-1}) dt$. How to prove this identity? I think that if we let $s=t^{-1}$, then ...
3
votes
0answers
31 views

Can Schroedinger equation be derived from the unitary representation of Galilean group? [migrated]

I have been trying to understand quantum mechanics as a unitary representation of spacetime symmetries. My first question is: Can Schroedinger equation be derived from the unitary representation of ...
4
votes
0answers
36 views

How to apply a double centralizer property on a faithful module of a self-injective Artin algebra?

Let all considered algebras be Artin algebras and let all considered modules be finitely generated. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: ...
5
votes
2answers
43 views

When does a representation of $H\subset G$ on $V$ extend to a representation of $G$ on $V$?

Let $G$ be a finite group, $H$ a subgroup, and $\varphi:H\rightarrow GL(V)$ a finite-dimensional representation of $H$ over a characteristic zero, algebraically closed field. Let $\chi$ be the ...
0
votes
1answer
27 views

Unitary invariant positive definite form

Let $G$ be a finite group and $V$ a finite-dimensional compex vector space which is a $G$-module. Define $(u,v)=\sum_{x\in G} h(ux,vx)$, where $h(u,v)$ is a positive-definite hermitian form. By ...
2
votes
1answer
34 views

How to show that a certain module is injective over an endomorphism algebra?

Let $A$ be a self-injective Artin algebra and $M\in\ \mathfrak{mod}\ A$ with the property $\mathfrak{add}\ _AA = \mathfrak{add}\ M$. Let $I$ be a finitely generated injective $A$-module. Why is ...
0
votes
0answers
17 views

Projector method for tensor and double groups

I'm currently trying to understand a computation in my script. The setup is the following: We are looking at the double group of $C_{3v}$, i.e. $C^D_{3v}$. The character table is given by the ...
3
votes
1answer
39 views

How to find irreducible representations of $\mathbb{C}S_2$ and $\mathbb{C}S_3$

I just starting to learn representation, so still have a lots of thing that is unclear. And here is a question what I wish to attempt. Let $S_n$ be the symmetric group and I want to find all the ...
0
votes
1answer
63 views

Always exists a representation $\rho$ for an arbitrary group?

I am studying the representations of the fundamental group of a fixed surface into PSL$_2\mathbb{R}$, and a simple question aries in me. If $G$ is a group, and $V$ a vector space over a field ...
0
votes
0answers
29 views

* representation of an algebra

Suppose $G$ is a finite group. Let $A(G)$ be the algebra of functions from $G$ to $C$. Now we define the convolution of $a$ and $b$ in $A(G)$ as $(a*b)(x)=\sum\limits_{y\in G} a(xy^{-1})b(y)$. Let $U$ ...
1
vote
1answer
56 views

What does the notation $U\mathfrak{sl}_2$ mean, and why is the $U$ written in a different typeface to the $\mathfrak{sl}$?

A representation theory homework problem asks me to determine the finite dimensional irreducible representations and the finite dimensional indecomposable representations of $U\mathfrak{sl}_2$. I ...
3
votes
1answer
35 views

Induction and Compact induction of representations

Let $H \leq G$ be a subgroup of a finite group, $G.$ Suppose $(\sigma, W)$ is a representation of $H.$ Then we know that $Ind_H^G \sigma $ and $ind_H^G \sigma $ are isomorphic, where $$Ind_H^G ...
1
vote
0answers
23 views

Why aren't all elements of the $45_a$ representation of $SO(10)$ zero?

We can write elements of the $45_a$, where $a$ denotes antisymmetric, as $10 \times 10 $ matrices, because $$ 10 \otimes 10 = 1_s \oplus 54_s \oplus 45_a$$ Here $10$ denotes the fundamental ...
2
votes
1answer
24 views

For a root system, why does $\beta\in\Delta_+\setminus\{\alpha_i\}$ imply $(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$?

Let $\mathfrak{g}(A)$ be a Kac-Moody algebra for a matrix $A$, with root basis $\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ...
1
vote
1answer
52 views

Why is $\mathfrak{g}(A)=\mathfrak{g}'(A)$ iff $\det(A)\neq 0$?

In many sources (Victor Kac, Zhexian Wan, etc.), it's stated as a remark that if $\mathfrak{g}(A)$ is the Kac-Moody algebra of a generalized Cartan matrix $A$, then $\mathfrak{g}(A)=\mathfrak{g}'(A)$, ...
0
votes
0answers
31 views

Quadratic Casimir of SO(5)

In the article A Four Dimensional Generalization of the Quantum Hall Effect, arXiv:cond-mat/0110572, by Zhang and Hu Quadratic Casimir operator for $SO(5)$ is given as $$p^2/2+q^2/2+2p+q .$$ When ...
2
votes
2answers
29 views

Free product of two algebras and actions of algebras.

Let $A, B$ be two algebras. Suppose that $A$, $B$ acts on $V$. Then we have two maps $$ \delta_1: A \otimes V \to V, \\ \delta_2: B \otimes V \to V, $$ which satisfy the axioms of actions. Do we ...
3
votes
0answers
43 views

Short pedagogical introduction to Young-tableaux and weight diagrams?

I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods ...
2
votes
1answer
25 views

Dual representation of $SL_n$ via Young diagram

Irreducible representations of $SL_n$ are encoded by Young diagrams with fixed number of rows not greater then $n-1$ (at least, I prefer this notation). There is an involution on these ...
3
votes
1answer
78 views

Decomposition of polynomial ring as $S_n$-module

I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy ...
0
votes
0answers
17 views

What are the character functions of $\mathbb{Z}_N \times \mathbb{Z}_N$ ?

$\mathbb{Z}_N \times \mathbb{Z}_N$ is an Abelian group which I can think of to consist of all tuples of the form $(\omega ^a, \omega^b)$ where $0 \leq a,b \leq (N-1)$ and $\omega = e^{ \frac{2 \pi i ...
1
vote
1answer
32 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
1
vote
0answers
23 views

Embedding of a finite group in a compact connected Lie group

How can one embed a finite group $G$ in a compact connected Lie group? I think if we take a faithful unitary representation of G , that will do the job.But if $G= Z/n$, then what should be the ...
0
votes
0answers
47 views

Are these functions characters?

In Ireland and Rosen are mentioned the functions $f,g:\mathbb{F}_{p^f}^*=G\to \mathbb{C}$, with $$f:x \mapsto \zeta_p^{x+x^2+\cdots + x^{p^{f-1}}}$$ $$g:x \mapsto \left( \frac{x}{P}\right)_m,$$ ...
3
votes
0answers
31 views

About irreducible group representations

Are the trivial representation, the alternating representation and the standard representation irreducible representations?! I can easily see that the trivial representation (which sends every ...
0
votes
1answer
29 views

Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix. The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots ...
2
votes
0answers
59 views

Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
2
votes
2answers
50 views

Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
1
vote
1answer
72 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
3
votes
2answers
96 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
5
votes
1answer
51 views

Recognizing action of semidirect product

I've been looking at some texts in representation theory and I see instances where the symmetric group $S_n$ and some other group, e.g., $GL(V_1) \times \ldots \times GL(V_n)$, act on a space. The ...
4
votes
1answer
39 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
1
vote
0answers
42 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
1
vote
1answer
43 views

In general, how do you construct a nontrivial representation of a group?

This is my first time studying representations. I'm not sure how to go about constructing a nontrivial representation of a group. Do I construct a function that satisfies the definition? Could you ...
0
votes
2answers
40 views

How do I construct a nontrivial linear representation of the group $G$=S' in $R^3$

$S'$ is defined as the unit circle. The product is defined as the sum of angles. How do I construct a linear representation in general? I don't know how to begin with this problem. Some ...
1
vote
1answer
36 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...
4
votes
0answers
40 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
2
votes
0answers
71 views

Fields of Research in Algebra [closed]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
2
votes
0answers
50 views

Dual space isomorphism and the dual representation

Let $V$ be a complex finite-dimensional vector space. Then there always exists an isomorphism $V \simeq V^*$, where $V^*$ is the dual space. The isomorphism can be fixed by choosing a non-degenerate ...
4
votes
1answer
71 views

The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
1
vote
0answers
20 views

Every representation of compact group is a direct sum of irreducible

Recently I asked about (references to) some results concerning representation theory of compact topological groups: here is the discussion Representation theory of locally compact groups In ...
0
votes
0answers
25 views

Stabilizer subgroup in adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
6
votes
1answer
52 views

Is there a systematic way to determine the irreducible representations of a finite group?

I was reading through fulton and harris's book on representation theory. I'm in the middle of chapter 3 and noticed their approach to finding irreducible representations of groups is pretty ...
1
vote
2answers
28 views

About the characters of representations of groups

I want to ask a question about the characters of representations of groups. we all know that the equivalent representations have the same character, and the character is a class function, so what ...
0
votes
0answers
7 views

Character degrees of 2B2(q^2)

Let $S \cong {}^2B_2(q^2), q^2 \ne 2$, and $\chi$ is the Steinberg character for $S$. What is the $\chi(1)$?
3
votes
1answer
34 views

Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...