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
37 views

Let $v\in V-0$, then $\varphi _{v}: k[x]\rightarrow V : f \mapsto f.v$ is a surjective $A$-module homomorphism.

Proposition. Let $A=k[x]$ and let $(V,\rho )$ be a finite dimensional irreducible $A$-module. Let $v\in V-0$, then $\varphi _{v}: k[x]\rightarrow V : f \mapsto f.v$ is a surjective $A$-module ...
3
votes
2answers
48 views

Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module.

Exercise: Show that $k[x]/(x^{2})$ is an indecomposable (1), but not irreducible (2) $k[x]$-module. I'm not sure about all different kind of modules, but this is a question of a book about ...
1
vote
0answers
27 views

multiplicities of irreducible representations

Let $G$ be a finite group and $G'$ be a subgroup. Let $\rho$ be a one-dimensional group of $G'$. Then define $\psi$ to be the induced action of $\rho$ - $\psi:= Ind_{G'}^G \rho$ Is there any general ...
1
vote
0answers
30 views

Spin Representations and Galois correspondence?

I have a vague question regarding the Spin representations. Is there a "quick" way of seeing that $Spin(2n)$ has exactly two irreducible representations which do not factor through $SO(2n)$, and one ...
0
votes
0answers
40 views

What are the non-linear representations of $SO(3,1)$?

The classification of the representations of the Lorentz group $SO(3,1)$ is well known, but the representations are usually expressed in linear form. My question is whether there is a framework to ...
0
votes
0answers
29 views

Iwasawa decomposition of $GL_n\times GL_m$

One knows that any reductive group, in particular GL$_n$, has an Iwasawa decomposition $G=NAK$. Is the Iwasawa decomposition of $GL_n\times GL_m$ simply the diagonal decomposition, $$GL_n\times ...
1
vote
1answer
43 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
3
votes
1answer
51 views

Every representation of a finite group is reducible?

I somehow "proved" that every representation of a finite group is reducible. While I'm fairly sure the error is something silly, I can't seem to place it. Could someone please help me figure out what ...
1
vote
0answers
45 views

Matrix representation and permutation matrices

In order to find the matrix representation associated to a permutation representation I identify each permutation with the corrisponding matrix representation. How can I prove that these matrices ...
2
votes
0answers
31 views

How does Fulton and Harris establish that the differential of a group hom respects ad?

Fulton and Harris, Representation Theory, Section 8.1 (pages 104 - 107 in my copy) is concerned with showing that group homomorphisms $\rho : G \to H$, where $G$ is connected, are completely ...
1
vote
0answers
19 views

limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals: $$\int_{G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & ...
0
votes
1answer
19 views

A question about positive forms on involutive algebras.

A linear form $f$ on an involutive algebra $A$ is said to be positive if $f(x^\ast x)\geq 0$ for every $x$ in $A$. To be useful, this definition requires that is not always possible to write ...
1
vote
1answer
50 views

Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
2
votes
1answer
26 views

Proving an Irreducible Representation

Consider the representation $$\pi\colon \mathbb R \to GL(\mathbb R^2)$$ by $$\theta \mapsto \text{rotation by }\theta.$$ I want to show that it is irreducible. I start with a non-zero invariant ...
0
votes
1answer
41 views

Show that if $V$ is isomorphic to $A/I$ for some left ideal $I$, then $V$ is a cyclic representation of $A$ over $k$

Suppose we have a representation $V$ of an algebra $A$ over a field $k$. Now assume that there exists a left ideal $I$ in $A$ such that $V$ is isomorphic to $A/I$. Now I have to show that $V$ is a ...
1
vote
1answer
54 views

Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe ...
2
votes
2answers
78 views

left regular representation of SU(2)

in Sepanski's book Compact Lie groups, he describes the representation theory of SU(2) as being isomorphic to $\mathbb{N}$ (SU(2) acts irreducibly on the (n+1)-dimensional space of homogeneous ...
5
votes
0answers
70 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
3
votes
0answers
36 views

Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
1
vote
1answer
92 views

Is every unitary irreducible representation an induced reperesentation?

I have recently read about induced representations and I have the following perhaps naive question about them. Let $G$ be a finite or infinite (Lie) group. Can we construct all irreducible unitary ...
0
votes
0answers
26 views

Finding inequivalent representations in a given group

I am studying characters of representations and how number of conjugacy classes is same as the number of inequivalent representations in a group. However, my question is, how do we actually find all ...
2
votes
2answers
64 views

Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
2
votes
1answer
40 views

Every representation of a finite group is completely reducible

Is this equivalent to saying that a representation is diagonalizable matrix in matrix form?
0
votes
0answers
21 views

Maximal Kostka Numbers

Let $\lambda\vdash n$ be a partition of $n$ and assume that $\lambda$ has $k$ parts. Then let $\mu$ run through all the other partitions of $n$ and consider the Kostka-number $K_{\lambda,\mu}$. Can ...
3
votes
3answers
67 views

Indecomposable representations of Lie algebra

Let $\mathfrak{g}$ be the nonabelian $2$-dimensional complex Lie algebra. It can be generated by two independent vectors $e_1,e_2$ such that $[e_1,e_2]=e_1$. Thus, $\mathfrak{g}$ is solvable and it ...
0
votes
1answer
37 views

Properties of Group representations, duality and the derived subgroup

I am trying to understand why 1) all finite-dimensional complex representations $V$ of $G$ are self dual, and 2) How the derived subgroup $[G,G]$ is a union of particular conjugacy classes. My ...
2
votes
1answer
36 views

Question on GL(n,F) representation

Let A be the group of all invertible n x n matrices over F, A+/- the subgroups of all upper/lower matrices. F^n as an A-module is irreducible? Is this because F^n has only one orbit under A? Why is ...
1
vote
1answer
40 views

The indecomposable projective A-modules

Let Q be the quiver bound by $αβ = 0$, $γδ = 0$. The indecomposable projective A-modules are given by where $A=KQ/I$. This an example in Assem-Simson-Skowronski book (Elements of the ...
1
vote
1answer
41 views

Finding a basis for $sp(4,\mathbb{C})$ and related basis.

Let $$L = so_4(\mathbb{C})= \{x \in End(\mathbb{C}^4)|^txS + Sx = 0 \} \text{ where }S = \left(\begin{array}{cc} 0 & I_2 \\ -I_2 & 0 \end{array}\right)$$ Letting $x = \left(\begin{array}{cc} ...
2
votes
1answer
88 views

quasi-split algebraic group

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...
1
vote
1answer
51 views

Computing quotient representations and Hom set fort wo representations

Consider the representation $M$ defined by We want to find all subrepresentations quotient representations of $M$, and $\mathrm{Hom}(M,N)$, where $N$ is a representation with $N \cong M$. I put B ...
1
vote
1answer
58 views

Exercise 5.8 from Lie Group, Daniel Bump

In the exercise 5.8 Bump has asked to prove that the group $Sp(4)$ over complex numbers, which is usual complex embedding $U(4)\cap Sp(4,\mathbb{C})$, can be described by, $$\left\{\begin{pmatrix} ...
1
vote
0answers
44 views

Questions about the bracket

In the map $\phi : L \mapsto \mathfrak {U}(L) $, where $ L $ is a lie algebra and $\mathfrak {U} $ is a universal enveloping algebra of $ L $. (1) Is the following relation true? If $[xy]=z$ in $ L ...
2
votes
2answers
162 views

Direct sum decomposition of weight spaces and relation to Tensor products.

There are 3 parts to the question that I am trying to understand, and while it is not homework it seems instrumental in decomposition modules into weight spaces and their relation to tensor products. ...
1
vote
0answers
22 views

P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
26
votes
2answers
501 views

What is the least $n$ such that it is possible to embed $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?

Let $\operatorname{GL}_2(\mathbb{F}_5)$ be the group of invertible $2\times 2$ matrices over $\mathbb{F}_5$, and $S_n$ be the group of permutations of $n$ objects. What is the least ...
0
votes
0answers
33 views

Convolution and Characters

I am confused about the purpose of the Formal Character, character functions, and the convolution in representation theory of Lie algebras. Is the Character function different than just the Character? ...
2
votes
1answer
45 views

Show that if $V$ is an irreducible finite dim. representation of $A$, then $z \in Z(A)$ acts in $V$ by multiplication by some scalar $\chi_V(v)$.

Let $A$ be an algebra over a field $k$. The center $Z(A)$ of $A$ is the set of all elements $z \in A$ which commute with all elements of $A$. For example, if $A$ is commutative, then $Z(A)=A$. ...
1
vote
1answer
65 views

Endomorphism ring of finite-dimensional representation

If $G$ is any group and $V$ is a finite-dimensional representation of $G$, then we can form the endomorphism ring $E = End_G(V)$. Suppose that $V$ is indecomposable, i.e. not a direct sum of ...
4
votes
0answers
41 views

Is there anywhere some explicit Bruhat decompositions are written down?

Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of ...
0
votes
0answers
34 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
1
vote
2answers
52 views

Show that the homomorphism $\lambda: k[X] \to End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ has a nontrivial kernel.

$\DeclareMathOperator{\End}{End}$ I'm trying to show that: Show that the homomorphism $\lambda: k[X] \to \End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ as in (see ...
0
votes
1answer
34 views

Quaternionic representation

Let $V$ be $G$-representation over quaternions $\mathbb{H}$. How to show that $$ \mathbb{H} \otimes_\mathbb{C} V $$ is canonically isomorphic to $V \oplus V$ as representation over $\mathbb{H}$? In ...
1
vote
1answer
45 views

Show that a simple ring is always an algebra over some field

Show that a simple ring $R$ is always an algebra over some field. So I need to show that there exist a field $k$ such that there exists a ring homomorphism $\phi : k \to Z(R) $. In an earlier ...
1
vote
1answer
53 views

Computing Path Algebra of a Quiver

Let $Q$ be a quiver over defined as follows Then $KQ\cong$ $\begin{pmatrix}K&K&K\\0&K&K\\0&0&K\end{pmatrix}$, where $KQ$ is just the path algebra. What the professor did was ...
3
votes
0answers
104 views

Tensor product of algebras which is Frobenius.

Let $A$ and $B$ be two finite dimensional algebras over a field $k$. Let us suppose that the $k$-algebra $A\otimes_{k} B$ is Frobenius (or symmetric). Is it true that $A$ and $B$ are two Frobenius ...
1
vote
0answers
25 views

How do I map Df(w) to it's [lie] group/algebra representation?

E.G. For $p,w\in(\mathbb{R}^3,+,\times_\vartheta)$ with $(\mathbb{R}^3,+)$ a vector space and with $p=(r,s,t)$, $w=(x,y,z)$ where we have $p\times_\vartheta ...
4
votes
0answers
94 views

Easy Introduction to Representation Theory

I have a student that is interested in reading up on representation theory in her own time. She knows a small amount of linear algebra, what you would expect in a simple sophomore linear algebra ...
2
votes
1answer
53 views

Confusion in Lie algebra notes

I'm self-studying through these notes, and I ran into a roadblock on the page 38, chapter $sl(2)$ and its irreducible representations. Right after defining $U(sl(2)) \otimes_{U(b^+)} \mathbb C$ ...
1
vote
1answer
43 views

“Powers” of injective representations “contain” all irreducibles

Let $G$ be a finite group and let $\rho : G \to GL(V)$ be an injective representation. I need to prove that each irreducible representation of $G$ is contained in $\otimes_{i=1}^{n} \rho$ for some $n ...