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)

3
votes
0answers
26 views

Finite-dimensional unitary representations of $SL_n(\mathbb{R})$

In Proposition 2.6.4 of his book Automorphic Forms and Representations, Bump is trying to prove that $SL_n(\mathbb{R})$ has no non-trivial finite-dimensional unitary representations. His argument is ...
2
votes
1answer
79 views

Rank of an action and definition of an orbital

Let $G$ be a group acting on a set $X$. In group theory sometimes it is helpful to consider the action of $G$ on $X\times X$; a good example is perhaps finding the dimension of ...
0
votes
1answer
27 views

Let $A=k[x]$ and $V=k[x]/((x-λ)^n)$. Find a filtration $V=V_0 ⊃ V_1 ⊃ \dots⊃ V_n=0$ such that the subsequent quotients $V_{i-1}/V_i$ are irreducible.

Let $A$ be the algebra $A=k[x]$ and let $V$ be the representation $V=k[x]/((x-\lambda)^n)$ for some $\lambda \in k$ and $n\in\Bbb N$. Find a filtration $V=V_0 \supset V_1 \supset \dots \supset ...
0
votes
1answer
27 views

Let $A=k[x]$ and let $V=k[x]/\big((x-λ )^{n} \big)$ for some $λ \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable.

Theorem. Let $A=k[x]$ and let $V=k[x]/\big((x-\lambda )^{n} \big)$ be a representation of $A$ for some $\lambda \in k$ and $n\in \Bbb{N}$. Then $V$ is indecomposable. This is a theorem in my book. ...
0
votes
1answer
24 views

A smooth non-stably trivial smooth vector bundle

This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some ...
3
votes
0answers
53 views

Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
1
vote
0answers
45 views

Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
8
votes
0answers
72 views

Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
0
votes
0answers
31 views

Question about the equivalence of two linear representations.

I would like to know if this approach is correct. I have two distinct permutation representations and I have to prove that the associated linear representations are equivalent. In order to do this I ...
0
votes
0answers
8 views

$\overline{\phi}: G/H \to GL(V)$ irreducible representation then $\phi= \overline{\phi}\circ \pi :G\to GL(V)$ it's irreducible

Let $H\trianglelefteq G$ be a normal subgroup of $G$ and let $\pi: G\to G/H$ be the canonical projection. Suppose that $\overline{\phi}: G/H \to GL(V)$ it's an irreducible representation. Define the ...
0
votes
0answers
24 views

Infinite Cyclic group representation

I am trying to learn Group representation and have a basic question regarding infinite cyclic groups. I am trying to find a representation of infinite cyclic group in $GL_n(\mathbb{C})$ and ...
1
vote
0answers
15 views

Restriction of a Specht module to the alternating group

Let $n\in\mathbf{N}$ and denote by $S_n$ the symmetric group on $n$ letters. For $\lambda\vdash n$ a partition of $n$ the Specht module $S^\lambda$ defines an irreducible representation. What ...
1
vote
1answer
21 views

Tensors furnish representations of the group

I'm bad at english, so what exactly does it mean in simple english that Tensors furnish representations of the group?
0
votes
0answers
31 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
39 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
24 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
23 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
24 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
21 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
31 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
34 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
39 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
26 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
17 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
16 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
41 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
22 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
30 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
48 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
37 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
47 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
29 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
45 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
19 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
48 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 ...
0
votes
0answers
19 views

dimension of the intertwiner (homomorphism) between equivalent irreducible representation is 1

How do I show this using the Schur's lemma? (Schur's lemma). Let $\phi, \rho$ be irreducible representations of $G$, and $T \in Hom_G(\phi,\rho)$. Then either $T$ is invertible or $T = 0$. ...
2
votes
1answer
24 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
16 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
46 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
26 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
32 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
31 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
33 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
43 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
41 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
47 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
43 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
42 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
8 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 ...
22
votes
2answers
364 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 ...