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)

4
votes
1answer
178 views

Irreps of $S_3=GL(2,2)$. Who is cuspidal?

$S_3 = GL(2, F_2)$. It has $3$ irreps - trivial, sign, standard 2d. Can one help me to understand general words about Irreps of $G(F_q)$ ("principal series", "parabolically induced", "cuspidal", ...
4
votes
1answer
294 views

Why do characters on a subgroup extend to the whole group?

As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally ...
1
vote
1answer
340 views

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. ...
7
votes
1answer
787 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
3
votes
3answers
291 views

A bit of direct sum confusion

So I'm reading through Serre's "Linear Representations of Finite Groups," and I'm a bit confused by what's probably a fairly minor point. However, subsequent proofs are hinging on it, so I figure ...
8
votes
2answers
345 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
1
vote
2answers
162 views

Product of two elements is identity implies they are mutual inverses..

Let $A$ be an associative unital n-dimensional algebra over field $F$. Show that if $ab=1$ for some $a,b \in A$ then $a=b^{-1}$
3
votes
1answer
307 views

Contragradient representation of a finite group

I am reading Serre's Linear Representations of Finite Groups and in an exercise in there he asks to show if $\rho$ is a representation of a finite group on $\textrm{GL}(V)$ with $V$ a finite ...
4
votes
1answer
553 views

Fundamental and the anti-fundamental representation of $U(n)$

I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate. ...
3
votes
0answers
168 views

Induction commutes with duals and tensor products

I need references for the following two results. Let $G$ be a finite group and let $H$ be a subgroup of $G$. Let $V$ be a finite-dimensional representation of $H$. $(\text{Ind}_H^G V)^{\ast} \cong ...
21
votes
0answers
431 views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...
2
votes
3answers
807 views

$so(4)=su(2)× su(2)$ contradiction

Modified question There was $sl(2,\mathbb R)$ used instead of $su(2)$ in the previous version. Thanks to MattE for pointing it out. I have seen it claimed many times that $so(4,\mathbb R)=su(2)\times ...
5
votes
1answer
492 views

Is there a difference between a model and a representation?

I'm thinking of models in logic here, vs. e.g. group representations. Is there a difference between a model and a representation? Could one not explain both at the same time? A model gives an ...
2
votes
1answer
1k views

Prerequisites for ‘Quantum field theory and representation theory: a sketch’ [arXiv:hep-th/0206135]

I'm interested in reading Dr. Peter Woit's article, Quantum field theory and representation theory: a sketch [hep-th/0206135]. What math and physics background would be needed? (A list of topics ...
1
vote
0answers
144 views

The universal enveloping algebra of a loop algebra as a quotient of the free associative algebra.

Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra and set by $\tilde{\mathfrak{g}}:=\mathfrak{g}\otimes_{\mathbb C} \mathbb{C}[t,t^{-1}]$ its loop algebra. How to express the ...
0
votes
1answer
273 views

Submodules of direct sums of simple modules

I'm reading these online notes on representation theory, and I don't fully understand this: Isn't $V\oplus(\bigoplus_{i\in I}S_i)$ a direct sum by definition, so we'd get $I=\{1,\cdots,n\}$? Can we ...
8
votes
1answer
210 views

Can $(\Bbb{R}^2,+)$ be given the structure of a matrix Lie group?

I have an assignment problem that is coming from Brian Hall's book Lie Groups, Lie Algebras and Representations: An Elementary Introduction. Suppose $G \subseteq GL(n_1;\Bbb{C})$ and $H ...
1
vote
1answer
118 views

The classification of irreducible representations of finite and compact Lie groups over $\mathbb{C}$

I just started reading a few lecture notes on representation theory and I was wondering about a big picture that one should keep in mind while reading through these lecture notes. Have all ...
5
votes
1answer
142 views

Dyslexia in group actions of $SL_2$ on binary cubic forms

The group $SL_2$ (say, $SL_2(\mathbb{C})$, but we could take $SL_2$ of anything else, or probably regard $SL_2$ as a group scheme) acts on binary cubic forms. (Or binary $n$-ic forms in general.) What ...
1
vote
1answer
118 views

Why is the Wedderburn formula in this case wrong?

in this question counterexample: degree of representation $\leq$ index of normal subgroup there was the answer (in the second comment under the answer), that the dihedral group $D_5$ hat exactly 3 ...
0
votes
2answers
39 views

Question on trace-weighted sums for irrep of finite group

For a finite group $G$, is the following true, where $\rho$ is a finite-dimensional complex unitary irreducible representation? $$\sum _{g \in G} \mathrm{Tr} (\rho(g)) \rho(g) = \frac{|G|}{n} ...
2
votes
1answer
148 views

counterexample: degree of representation $\leq$ index of normal subgroup

if I have a finite group $G$ with an abelian normal subgroup $N$ and an irreducible representation $\pi$ of $G$ over $K$. Then I know, that $deg(\pi) \leq [G:N]$, if $K$ has positive characteristic ...
1
vote
1answer
54 views

Squaring Class Sums of $S_n$ over characteristic $2$

I'm trying to find examples of groups $G$ where there is a non-Brauer pairs have an interesting conjugation action on it. I am currently trying the symmetric group in characteristic 2. The center of ...
6
votes
1answer
126 views

Why need two directions to make $\sim_{\rm wa}$ an equivalence relation?

Let $\pi$ and $\sigma$ be representations of a $C^*$-algebra $\mathcal{A}$. They are weak approximately equivalent ($\pi\mathbin{\sim_{\rm wa}}\sigma$) if there are sequences of unitary operators ...
2
votes
2answers
350 views

Finite Subgroups of $GL(n,\mathbb{C})$

In the Artin's book on Algebra, the author stated a theorem (Ch.9, Thm. 2.2): "A finite subgroup $G$ of $GL(n,\mathbb{C})$ is conjugate to a subgroup of $U(n)$. Here, $U(n)$ is the unitary group, ...
2
votes
2answers
187 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
4
votes
1answer
119 views

recovering a representation from its character

The character of a representation of a finite group or a finite-dimensional Lie group determines the representation up to isomorphism. Is there an algorithmic way of recovering the representation ...
1
vote
2answers
274 views

Antisymmetric powers of $SO(n)$ representation.

I am particularly interested for $SO(3)$. Let us say that I start with the natural/defining $3$-real-dimensional vectorial representation of $SO(3)$ and I choose the generator of rotation in the ...
2
votes
1answer
562 views

Weyl's unitarity trick

Weyl's unitarity trick creates from an irreducible representation of a compact group a unitary representation by averaging with a Haar measure. Does anyone know a reference to the paper (or book, ...
2
votes
2answers
110 views

Projective representations of loop groups

If $G$ is a Lie group and we take its loop group $LG$ why do we deal with projective representations of $LG$ and central extensions thereof? Where does the extra complexity come in to require us to ...
2
votes
2answers
73 views

How can I prove that $V$ is irreducible as a representation of $O(V)$?

Let $V$ be a finite dimentional vector space over a field $k$. Let $g(\cdot,\cdot)$ be a nondegenerate symmetric bilinear form on $V$. Let $O(V)$ be the subgroup of $GL(V)$ that preserves $g$. Then ...
2
votes
1answer
136 views

Normalizer of regular action made linear

For a finite group $G$ the regular action $\rho$ of $G$ on itself (by right multiplication) has the property that the normalizer of $\rho(G)$ in the symmetric group $S_G$ is isomorphic to the ...
1
vote
0answers
61 views

What is the relation between the representation of a Hecke algebra and the representation of the Coxeter group?

Given a Coxeter group $W$, there is a corresponding Hecke algebra (Iwahori-Hecke algebra). There are many results on the representation of the Hecke algebra. But why is this motivated? How is the ...
3
votes
1answer
469 views

Faithful irreducible representations of cyclic and dihedral groups over finite fields

How to determine all the faithful irreducible representations of $\mathbb Z_n$ and $D_{2n}$ over $GF(p)$, where $p$ is a prime not dividing $n$?
2
votes
1answer
151 views

Limit of discrete series representation $GL(2,\mathbb{R})$

I have a couple of elementary question about the limit of discrete series representation (=LDS) of $GL(2, \mathbb{R})$. I find the topic fairly confusing. Write $I(\mu_1, \mu_2, s)$ the prinicpal ...
4
votes
0answers
270 views

Correspondence between summands of a module and primitive idempotents of the endomorphism ring of the module

I am reading this paper and have the following questions. Let $A$ be a finite-dimensional algebra over a fixed field $k$. Let the finitely-generated $A$-module $M$ be a generator–cogenerator for ...
2
votes
0answers
252 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
4
votes
1answer
302 views

is this “adjoint” representation of $\mathfrak{gl}_2(\mathbb{F}_p)$ irreducible?

Let $\mathbb{F}_p$ be a finite field. There is an action (by conjugation) of $\text{GL}_2(\mathbb{F}_p)$ on the vector space of $2 \times 2$ matrices with coefficients in $\mathbb{F}_p$ that have ...
4
votes
1answer
154 views

Question about a question (irreducible representations of a semidirect product)

In the question Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$ the author is talking about irreducible representations of a semi-direct ...
3
votes
1answer
143 views

Lie algebra representation induced from homomorphism between spin group and SO(n,n)

Consider the spin group, we know it is a double cover with the map: $\rho: Spin(n,n)\longrightarrow SO(n,n)$ s.t $\rho(x)(v)= xvx^{-1}$ where $v$ is an element of 2n dimensional vector space V and ...
1
vote
1answer
127 views

Question about Fourier series

The Fourier series of a function $f: G \to \mathbb C$ where $G$ is a group is the representation of $f$ in terms of characters $\chi_g \in \mathrm{Hom}(G, S^1)$ of $G$. I understand the case where ...
0
votes
0answers
206 views

Representation theory for linear algebraic groups

In representation theory of linear algebraic groups, we consider the "irreducible" and "completely reducible" types of representations $(V, \rho)$, a $G$-representation is irreducible if {0},V are ...
3
votes
0answers
75 views

Conditions for equivalence of $A$-modules, given equivalence of $B$-modules for $B$ a subalgebra of $A$.

Pardon me if my terminology is messed up and I must admit that the following question is rather general, but here goes: Let $A$ be an algebra and $B$ a proper subalgebra of $A$. Suppose that, as ...
0
votes
1answer
149 views

Question about the socle of a finite-dimensional algebra

Let $n\in \mathbb{N}$ and $k$ be an arbitrary field. Is the socle of the algebra $k[x,y]/\langle x^2,y^{n+2}\rangle$ isomorphic to $k$? Is $k[x,y]/\langle x^2,y^{n+2}\rangle$ a symmetric algebra or ...
4
votes
0answers
434 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
2
votes
0answers
77 views

indecomposable $K[G]$ modules -> get irreducible $K[G]$ modules

if I found indecomposable $K[G]$ modules, are there any techniques to get from this irreducible $K[G]$ modules? (e.j. for $k=\mathbb{Z}/p \mathbb{Z}$ and $G=C_p$) regards, Khanna
4
votes
1answer
466 views

Motivation for studying quadratic algebras, Koszul algebras, Koszul duality

I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but ...
3
votes
1answer
80 views

Subspace spanned by powers of a faithful character

The following well known theorem can be found in many books on character theory: Let $\chi$ be a faithful character of a finite group $G$ and suppose that $\chi(g)$ takes on exactly $m$ different ...
3
votes
0answers
62 views

Concrete description of $T^*(G/B)$

Let $G=GL(n,\mathbb{C})$ and $B$ be a Borel in $G$. The variety $G/B$ is known to be a moduli of flags (i.e., a variety of Borels) with a bijection between a class of Borels and points of $G/B$. ...
3
votes
0answers
311 views

How many irreducible 3-dimensional representations does SO(4) have?

EDITED: I learned something from Qiaochu Yuan's answer to my other question on SO(4). I clarify my original question with more information below. My original question is: Does $SO(4)$ have exactly two ...