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)

1
vote
1answer
29 views

Representation of indefinite Kac-Moody algebras

The Kac-Moody algebras are divided in three very distinct classes: finite-dimensional, affine and indefinite type. For the first class the finite-dimensional representation theory is very known. For ...
1
vote
0answers
38 views

Surjective quadratic mapping

Are there any known values of $n$ for which there exists a surjective quadratic mapping $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ with non-trivial zeroes?
0
votes
1answer
43 views

Question about Schur's lemma and irreducible representations of $S_n$

Schur's lemma says that if $M,N$ are two irreducible representations of a group $G$, then either $Hom_G(M,N)=0$ if $M,N$ are not isomorphic, or every $\varphi\in Hom_G(M,N) $ is invertible if they are ...
2
votes
2answers
40 views

A question on the decomposition of the group algebra $\mathbb{C}[G]$ of a finite group $G$

I am am very confused about a fundamental result in representation theory of finite groups. Please let me first introduce the setting. Let $G$ be a finite group. The group algebra $\mathbb{C}[G]$ is ...
1
vote
0answers
17 views

Help with a stage in Peter-Weyl proof: that “matrix entry” functions separate points

Edit The question as originally phrased was clumsy. What I really need is the simplest proof, or reference, anyone can rustle up of this: "for $G$ a compact Lie group, and $g$ and $h$ distinct ...
0
votes
0answers
18 views

Induced representation via diagonal embedding

I've been working on a couple of problems from Fulton-Harris' Representation theory book. In particular, for 6.11, we want to prove that $\mathbb{S}_v(V\otimes W)=\oplus C_{v\mu\lambda} ...
1
vote
1answer
42 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
0
votes
0answers
30 views

The Jacobson radical of an algebra contains every maximal $A$-submodule of an $A$-module

Let $A$ be an algebra over the field $F$. For an $A$-module define $\mathcal A(V) = \{ a \in A \mid Va = 0 \}$, the annihilator of $V$. Denote by $\mathcal M(A)$ a set which contains one isomorphic ...
0
votes
1answer
21 views

Representation of the full ring of $n\times n$ matrices

I am now reading the book 'representations of groups' by Boerner. On page 72, he states the theorem: Every representation of $M_n$ is completely reducible; every irreducible representation is ...
2
votes
0answers
37 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
0
votes
1answer
20 views

For a semisimple algebra and two $M$- and $W$-homogeneous parts for $M \ncong W$, why we have $M(A)W(A) = 0$.

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
2
votes
1answer
17 views

Problem with $SL(2)$ isometric action on a compact homogeneous space

Let $G=SL(2,\mathbb{R})$, fix any left-invariant Riemannian metric $g$ on $G$. Let $\Gamma$ be a cocompact discrete subgroup of $G$ and $X=G/\Gamma$. Because $\Gamma$ acts by isometries $g$ descends ...
7
votes
1answer
90 views

Symmetric power of tautological representation of $U(n)$

Let $S^kV$ be the $k$-th symmetric power of tautological representation of $U(n)$ how to see that it's irreducible? I'm trying to do it using weight, but with no benefits..
2
votes
0answers
64 views

Why there is an isomorphism $D(soc^{i}M) \cong DM/rad^{i}DM$

I am reading the book Elements of the Representation Theory of Associative Algebras, volume 1, by Assem et al. On page 162, it is written $D(soc^{i}M)\cong DM/rad^{i}DM$, where $DM=\mathrm{Hom}(M,k)$. ...
1
vote
1answer
45 views

About a Corollary of Yoneda's Lemma

I am reading Assem-Simson-Skowronski's book "Elements of The Representation Theory of Associative Algebras". I do not understand a Corollary 6.2, (IV. 6.2, Functorial Aproach to almost split). It says ...
0
votes
0answers
20 views

Reference request: bounded derived categories and their Auslander-Reiten quivers

I have some knowledge of Auslander-Reiten theory, tilting theory, derived categories and triangulated categories though I still find most proofs using derived categories in "Tilting Theory and Cluster ...
7
votes
1answer
79 views

Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ is a nilpotent ideal ($p$ is a prime, $G$ is a $p$-group)

Let $p$ be a prime and let $G$ be a finite group of order a power of $p$ (i.e., a $p$-group). Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ (to be read as $\left( ...
0
votes
0answers
22 views

Is there a general formula for the following Lie algebra quantity?

Consider the generators of $SO(n)$, written as $M_{\mu\nu} = - M_{\nu\mu}$ and they satisfy $$ \left[ M_{\mu\nu} , M_{\rho\sigma} \right] = i \left( \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\mu\sigma} ...
0
votes
0answers
11 views

$\text{Hom}$ to a projective $D[G]$-module for a complete DVR $D$

Suppose you have a complete DVR $D$ and a finite group $G$ with $D[G]$-modules $A$ and $B$. Does $B$ being projective imply that $\text{Hom}_{D[G]}(A,B)$ is $D$-free? Or should it be $A$ that's ...
0
votes
2answers
34 views

Representation of a group on a vector space induces a representation on another representation space?

Caveat: this is a very basic question. Suppose you have a representation of a group $G$ on a vector space $V$, let's say to be concrete $\mathbb{R}^n$. How is this representation related to the one ...
1
vote
0answers
38 views

Primitive of the matrix elements of irreducible representations of Lie groups

I am interested in the matrix coefficients $U_{ij}(g)$ of unitary irreducible representations of a Lie group $G$. In my case, these coefficients arise from the Peter-Weyl theorem. I would like to ...
1
vote
0answers
26 views

Tensor product of $Spin(2k)$ representations

I am trying to find the tensor product of spinor representations of $SO(2k)$. Labels are given as $$(n+I/2,I/2,\ldots,I/2,s)\otimes(I/2,\ldots,I/2).$$ Where $I$ and $n$ positive integers. How can ...
-1
votes
2answers
39 views

$G$ and $G/H$ representations

It is known that if a group $G$ has an invariant subgroup $H$ and the factor group $G/H$ has a known representation then this representation is also a representation of group $G$. But, how can we ...
0
votes
1answer
33 views

Bases for irreducible representations $V$ and $W$ are linearly independent implies basis for $V \oplus W$ is linearly independent.

Let $\rho: G \to GL(U)$ be a reducible representation with dimension $n$ of a finite group $G$ such that $U= V \oplus W$, with $V$ and $W$ irreducible. If $\{v_1, v_2, ..., v_k\}$ and $\{w_{k+1}, ...
0
votes
0answers
20 views

Integration on compact group

Let $K$ be a compact topological group, and let $(V,\pi)$ be a continuous representation of $K$ over the complex field $\mathbb{C}$. Denote by $\mathrm{d}$ the Haar measure on $K$. If $v\in V$ ...
1
vote
0answers
23 views

Young tableaux to Specht polynomial to Irreducible representation for $(1,3,5) \in S_5$

What I am trying to do? Work out the irreducible representation of the group element $(1,3,5) \in S_5$ for the partition $2+2+1$ . Motivation: Learn how to calculate irreducible representation from ...
4
votes
1answer
28 views

Description of the algebra of $G$-invariant polynomials by generators and relations

Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = ...
1
vote
1answer
29 views

Representation from Young tabloids

I am following the note Young Tableaux and the Representations of the Symmetric Group to work out a representation from a Young tableau for $S_n$. Here $\lambda$ is a partition of an integer $n$. In ...
1
vote
1answer
30 views

Notation in quantum groups.

The quantum group $U_q(sl_3)$ is generated by $E_1, E_2, F_1, F_2, K_1, K_2, K_1^{-1}, K_2^{-1}$ subject to some relations. I read some papers and there is a notation $K_{\lambda}$, where $\lambda$ is ...
5
votes
0answers
49 views

Construction of vector space isomorphism where $f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G$

Let $G$ be a finite group, $R = \mathbb{C}G$ the regular representation of $G$, and $\rho : G \to \text{GL}(V)$ a finite dimensional representation of $G$. Write $\sigma: G \to \text{GL}(V \otimes R)$ ...
0
votes
0answers
30 views

Young tableaux from Young diagram for $S_5$

In this question I have computed the Young diagram for the symmetric group $S_5$. Now I am trying to compute the Young tableaux which should be straightforward as follows. My questions: Am I ...
1
vote
1answer
21 views

Definition for non-degenerate module

[QUESTION] If $R$ be a ring, what is the meaning of a non-degenerate $R$-module? In a previous question post at (What is a non-degenerate module?), some experts said that if $M$ is a $R$-module such ...
2
votes
1answer
32 views

Confusing partitions of $S_5$ in two different sources

I am trying to understand the partitions of $S_5$ created by it's conjugacy classes but two sources have two different partitions. Source 1: Source 2: So, for example, in the first table, the ...
1
vote
0answers
59 views

Young diagram for $S_5$

I am trying to draw the Young diagram for $S_5$. I know the following pieces of information about $S_5$. The order of the group is $120$. The number of conjugacy classes and so partitions is $7$. ...
0
votes
1answer
40 views

Does anyone know examples on valuation field? [closed]

Does anyone know examples on valuation field? Also, I need some a good reference on that subject.
0
votes
0answers
42 views

Product of standard and sign representation of $S_5$

I am able to work out the sign representation of $S_5$ and standard representation of $S_5$. How do I compute the product of standard and sign representation of $S_5$? What kind of product do I need ...
2
votes
0answers
25 views

Archimedean Hecke Algebra for number fields

Suppose we have a $GL_1$ over a number field $F$. I am interested in a description for the archimedean Hecke algebra (always taking the maximal compact subgroup). We know will be the tensor product ...
0
votes
2answers
41 views

Standard representation of $S_5$

I am trying to determine the standard representation of $S_5$. I understand that it will be a map from group elements to $\mathbb{C}^4$. The character table is as follows. I understand that the ...
0
votes
1answer
33 views

$\text{Hom}_k(M,N)\cong M^*\otimes_k N$ as Hopf-algebra modules.

I'm reading Representations and Cohomology by D.J. Benson. At the beginning of the third chapter the following is explained: Let $\Lambda$ be a bialgebra over $R$ and $M,N$ left $\Lambda$-modules. We ...
2
votes
1answer
50 views

Character table for $G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle$

Let $G$ be the group of order $20$ defined in terms of generators and relations: $$G:= \langle x,y \mid x^5=y^4=1\text{ and }yx=x^2y\rangle.$$ Can anyone help me to derive the character table? ...
0
votes
1answer
24 views

One dimensional simple modules

Let $A$ be a finite dimensional algebra over an algebraically closed field and let $mod A$ denote the category of finite dimensional A-modules. Problem Suppose that every simple $A$-module is ...
1
vote
0answers
16 views

How to find the absolutely irreducible representations of $D_4$ dihedral group

I want to calculate the irreducible representations of $D_4$ and ultimately the absolutely irreducible representations. Right now what I do is since I know the group order is $2n=2(4)=8$ write down ...
1
vote
0answers
21 views

Irreducible projective representations of $\mathbb{Z}_2\times\mathbb{Z}_2$

I am currently learning about projective representations of finite groups and their reducibility. According to Schur's theorem (Schur 1904) the degree (dimension) of each irreducible projective ...
1
vote
1answer
14 views

Irreducible Components of Standard representation of SO(2) on $\mathbb{C}^2$

We denote by $SO(2)$ the group of $2 \times 2$ orthogonal matrices of determinant $1$ with real entries. We have a natural representation of $SO(2)$ on $\mathbb{C}^2$ given by matrix multiplication: ...
2
votes
1answer
52 views

Probability distribution on finite group

I'm preparing for finals and this is a practice question. I'm not really sure how to start, so any solutions/hints/starting points are appreciated. Suppose $P = \sum a_gg$ were a probability ...
0
votes
0answers
22 views

The property of a module that has a simple socle

Let M be a module that has a simple socle.I can get that M is indecomposable and all submodules of M contain socM. Are there any other properties of M? Can we character the structure of M? And is ...
1
vote
0answers
30 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
6
votes
1answer
65 views

$SL_2(\mathbb{F})$, decomposing $\mathbb{C}\{X\}$ into irreducible $G$-representations and dimensions

Let $\mathbb{F}$ be a finite field with $q$ elements and $H = \mathbb{F}^\times$, the multiplicative group of $\mathbb{F}$. It is known that $H$ is a cyclic group of order $q - 1$, so $\widehat{H} = ...
5
votes
1answer
28 views

Name for the module corresponding to a square matrix

I recently learned that for each $n \times n$ matrix $A$ with entries in some field $F$, there is a corresponding $F[x]$-module $M_A$. Namely, $M_A$ is the set $F^n$ with vector addition defined as ...
2
votes
0answers
40 views

Decompose $Sym^2 (V)$ into direct sum of irreducible $S_n$-subrepresentations, where $V$ is the $2$-dimensional representation

Decompose $\operatorname{Sym}^2 (V)$ into direct sum of irreducible sub representations. (Hint: Again consider the action on basis vectors.)" Here, $V=\Bbb C^2$, with its standard basis, and the ...