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
1answer
33 views

Representations of two-dimensional Lie algebra

It is widely known that there is only one $2$-dimensional non-abelian Lie algebra: it can be generated by two vectors $e_1$ and $e_2$ such that $[e_1,e_2]=e_1$. Let us lenote it by $L$. The question ...
5
votes
1answer
42 views

Weight space of a representation of ${\frak sl}(2,\mathbb C)$

Suppose $(\pi,V)$ is a finite representation of $SU(2)$. Then there's an induced representation $(\pi_*,V)$ of the complexified ${\frak su}^\mathbb C(2) = {\frak sl}(2,\mathbb C)$. Show that the ...
2
votes
2answers
66 views

Nilpotent groups are monomial

I'm trying to show that a nilpotent group $G$ is monomial; i.e., that every irreducible representation $\rho$ of $G$ satisfies $\rho = \text{Ind}_H^G(\tau)$ for some $H \leq G$, $\tau$ a one ...
1
vote
1answer
68 views

Representation of $GL(V)$ on exterior algebra

I have a couple ideas for the following problem and would like verification, since I am still shaky with representation theory. Let $V$ be a $n$-dimensional vector space over a field $k$ and let ...
2
votes
0answers
43 views

Divided powers in the context of elements of the Schur algebra

I am currently reading through the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl}_2$. Here it defines the following matrices $e := ...
0
votes
1answer
46 views

Every unitary representation is a direct sum of cyclic representations.

Every unitary representation is a direct sum of cyclic representations. it can be proved without the Zorn's Lemma ?
0
votes
1answer
46 views

Character table from a representation?

Can anybody explain how to contruct a character table. A good explained example will be fantastic to me. For example, the character table of $S_4$. I'm quite desperate about representation theory!!
3
votes
1answer
23 views

A class function $f$ is a character if and only if $(f,\chi_{q_i})_G $ is a non-negative integer, for all irreducible characters $\chi_{q_i}$

I'm currently revising representation theory and I'm a bit stuck trying to prove the converse of the above statement. $(\Rightarrow)$ is straight forward because if $f$ is the character of a ...
1
vote
1answer
35 views

Showing $V\cong W$ if $dim V^H=dimW^H$

I am trying to show that if $W$ and $V$ are to $\mathbb{Q}[G]$ modules then $V\cong W$ if $dim V^H=dim W^H$ for all cyclic $H\leq G$ ( where $V^H$ denotes the invariant subspace under $H$ So I have ...
1
vote
2answers
44 views

Non-unitary representation

How to prove $\pi :\mathbb R\to \mathbb C^2$, defined by $t\mapsto \begin{pmatrix} 1 & t\\ 0 & 1\end{pmatrix}$ is a non-unitary representation? Is the following correct? $\pi$ is a ...
1
vote
1answer
24 views

Prove that a representation have a base and it's irreductible

I'm quite new in representations and I'm trying to do next problem: (It's supposed that I don't know anything about characters theory) We want to study $S_3=(\tau=(123),\sigma=(1,2)\,|\, ...
1
vote
0answers
32 views

$\mathbb{1}\uparrow_H^{G}$ is the permutation representation on $G/H$

Is the following correct? If we have $G$ is a group with $H\leq G$ and we take $\mathbb{1}$ to be the trivial character on $H$ then I am trying to show that $\mathbb{1}\uparrow_H^{G}$ is then the ...
1
vote
2answers
57 views

the presentation of $SL(2,\mathbb{Z})$

There is a natural presentation $SL(2,\mathbb{Z})\hookrightarrow GL(2,\mathbb{R})$, are there other presentations in real dimension 2? Or there is a classification of all the presentation of ...
18
votes
5answers
265 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
1
vote
0answers
37 views

Showing that $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real

I am trying to show that for a finite group $G$ and $g\in G$, $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real for all $\chi$ irreducible characters of $G$. I have the following: I first want ...
1
vote
1answer
58 views

Group representations and short exact sequences

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of groups. What can be said about group representations of $B$ if we assume a complete classification of the ...
0
votes
2answers
28 views

Showing 1-dim representations factor through $G/G'$

I have a question that is as follows: Show that the 1-dim complex representations of $G$ are those that factor through $G/G'$. Now I am a bit confused by this question, what exactly does it mean ...
0
votes
0answers
34 views

Why is character sum of eigenvalues?

Working my way through a first course in Representation theory, I run into some difficulties (due to bad knowledge of linear algebra) with that said I am wondering about the following. Let $\Theta : ...
2
votes
0answers
44 views

Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal ...
1
vote
0answers
15 views

restriction of spin representation to block diagonal subgroup

What is the restriction of the (complex) spin representation of $so(n+m)$ to the block diagonal subalgebra $so(n)\times so(m)$? A naive guess is that it is the (complex) tensor product of the two ...
0
votes
1answer
35 views

Question on Frobenious Reciprocity

I have in my notes the statement of frobenoius reciprocity in the following two ways: If $H\leq G$ and suppose that we have $\chi_1$ a character of $G$ and $\chi_2$ a character of $H$. Then: ...
1
vote
2answers
43 views

Representation of dense Subset

let $\mathcal B \subset \mathcal A$ a dense subset of a C*-algebra $\mathcal A$. I have a representation for $\mathcal B$. Can I then conclude that this is somehow also a representation for ...
0
votes
1answer
45 views

Generalized Clifford's Theorem

A typical statement of Clifford's theorem is the following: Let V be a finite dimensional irreducible representation of a group G, and let N be a normal subgroup of finite index in G. Then the ...
0
votes
0answers
61 views

decomposition of representation kG of G

Decompose $kG$ in to indecomposable representations and decide which summands are irreducible. (a)$G=S_2,k=\mathbb{C}$ (b)$G=\mathbb{Z}/3\mathbb{Z},k=\mathbb{C}$ ...
0
votes
1answer
79 views

The number of a set of irreducible projective characters vs the number of the ordinary characters of a finite group G.

I need valid references to show that the number of a set of irreducible projective characters with non-trivial factor set is always strictly less than the number of the ordinary characters of a ...
4
votes
0answers
21 views

“twisted” powers in symmetric monoidal categories

Suppose $C$ is a symmetric monoidal category with monoidal product $\wedge$, $X$ is a $G$-object for some finite group $G$ (say), and $T$ is a finite $G$-set of size $n$. The $n$-fold monoidal power ...
0
votes
0answers
39 views

Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
2
votes
1answer
42 views

An $\mathrm{Ad}$-invariant inner product that agrees with the trace

Let $\mathfrak{g}$ be a real semisimple Lie algebra. Then, we have an obvious $\mathrm{Ad}$-invariant inner product (I don't care about positive definiteness) on $\mathfrak{g}$, namely the Killing ...
0
votes
0answers
48 views

Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
3
votes
4answers
76 views

nonsemisimple $k$-algebra

Say $k$ is a field and is the $k$-algebra $A:=\prod_{i\in \mathbb N} k$ (multiplication is defined componentwise) semisimple? If not, what would be a submodule of the regular representation , that is ...
1
vote
0answers
41 views

Lie group representatiom - quasi-equivalent representations

Let $T$ and $U$ be unitary representations of a conected simply conected nilpotent Lie group, such that all irreducible subrepresentations of $T$ and $U$ are the same. If $T$ and $U$ are finite, then ...
1
vote
0answers
303 views

Fourier Transform on compact groups

I'm trying to get my head around the concept of Fourier Transform on a compact group. The standard definition is $$\widehat{f}(\pi)=\int_Gdg\,f(g)\pi(g)$$ where $\pi\in\widehat G$, the Pontryagin ...
0
votes
0answers
22 views

Representing numbers in unique p-adic to matrix representation, is there a way?

What I am wanting to do, if it is possible is find unique matrix representations for the p-adic representation of numbers. So for example, say I have the number 1365 = 3*5*7*13. Now I could take ...
0
votes
1answer
40 views

The sign representation of the Symmetric Group

I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. On page 11, after ...
2
votes
0answers
27 views

What is a weight vector in layman's terms?

I am trying to understand what a weight vector is but the wikipedia is in greek. What is the jist?
0
votes
0answers
19 views

Representation vector expansion in subgroup representation vectors

Imagine I have a Group $G$ and a representation of it acting on a vector space $V_G$. Imagine that I have a subgroup of $G$ named $H$ and a representation of $H$ acting on $V_H$. Consider a vector ...
1
vote
1answer
47 views

isomorphism $\mathrm{Hom}_G(k,V)\to V^G$

Let $V$ be a finite dimensional representation of $G$ over a field $k$ and let $G$ act on $k$ trivially. The evaluation at $1\in k$ gives an isomorphism $\mathrm{Hom}_G(k,V)\to V^G$, where ...
0
votes
1answer
12 views

Consider the action of $S_3$ on $C^3 = \{ (x,y,z) | x + y + z = 0\}$. Show that $\rho$ is irreducible.

The action is defined as $\rho_g (x_1, x_2, x_3) = (x_{g(1)}, x_{g(2)}, x_{g(3)})$. For example: if $g=(12)$, then $g(2,3,-5) = (3,2,-5)$. I understand that the action just permutes the elements, ...
2
votes
0answers
38 views

Representations of Nilpotent Lie Algebras

Let $\mathfrak{g}$ be a rational, nilpotent Lie algebra. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. But this representation generally is not ...
0
votes
0answers
25 views

About First Orthogonality theorem

Let $G$ be a finite group, $(U,\theta_1)$ and $(V,\theta_2)$ be irreducible $k$-representations, $m=\dim_k U$ and $n=\dim_kV$. By the way, $K$ is an algebraically closed field. Let ...
1
vote
1answer
21 views

Dimension of intertwining space of unitary representation

I'm currently trying to read through an article by Poguntke, to be found here. The main theorem of the article is the following: Theorem. Let $\pi$ and $\pi'$ be irreducible unitary ...
2
votes
0answers
29 views

The embedding $L^2(\Gamma(N)\backslash\text{SL}_2(\mathbb{R})) \hookrightarrow L^2(\text{SL}_2(\mathbb{Q})\backslash \text{SL}_2(\mathbb{A})))$?

Let $\mathbb{A}$ be the ring of adeles. Let $N$ be a natural number. Let $\Gamma(N)=\ker(\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})))$. I saw the following embedding of ...
0
votes
0answers
30 views

$C[G]$-module map

Suppose: $z$ is an element of the center of $G$, and let $V$ be a $C[G]$-module, and let $T_z$ be the linear transform that arises from multiplication by $z$ ($T_z(v) = zv$). Then I want to show that ...
1
vote
1answer
45 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
1
vote
0answers
24 views

Universality of restricted representations

Suppose that $H$ is a subgroup of a finite group $G$. Given an irreducible representation $\rho$ of $G$, this creates a (possibly reducible) representation $\rho'$ of $H$ obtained by restricting ...
0
votes
0answers
29 views

Subring of $\mathbb{C}[S_4]$

I'm doing a question from an old exam paper and I'm stuck on the following: Does $\mathbb{C}[S_4]$ contain a subring isomorphic to $M_2(\mathbb{C})$? Here subring doesnt need to contain the unit 1. I ...
1
vote
1answer
65 views

Intertwining map in Schur's Lemma

I am learning Schur's Lemma from page 4 here. It says Schur's Lemma 1. If $(\rho_1, V_1)$ and $(\rho_2, V_2)$ are irreducible representations of a group $G$, then any nonzero homomorphism $\phi : ...
0
votes
1answer
29 views

When does a sequence of finitely generated $k[G]$ modules split?

I am self studying some non-commutative algebra, and I want to make sure I don't confuse myself. Here is what I am thinking: Let A and B be finitely generated $k[G]$-algebras, for $G$ a finite group ...
0
votes
0answers
28 views

Quotient braid group as a representation of SU(n)

I am working with the quotient braid group $B_3 (3) = B_3 / \langle\sigma_1 ^3\rangle$, where I construct a vector space $V$ so that every element $a \in B_3 (3)$ has a corresponding basis vector ...
1
vote
1answer
54 views

Group algebra is a tensor product?

Am I correct in describing the group algebra $R[G]$ as $R \otimes_{Z} G$? (As a tensor product of $Z$-algebras.) There is clearly a map $R \times G$ to $R[G]$, just by sending $(r,g)$ to $rg$, and ...