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)

2
votes
1answer
29 views

How can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\mathrm{End}V)[[x,x^{-1}]]$?

In the context of vertex operator algebras, if $V$ is a vector space, how can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\operatorname{End}V)[[x,x^{-1}]]$? The notation $V((x))$ is the ...
2
votes
1answer
70 views

Pushforward of a representation?

Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, ...
0
votes
0answers
16 views

Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$?

Let $g$ be a Lie algebra. Does $r \in \Lambda^2 g$ imply that $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] \in \Lambda^3 g$? Thank you very much.
1
vote
2answers
53 views

$G$-invariant complement to an infinite dimensional vector space

Let $G$ be a finite group and let $$\rho : G \to GL(V)$$ be a complex representation of $G$. Suppose we have an internal direct sum decomposition $$V=W \oplus U$$ where $W$ is infinite dimensional ...
0
votes
0answers
39 views

Closure of algebraic groups

Let $\phi: G\rightarrow V$ an embedding, with $G$ a complex algebraic group and $V$ a vector space (actually a $G$-representation). Is it true that the closure (in the Zariski topology) of $\phi(G)$ ...
0
votes
1answer
27 views

Linear representation of finite group.

Let $G=\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$ and $K=\mathbb C$ a field. I have to give the non equivalent $K-$ linear representation of degree 1. I can do it, but I wanted to know ...
0
votes
1answer
21 views

$G$ is a group of order $12$ admitting an irreducible $3-$dimensional reprsentaion. What are the dimensions of its irreducible representaions?

Given $G$ is a group of order $12$ admitting an irreducible $3-$dimensional representaion. What are the dimensions of its irreducible representaions? Is there a theorem that gives an answer? I am ...
1
vote
1answer
25 views

How to calculate the weights and weight vectors of $Sym^n(V)$

I am wondering how to calculate the weights and weight vectors of $Sym^n(V)$ note: I am working with in $\mathfrak{sl}_2$ From my lecture notes, I know that the weight vectors $Sym^2(V)$ are $v_j$ ...
0
votes
1answer
49 views

Composition factors of injective indecomposable and projective indecomposable modules

Let $A$ be a finite-dimensional algebra over an arbitrary field $K$. Let $L_1$ and $L_2$ be simple modules such that $L_1 \not \cong L_2$. Let the $A$-module $Q_1$ be the injective hull of $L_1$, ...
1
vote
1answer
38 views

How to determine irreducible modules of $\text{C}_5$ over $\mathbb{Z}_2$?

Since $2\nmid 5$ and $5=1^2+2^2$, we can apply Maschke's theorem: $\mathbb{Z}_2\text{C}_5=V_1\oplus(V_2\oplus V_3)$ with $V_1$ the trivial module and $V_2,V_3$ 2-dimension irreducible modules. But I ...
0
votes
1answer
52 views

Representation of a linear map as a matrix.

Let $I = \left < f_1, \dots, f_n \right > \subset R$ be an ideal generated by homogeneous elements where $\deg(f_i) = d_i$ and $\phi$ be the graded $R$-mod homomorphism $$\phi: R(-d_1) \oplus ...
3
votes
1answer
41 views

finding high weight vector in Verma module

Let $\frak{g}$ be a (semi-)simple lie algebra. Let $\lambda$ be a dominant integral weight. Denote $L(\lambda)$ to be the irreducible representation of highest weight $\lambda$. From BGG resolution, ...
4
votes
1answer
144 views

Representations of a group and its normal subgroup

While I was thinking the problem that I asked previously, I encountered this exercise problem in the book Tensor categories. The problems (Exercise 4.15.3) is that Let $N$ be a normal subgroup of ...
2
votes
1answer
38 views

Representations of a group and its subgroup

Let $G$ be a finite group. Let $(V, \pi)$ be a representation of $G$, where $V$ is a finite dimensional vector space and $\pi:V\to V$ is an automorphism. Restricting this action to a subgroup $H$ of ...
2
votes
1answer
21 views

irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
4
votes
1answer
34 views

Apparent Contradiction to Weyl's Theorem

Let $L$ be $sl(2)$, i.e., $L=span\{h,e,f\}$, where $[h,e]=2e$,$[h,f]=-2f$,$[e,f]=h$. This is semi-simple. Suppose I create a module $V=span\{v_1,v_2,v_3\}$ and define actions as follows: ...
1
vote
1answer
32 views

Wedderburn component of $\mathbb C[G]$ corresponding to a contragredient character

Let $G$ be a finite group and let $\phi$ be an irreducible character of $G$ over $\mathbb C$. How does the Wedderburn component of $\mathbb C[G]$ corresponding to the contragredient character of ...
2
votes
0answers
30 views

A certain generalization of flag varieties

The most standard notion of a (partial) flag is the sequence of vector subspaces ordered by inclusion, $$ V_1\subset V_2\subset \ldots \subset V_k=V. $$ Given the dimensions of the subspaces $\dim ...
0
votes
1answer
19 views

Find all irreducible representations $\pi$ of G and matrices $\pi$(x) and $\pi$(y) with respect to suitable basis

Here is the information I have: G is the group of order 21 it is generated by two elements $x$ and $y$ $x^7=y^3=e$ $xy=yx^2$ I want to do two things: Construct all irreducible representations ...
1
vote
1answer
50 views

Compute the character of $\pi$ and decompose into irreducible representations

$V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ $\mathbb{C}$ has standard basis $e_1, e_2$ and V has basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ $\pi$ is a representation of ...
1
vote
1answer
24 views

Irreducible rep, group centre: $\pi$(z) $=\lambda$(z)v

Note: not sure if title is displaying well; formula is directly below lambda is a scalar that I need to show exists $\pi$(z) $=\lambda$(z)v lambda is a scalar that I need to show exists I want to ...
2
votes
1answer
32 views

Let ($\pi, V$) be a representation of $G$ with character $X$: if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations

Let $(\pi, V)$ be a representation of $G$ with character $X$. Prove that if $\langle X, X\rangle=2$ then $V$ is the sum of two irreducible representations I was under the impression that the inner ...
0
votes
0answers
19 views

Why are two tabloids of the same shape column permutations of each other?

Let $\{T_1\}$ and $\{T_2\}$ be two tabloids of the same shape. My professor today said in class that this implies $\{T_2\}=\pi\{T_1\}$ for some $\pi\in C_{T_1}$. What does this mean? Does $\pi$ act ...
0
votes
0answers
21 views

Procedure to construct a map from the automorphism group of a graph to the natural permutation representation

Let $\Gamma$ be a graph with $n$ vertices. Let $\varphi_\Gamma$ be the map from the symmetric group $S_n$ to the space of natural permutation representation $\text{Mat} \left(n, \mathbb{C}\right)$ ...
1
vote
0answers
14 views

Representation of Weyl algebra

Let's consider an algebra $W$, generated by a family of differential operators of type $$\sum_{k=0}^{n}{a_{k}(x) \cdot \frac{d^{k}}{dx^{k}}}$$ (may also known as Weyl algebra). I would like to prove ...
2
votes
0answers
66 views

Weyl group of complex Lie group

Let $G$ be a compact connected Lie group with maximal torus $T$. The Weyl group is defined by $$W:=N_G(T)/T.$$ Now, $G$ has a complexification $G_{\Bbb C}$ with maximal torus $T_{\Bbb C}$ which is the ...
0
votes
1answer
38 views

How to show that $\frac{1}{u} q^{2u\frac{d}{du}} =q^2 q^{2u\frac{d}{du}} \frac{1}{u} $?

Let $q$ be a complex number. Let $u$ be a variable. How to show that $\frac{1}{u} q^{2u\frac{d}{du}} =q^2 q^{2u\frac{d}{du}} \frac{1}{u} $? I think that it suffices to show that $q^2 u ...
5
votes
1answer
54 views

Relation between reflection group and coxeter group

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, ...
0
votes
1answer
34 views

Prove that $(\mathfrak{su}(2))^* \cong \mathfrak{sb}(2)$

Let $\mathfrak{su}(2)$ be the Lie algebra with basis elements $$ e_1=\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} , \quad e_2=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} , ...
2
votes
1answer
38 views

Finite linear Representations are necessarily distance preserving

Suppose I have a linear representation of a finite group $G$. That is a homomorphism $$ \pi: G \rightarrow GL(\mathbb{R}^n) $$ Meaning a collection of matrices, which under matrix multiplication form ...
1
vote
1answer
24 views

Ideal as subrepresentation of algebra

I am confused the statement that ideal is a subrepresentation of regular representation of an algebra. Actually I am not very clear about the definition of regular representation of an algebra, ...
1
vote
1answer
18 views

How to define a quiver of basic and non-connected associative algebras

Recently, I am reading a book "Elements of the Representation Theory of Associative Algebras". Let $A$ be a basic and connected finite dimensional $~\mathbb{K}$-algebra and $\{e_1, e_2, \cdots, ...
0
votes
1answer
36 views

A quotient module of a Lie algebra

Let $L$ be a Lie algebra. If $A$ and $B$ are $L$-submodules of an $L$-module $V$, such that $A\subset B$ and $I\cdot B\subset A$ for some ideal $A$ in $L$. I want to understand why this implies that ...
5
votes
1answer
62 views

Linear independence for functions on Z/m

Let $p$ be prime and consider the functions $f_k:(\mathbb Z/p)\backslash\{0\}\rightarrow\mathbb R$ defined by $f_k(x)=\csc^2\left(\frac{k\pi x}{p}\right)$. Question: How might I show that the ...
29
votes
2answers
454 views

Haar Measure of a Topological Ring

A topological ring is a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and ...
3
votes
1answer
72 views

Homorphism from $B(G)$ to $\mathbb{Z}$

Let $G$ be a finite group, and $B(G)$ be its Burnside ring. Show that each ring homorphism $\varphi:B(G)\to\mathbb{Z}$ is the mark of some $H\le G$, i.e. it maps to an equivalent class of finite ...
0
votes
0answers
19 views

Dimension of character in square equals index of center.

Let $G$ be a finite group and let $\chi$ be an irreducible character. Assume that $G/Z(\chi)$ is abelian, how can I prove that then $\chi(1)^2=\mid G:Z(\chi) \mid$? Note that $Z(\chi)= \{g \in G : ...
0
votes
1answer
17 views

Definition of multiplicity of trivial representation

Suppose we have a group representation $$\rho: G \to GL(V)$$ where $V$ is a finite-dimensional complex vector space and $|G| < \infty$. I have been confusing myself about the definition of ...
0
votes
1answer
41 views

Confused by two different perspectives on $G$-vector bundles

I'm trying to understand how these two perspectives on vector bundle with a $G$-action come together. Perspective 1: Let $P \to X$ be a principal $G$-bundle. The associated bundle construction gives ...
4
votes
1answer
52 views

Question on irreducible character.

Suppose that $\chi \text{Irr}(G)$, i.e $\chi$ is an irreducible character, and assume that $G/Z(\chi)$ is abelian, where $Z(\chi)=\{g \in G : \mid\chi(g)\mid = \chi(1) \}$. How can I prove thet ...
1
vote
1answer
55 views

Congruence subgroup of $\mathbb{GL}_n(\mathbb{Z}_p)$

In course of my research I met the following situation : 1) I have a bunch of open subgroup (so of finite index) in $\mathbb{GL}_{n}(\mathbb{Z}_p)$. 2) My groups arises naturally as stabilizers of ...
1
vote
1answer
51 views

Tensor product group representations and spaces of intertwiners.

Let $V_{1}$, $V_{2}$, $W_{1}$, and $W_{2}$ be the carrier spaces of representations of some finite group $G$. Suppose also that $G$ acts trivially on $V_{1}$ and $V_{2}$. I would like to prove the ...
0
votes
1answer
29 views

Second symmetric power of sum of representations

Let $\mathfrak{g}$ be a complex Lie algebra of type $A_{n-1}$. Consider representation of $\mathfrak{g}$ on direct sum of complex vector spaces which is given by the highest weight ...
1
vote
1answer
23 views

Show $\rho_1 \oplus \rho_2 : G → GL(V \oplus W)$ is a homomorphism

Show that the direct sum $V \oplus W$ is a representation of the finite group $G$. Given that $V, W $ are representations. attempt: Suppose that $V, W$ are vector spaces. Then define $\rho_1: G → ...
1
vote
1answer
34 views

If a group algebra acts regularly on a module, can this module be identified as a left ideal?

To be more specific, I am looking at $F_2[D_p]$, where $D_p$ is the dihedral group of order $2p$. If this group acts regularly on the basis of a vector space $F_2^{2p}$, and there is a subspace of ...
2
votes
0answers
28 views

Sum of elements in row of character table is positive integer.

If $G$ is a (finite) group, how can I prove that in the corresponding character table, the sum of the elements in any row is a non-negative integer? The hint in the book says that I should let $G$ act ...
2
votes
1answer
20 views

Compute the sum of number of fixed points

$\newcommand{\def}{\mathrm{def}}\newcommand{\std}{\mathrm{std}}\newcommand{\triv}{\mathrm{triv}}$Suppose $V_\def, V_\std, V_\triv$ are the defining , standard and trivial representations of the ...
0
votes
0answers
23 views

Show $\phi$ is a isomorphism as a lie algebra homomorphism

Show $\phi$ is a isomorphism as a lie algebra homomorphism $\phi: \textbf{su}_2 \bigotimes_{\mathbb{R}} \mathbb{C}\rightarrow sl_2(\mathbb{C})$ and $\phi: a(I \bigotimes 1)+b(J \bigotimes 1)+c(K ...
1
vote
1answer
24 views

Schur-Weyl duality for general representations

The classical Schur-Weyl duality deals with the decomposition of $V^{\otimes k}$ into irreps of $S_k\times GL(n)$, where $V=\mathbb{C}^n$ is the defining irrep of $GL(n)$. Is there a version of the ...
0
votes
0answers
25 views

Compute $\Sigma_{\pi \in S_n} f(\pi)$ and $\Sigma_{\pi \in S_n} f(\pi)^2$.

Suppose $V_{def}, V_{std}, V_{triv}$ are the defining , standard and trivial representations of the symmetric group $S_n$. And let $V_{def} \cong V_{std} \oplus V_{triv}$, and suppose the characters ...