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
4answers
70 views

What is a representation?

I know the definition is given as follows: A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means Can someone help me gain some ...
2
votes
0answers
36 views

A question on Auslander-Bridger transpose

I am learning Auslander-Reiten Theory. When I read the book Frobenius Algebras I. Basic Representation Theory, I have some problems. On page 236-237, The Proposition Proposition 4.5. Let $M$ and ...
1
vote
0answers
14 views

Finite dimensional, irreducible representations of the Lie superalgebra gl(1|1)

I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial ...
0
votes
0answers
31 views

On a theorem of R. Brauer about character theory.

Let $\operatorname{Cl}(g_1),\ldots,\operatorname{Cl}(g_r)$ be the conjugacy class of a finite group $G$ and let $C_i \in C[G]$ be the sum of the element in $\operatorname{Cl}(g_i)$ where $C[G]$ is the ...
1
vote
1answer
15 views

The Lie algebra of the generalized unitary group $\{g \in GL_n(\mathbb{C}) : gS\bar{g}^t=S\}$ is $\{XS+S\overline{X}{}^t=0\}$

Let $ S \in M_n(\mathbb{C}) $ be a square matrix and let $ X$ be in the Lie algebra $\mathbb{\mu(S)} $ of the generalized unitary group, $$U(S):=\{g \in GL_n(\mathbb{C}); gS\bar{g}^t=S\} .$$ ...
1
vote
1answer
26 views

Adjoint matrix in $\mathbb{so_3}$

$\mathbb{so_3}$ has the following basis: $X_1=\begin{bmatrix} 0 & & \\ & &1 \\ & -1 & \end{bmatrix}$, s: $X_2=\begin{bmatrix} & & 1\\ & 0& \\ -1 & & ...
0
votes
1answer
31 views

Show that $ (\pi(g)\phi)(v)=\phi({^t}gv) $ defines a representation

Let $ G=SL_2(\mathbb{C}) $ and consider the action of $ G $ on the space of smooth functions on column vectors $ v \in \mathbb{C^2} $ given by: $ (\pi(g)\phi)(v)=\phi({^t}gv) $ Question 1: Show that ...
1
vote
1answer
30 views

Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g} $ we can define the adjoint representation as: $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $ I am ...
0
votes
0answers
22 views

Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R}) $ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
1
vote
2answers
65 views

Show that group has a nontrivial normal subgroup.

Let $G$ be a group and assume that it has an irreducible (complex) character of degree $2$. How can I prove that then $G$ has a non-trivial normal subgroup? I tried to prove that for the ...
1
vote
0answers
31 views

Quick question on Pauli matrices and u(2)

The wiki page for Pauli matrices states "Together with the 2 × 2 identity matrix I (sometimes written as σ0), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the ...
3
votes
1answer
61 views

Show that any representation of $\mathfrak{sl}(2,\mathbb C)$ is a subrepresentation of $V^{\otimes m} \oplus V^{\otimes {(m+1)}}$ for some $m$

Suppose $M$ be a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb C)$, then there is a positive integer $m$ such that $M$ is isomorphic to a subrepresentation of $V^{\otimes m} \oplus ...
1
vote
1answer
40 views

How to check if two group representations are equivalent

If we have two representations of the same group $G$, say $\phi$ and $\psi$, they are called equivalent if a $U$ exists such that $U^{-1} \phi(g) U=\psi(g)$ holds for all $g$ in $G$. However, given ...
1
vote
2answers
18 views

Show $v\in FS_n$ is an $F$-multiple.

This is coming from Exercise 8 in Section 18.1 of Dummit and Foote. We are talking about representation theory and in particular focusing on Example 3 and 10 in this section. Let $n\in ...
3
votes
0answers
24 views

Question surrounding young symmetrizers

Let $\lambda$ be a partition of $n$, and let $T$ be the standard tableau associated to $\lambda$ (write the Young diagram of $\lambda$ down and fill in the boxes with $1$ through $n$ left to right, ...
1
vote
1answer
10 views

Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \oplus_{J}\textbf{r}_J $$ where $\textbf{r}_J$ are ...
0
votes
1answer
36 views

Dual representations of fundamental representations of a Lie algebra.

Let $g$ be a Lie algebra. Let $V(\omega_i)$, $i=1,\ldots,n$, be the fundamental representations. Are the dual representations $V(\omega_i)^*$ highest weight representations? The dual representation ...
1
vote
1answer
19 views

Relations of $S^2 V$ and heighest weight representations of Lie algebras.

Let $V$ be the natural representation of $sl_n$. Then $V = V(\omega_1)$, where $\omega_1$ is the first fundamental weight. We have $\Lambda^2 V = V(\omega_2)$. Is $S^2 V = V(\lambda)$ for some weight ...
0
votes
1answer
53 views

Do we have $\mathbb{C}[V^*] \cong S(V)$ or $\mathbb{C}[V] \cong S(V)$? [closed]

Let $V$ be a vector space over $\mathbb{C}$ and $V^*$ its dual vector space. Let $\mathbb{C}[V^*]$ (resp. $\mathbb{C}[V]$) be the coordinate ring of $V^*$ (resp. $V$) and $S(V)$ the symmetric algebra. ...
0
votes
0answers
25 views

Definition of coroots

I'm having a bit of trouble with the definition of coroots. From textbooks, we know that given a root system $\Phi$ with $\alpha \in \Phi$, there exists a coroot system $\Phi^{\vee}$ with ...
1
vote
1answer
20 views

$G \ne [G,G]$ from irreducible representations

For a group $G$ of order 24, how can I prove using restrictions on possible irreducible reps, that $G \ne [G,G]$? A priori of knowing how many conjugacy classes there are, I can get to $24 = 1^2 + 1^2 ...
1
vote
0answers
38 views

Prove it exists integers so to write character as linear combination.

Let $G$ be a group and suppose that $\chi$ is a character on $G$. Furthermore, suppose that $\chi(g)$ is constant for all $g \not=1$, prove it exists integers $a,b$ such that $$\chi=a1_G + b\chi_0$$ ...
0
votes
0answers
23 views

Prove $V$ is simple $\iff$ all non-zero vectors are cyclic

I am working on some Representation Theory practice questions and I think I have given a valid proof of : Prove $V \ne 0$ is a simple A-Module$\iff$ all non-zero vectors are cyclic $"\leftarrow"$ ...
3
votes
1answer
42 views

Find conjugacy classes of $G= \left\langle a, b \mid a^4, b^2=a^4, aba=b \right\rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, ...
0
votes
1answer
43 views

Character of $Sym^2(V)$ and decomposition into irreducible representations

Let $G=S_3$ be the symmetric group on three elements, whose character table is given as follows: Let $V$ be the unique irreducible representation of dimension $2$ Question 1: Compute the character ...
2
votes
1answer
26 views

Character regular representation

Consider the regular representation of a finite group $G$ and let $X_{reg}$ be its character. Let $(\pi, V)$ be any finite dimensional representation of $G$ with character $X$. Show that ...
0
votes
1answer
30 views

Character Table, Row and Column orthogonality, Conjugacy Classes

Let $G$ be a finite group with conjugacy classes $C_1, C_2, ..., C_k$ and let $g_i \in C_i$ be an element for each $i=1, ..., k$ Part 1: State the theorems on row and column orthogonality in the ...
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
35 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 ...