Tagged Questions

4
votes
0answers
26 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
2
votes
0answers
29 views

Character of half-spin representation

Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm ...
0
votes
0answers
26 views

Classifying all rank 2 and 3 root systems

I am working with the representation theory of complex simple Lie algebras, and have a question: It is intuitively clear that the root systems $A_1\times A_1$, $A_2$, $B_2$, and $G_2$ comprise all ...
7
votes
0answers
78 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
0
votes
0answers
30 views

Subalgebra generated by Cartan subalgebra and root spaces

Let $\alpha_1,...,\alpha_k$ be the roots of a semisimple Lie algebra $\mathfrak{g}$ and $\mathfrak{g}_{\alpha_i}\subset \mathfrak{g}$ the corresponding root spaces. Then the subalgebra of ...
0
votes
0answers
41 views

A construction of the antipode in $GL_q(2)$

I am reading Quantum groups by Kassel, page 84, where the structure of Hopf algebra on $GL_q(2)=M_q(2)[t]/(t \det_q -1)$ is defined. To construct the antipode, set $S'$ as follows. $S'(a)=d, ...
4
votes
1answer
74 views

Characters of diagonalizable algebraic groups with no p-torsion

Let $G$ be a diagonalizable algebraic group and $X$ be the character group of $G$. Let $Y$ be a subgroup of $X$. We define $Y^{\perp}$ to be all the $x\in G$ such that $\chi(x)=1$ for all $\chi\in Y$. ...
1
vote
1answer
75 views

Determining which maps are isomorphisms

1) Let $G=G'=\{(a,b)\mid a,b \in \mathbb{R}, a, b \ne 0\}$ with group operation $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$. Let $\phi (a,b) = (b^{-1}, ab^2)$. My solution: 1-1: Suppose ...
1
vote
1answer
48 views

Invariant form on Lie algebra

Does anyone have a reference for the following fact? Let $G \subset GL(n)$ be a compact Lie group. Then the form $$f(A)=-Tr(A^2)$$ defined for $A \in T_e G \subset \mathfrak{gl}(n)$ is positive ...
2
votes
1answer
101 views

Conjugate Representations

Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
1
vote
1answer
40 views

When can you build up all representations from the fundamental and antifundamental ones?

Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I ...
1
vote
0answers
58 views

Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
2
votes
1answer
58 views

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?
2
votes
1answer
33 views

How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
8
votes
1answer
220 views

On 'backslash-forward slash' notation

I am curious about a notation that I have seen, but I have only seen it in contexts beyond my current level of ability and so haven't learned its meaning. Also, it's often difficult to search for the ...
7
votes
1answer
105 views

What is reductive group intuitively?

I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite ...
6
votes
2answers
218 views

Why is the name general “linear” group?

Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?
5
votes
0answers
123 views

Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
4
votes
1answer
117 views

Two Lie groups which are isomorphic but not homeomorphic

I am looking for an example of two Lie groups which are isomorphic as groups but not homeomorphic as topological spaces. Or, even more interestingly, a proof that two such groups cannot exist. Does ...
0
votes
0answers
147 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 ...
4
votes
3answers
108 views

When does the Commensurator of a subgroup of a group $G$ not equal $G$?

Let $H\leq G$ be two groups. I'm interested in the Commensurator $$\mathrm{comm}_G(H)=\{g\in G : gHg^{-1} \cap H \text{ has finite index in both}\}.$$ Obviously, $\mathrm{comm}_G(H)\leq G$. I read on ...
5
votes
2answers
207 views

$SU(2)$ Lie group

I have been studying Lie groups for a bit of fun for a while now and think they are fascinating. I have recently been told that $SU(2)$ can be used in some way to keep track of navigational systems in ...
2
votes
0answers
95 views

How to prove the lie algebra of $n\times n$ traceless matrices is semi-simple?

The Lie algebra of all the $n \times n$ matrices is not semi-simple. However, if we restrict ourselves to traceless $n\times n$ matrices, we do obtain a semi-simple (in fact, simple) Lie algebra which ...
5
votes
1answer
218 views

Subgroups of a vector space

I would like to have an overview of how a subgroup of a vector space over $\mathbb R$ of dimension $n$ can look like. Is there a complete classification available? I know that there are for examples ...
0
votes
0answers
89 views

How can I generate $\mathrm{SL}(n,\mathbb Z)$ by the subgroup $\mathrm{SL}(n-1,\mathbb Z)$ and another Element of $\mathrm{SL}(n,\mathbb Z)$?

Let $\{z_1,...,z_n\}$ be the canonical Basis of $\mathbb{Z}^n$, such that $z_i$ equals the vector $(0,\dotsc,0,i,0,\dotsc,0)$ with a 1 in the $i$th position. I want to show that the ...
4
votes
2answers
152 views

A trivial question concerning $sl_{n}\mathbb{C}$ representations

The question is, does the fact $$ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)^{2}=0, \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 ...
0
votes
1answer
73 views

Lie group multiplication/Parameter space

So I have a set P={ $p(\alpha,\beta,\gamma)=\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$ $|$ $\alpha,\beta,\gamma$ $\in R$}. I needed to show P is a Lie group, which I have ...
4
votes
1answer
94 views

Continuous transformations of a triangle bound on $S_1$

Suppose we take the circle $S_1$ and three points on this circle, which defines a triangle. By moving the points continuously on $S_1$, we obtain a continuous transformation of the triangle. I was ...