All Questions
Tagged with rt.representation-theory lie-groups
832 questions
12
votes
2
answers
887
views
Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)
$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism:
$$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$
which in our case ...
- 12.9k
1
vote
0
answers
125
views
A basic application of Mackey's theorem
Let $G=GL(2,\mathbb {F}_q)$ and $B=\left\{\begin{pmatrix}
* & * \\
0 & *
\end{pmatrix}\right\}$ the
Borel subgroup of upper triangular matrices. Let $\chi_1$ and $\chi_2$ be the characters ...
- 155
6
votes
1
answer
445
views
Is every finite subgroup the integer points of a linear algebraic group?
Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group?
Let $ ...
- 11.2k
5
votes
3
answers
1k
views
Reg the motivation behind Lusztig-Vogan bijection
Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, ...
- 2,821
3
votes
0
answers
167
views
The subalgebras of $\mathfrak{su}(2^n)$
$\DeclareMathOperator\su{\mathfrak{su}}$I want to find out all the subalgebras of $\su(N)$, in particular, $N=2^n$, which is the Lie algebra of $n$-qubits.
I don't know whether this is a hard question ...
- 63.9k
0
votes
0
answers
228
views
How do I detect whether a representation is (or is not) the adjoint representation?
Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. There is a God-given representation of $\mathfrak{g}$, namely, the adjoint representation $\operatorname{ad} : \mathfrak{g} \to \operatorname{Der}(\...
- 12.9k
4
votes
2
answers
332
views
Does the maximal compact subgroup always act transitively on a compact homogeneous space?
Let $ G $ be a Lie group, $ H $ a closed subgroup, and $ G/H $ compact. Under what conditions do we have that
$$
G/H \cong K/(K\cap H)
$$
where $ K $ is a maximal compact subgroup of $ G $? Obviously ...
- 4,081
2
votes
0
answers
69
views
Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group
Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
- 12.9k
1
vote
1
answer
207
views
Character formula for the fundamental representations of $\frak{sl}_n$
For the Lie algebra $\frak{sl}_{n+1}$ we denote its fundamental irreducible representations by $V(\pi_i)$, with $i=1, \dots, n$. Where can I find a table of the character formula (in other words a ...
- 8,597
2
votes
0
answers
107
views
The density of the image of a unitary irrep (a generalization of Burnside's theorem)
I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...
- 12.9k
2
votes
0
answers
81
views
Fourier transform in the complex motion group
I am looking for a reference that deals with the unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$ i.e., the semi-direct product of $\mathbb C^2$ with the special unitary group $K=...
- 650
24
votes
2
answers
2k
views
Is it possible to realize the Moebius strip as a linear group orbit?
On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO:
Is the Moebius strip a linear group orbit? In other words:
Does there exists a Lie group $ ...
- 12.9k
8
votes
0
answers
267
views
A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
- 12.9k
6
votes
1
answer
925
views
Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we ...
- 1
2
votes
0
answers
127
views
Is the image of the exponential map of a complex semisimple group Zariski open?
Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map
$$\exp : \mathfrak{g} \to G$$
Zariski open in $G$?
- 443
23
votes
3
answers
2k
views
How bad can $\pi_1$ of a linear group orbit be?
Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
- 5,429
1
vote
1
answer
137
views
Does every locally compact, simply connected group admit enough finite dimensional representations?
Given a simply connected locally compact group $G$, is it true that $G$ admits enough finite dimensional representations (over any field and not necessarily continuous) to separate points in $G$, what ...
- 12.9k
0
votes
1
answer
169
views
Representations of simply connected Lie groups [closed]
Let $G$ be a simply connected Lie group. Is it true that any finite dimensional representation of its Lie algebra is the differential of a representation of $G$?
A reference would be helpful.
Sorry if ...
- 126
3
votes
2
answers
649
views
Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$ [duplicate]
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible ...
- 954
5
votes
1
answer
356
views
Density of matrix coefficients of unitary representations of a locally compact group
Let $G$ be a locally compact group, $C_0(G)$ the $C^*$-algebra of continuous functions on $G$ that vanish at infinity, $C_b(G)$ the $C^*$-algebra of bounded continuous functions on $G$. We know that $...
- 25.8k
1
vote
0
answers
102
views
Bounding the dimensions of faithful representations of a quotient group
For $G$ a compact Lie group, let $\operatorname{mdfr}(G)$ be the minimum dimension of a faithful complex representation of $G$. Is there a bound on $\operatorname{mdfr}(N(H)/H)$ for $H$ a subgroup of ...
- 63.9k
2
votes
1
answer
261
views
Semisimple Lie algebra modules with $1$-dimensional weight spaces
Given a semisimple complex Lie algebra $\frak{g}$ of rank $r$, with Chevally generators $E_i,F_i,K_i$. Let $V$ be a finite dimensional representation of $\mathfrak{g}$ such that each weight space of $...
- 8,597
6
votes
2
answers
380
views
Rank one adjoint operators on a Lie algebra
Let $\mathfrak{g}$ be a (finite dimensional) semi-simple Lie algebra over a field $k$ and let $x \in \mathfrak{g}$. By definition, we have the equivalence:
$$ \mathrm{rk}(\mathrm{ad}_x) = 0 \iff x = 0,...
- 63.9k
2
votes
0
answers
187
views
Counting fixed points on flag variety and Deligne-Lusztig functors
Let $G=GL_n(q)$ be the general linear group over $\mathbb{F}_q$ and $T$ be the torus of diagonal matrices. We also pick a Levi subgroup of the form $L=GL_{n_1}(q)\times GL_{n_2}(q) \times \cdots \...
- 1,061
6
votes
1
answer
1k
views
Centralizers of nilpotent elements in semisimple Lie algebras
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $...
- 63.9k
6
votes
1
answer
351
views
Necessary and sufficient conditions for Littlewood Richardson coefficients to be non zero
Is there any necessary and sufficient conditions for $V(\tau)$ to be an irreducible component of the tensor product of two irreducible representations $V(\lambda)$ and $V(\mu)$ of a simple lie algebra ...
- 24.2k
6
votes
0
answers
201
views
Hall-Littlewood polynomials of non-dominant weights
$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
- 528
3
votes
1
answer
258
views
Symplectic orbits in projective Hilbert spaces are simply connected
Let $G$ be a connected Lie group and let $(\pi, \mathcal{H})$ be an irreducible unitary representation of $G$ on an infinite-dimensional Hilbert space $\mathcal{H}$. Denote by $\mathcal{H}^{\infty}$ ...
- 131
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
7
votes
2
answers
775
views
Why is the generalized flag variety a “variety”?
In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\...
- 15.5k
0
votes
1
answer
187
views
Matrix representations of Lie groups of type $B_n$
For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices:
\begin{align}
\left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
- 12.9k
2
votes
0
answers
76
views
Induced representations of Lie groups in case of compact quotient
Suppose you are given a split exact sequence of Lie groups
$$
1\to N\to G\to K\to 1,
$$
where $K$ is compact abelian. If $(\sigma,V_\sigma)$ is an irreducible unitary representation of $N$, is it true ...
CommunityBot
- 1
3
votes
1
answer
483
views
Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More ...
- 156k
3
votes
1
answer
176
views
Example of a supersolvable Lie group/algebra whose nilradical does not have a complement
What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is ...
- 12.9k
2
votes
1
answer
220
views
Set of $\mathrm{SU}(6)$ matrices which conjugate $\mathbb{1}_3 \otimes \sigma^3$ subalgebra element into $\mathfrak{su}(2)$
$\DeclareMathOperator\SU{SU}$Consider the Lie group $\SU(6)$, its Lie algebra $\mathfrak{su}(6)$ and the $\mathfrak{su}(2)$ subalgebra spanned by $\mathbb{1}_3 \otimes \sigma^i$, where $\sigma^i$ are ...
- 15.5k
2
votes
1
answer
206
views
Extending representations of Lie subalgebras to the whole Lie algebra
Let $\frak{g}$ be a complex simple Lie algebra and let $\frak{k}$ be a non-zero semisimple Lie subalgebra of $\frak{g}$. Is it possible to realize every simple $\frak{k}$-module $W$ as a $\frak{k}$-...
- 435
2
votes
1
answer
191
views
Normalizer of SU$(2)$ in SU$(6)$
Consider the $\mathfrak{su}(2)$ subalgebra of $\mathfrak{su}(6)$ embedded as
$$\mathfrak{su(2)}=\text{Span}\{\mathbb{1}_3 \times \sigma^i\}, \quad i=1,2,3$$
with $\sigma^i$ the Pauli matrices and $\...
- 11.2k
6
votes
1
answer
242
views
Do weight vectors live between the highest and lowest weights?
For a simple complex Lie algebra $\frak{g}$, let $V$ be an irreducible $\frak{g}$-module. Is it true that the weights of the non-zero weight vectors in $V$ are less than the highest weight vector and ...
- 12.9k
2
votes
1
answer
114
views
Set of $U(6)$ elements which leave a Lie-algebra element invariant under conjugation
Consider the specific element of the corresponding Lie algebra $\mathbb{1}_3 \times \sigma^3$, where $\mathbb{1}_3$ is the unit matrix in 3 dimensions, $\sigma^3$ is the 3rd Pauli matrix and $\times$ ...
- 31.5k
5
votes
2
answers
390
views
Dimensions of Jordan blocks associated to representations
Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes
into generally several Jordan blocks ...
- 6,349
9
votes
2
answers
522
views
Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup?
Let $ G $ be a simple linear algebraic group. Let $ G_\mathbb{R} $ be the real points of $ G $. Let $ G_\mathbb{Z} $ be the integer points of $ G $. Is $ G_\mathbb{Z} $ a maximal closed subgroup? In ...
- 11.2k
2
votes
0
answers
66
views
Smooth representations of a direct product of groups
Let $G_1, G_2$ be Lie groups, and let $E_i$, $i=1,2$ be a smooth representation of $G_i$ in a locally convex complete Hausdorff TVS. Then $E_1\hat\otimes E_2$ is a smooth representation of $G_1\times ...
- 1,488
3
votes
0
answers
84
views
Bilinear maps on smooth vectors of unitary representations
Let $G$ be a connected semi-simple real Lie group with finite center. Let $R_i$ ($i=1,2,3$) be unitary irreducible representations of $G$. Let $R_i^\infty$ be the corresponding representations of $G$ ...
- 1,488
2
votes
1
answer
223
views
Smallest dimension for faithful orthogonal representation
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$The compact simple Lie groups $\SO_8(\mathbb{R}) $ and $\SO_9(\mathbb{R}) $ both have rank 4. The group
$$
G=\SU_3 \times \SU_2 \times \...
- 12.9k
11
votes
1
answer
258
views
Algorithmic Borel finiteness for hyperbolic manifolds
It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
- 63.9k
4
votes
2
answers
120
views
When representations of reductive Lie group in a Banach space and in its Garding space have the same length?
Let $G$ be a real reductive Lie group (e.g. $G=\operatorname{GL}(n,\mathbb{R})$). Let $\rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^\infty\subset V$ be the subspace of ...
CommunityBot
- 1
6
votes
1
answer
255
views
A weight generalization of root systems?
For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...
- 24.2k
2
votes
1
answer
229
views
Action of the negative Cartan-Weyl generators on a highest weight element
Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha_1, \cdots, \alpha_r$ be the corresponding choice of simple roots, $X_{\alpha_i}, ...
- 123
3
votes
0
answers
101
views
Character formula for real representations
For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. The real representations of a real semisimple Lie algebra are classified using their ...
- 103
1
vote
1
answer
244
views
Irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$
I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ of finite dimension (in the following by "representation" I mean a representation of ...
- 63.9k