All Questions
50 questions
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
31
votes
3
answers
3k
views
Rep Theory Consequences of Bott--Weil--Borel
I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
- 3,399
27
votes
2
answers
2k
views
Intuition for symplectic groups
My question essentially breaks down to
How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
- 779
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 $ ...
- 4,081
22
votes
1
answer
720
views
Does $E_8$ know $Spin(7)$?
One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...
- 54.6k
16
votes
1
answer
2k
views
A careful roadtrip from locally symmetric spaces to algebra
I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
- 7,789
14
votes
1
answer
681
views
If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?
This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$
$\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
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 ...
- 7,789
9
votes
0
answers
366
views
Mappings of the sphere (to itself) defined by homogeneous polynomials
Preamble
$\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that
If $G$ is a subgroup of $\SO(m+1)$ ...
- 39k
8
votes
1
answer
3k
views
The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$
The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia article. Am I correct in reading from this that any given ...
- 449
8
votes
2
answers
462
views
The action of $GL_{\infty}$ on the infinite wedge space
This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...
- 671
7
votes
1
answer
1k
views
Morphisms of principal bundles with different structure groups and associated bundles
Consider a pair of principal bundles $P \to M$ and $P' \to M'$ with groups $G$ and $G'$, respectively. A morphism from $P$ to $P'$ is a pair $(\Phi, \phi)$ where $\phi: G \to G'$ is a Lie group ...
- 6,025
7
votes
2
answers
668
views
Branching laws for $SO(n)$
The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article:
https://en.wikipedia.org/wiki/Restricted_representation#...
- 71
7
votes
0
answers
1k
views
What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
6
votes
2
answers
729
views
Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?
Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is $\text{Lie}(\...
- 6,025
6
votes
1
answer
596
views
Vector fields, diffeomorphism subgroups and lie group actions
Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...
- 7,789
6
votes
1
answer
1k
views
An integral with respect to the Haar measure on a unitary group
Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...
- 141
6
votes
0
answers
304
views
Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)
For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...
- 3,399
5
votes
1
answer
392
views
Equivariant $K$-theory, singular vectors, and flag manifolds
For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
- 3,399
5
votes
0
answers
135
views
Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
4
votes
1
answer
714
views
Differential equations and Lie groups
I am a physicist and I am pondering over a particular generalization of Stokes' theorem and Maxwell's equations. They apply to vector fields like the electric or magnetic one. However if the vectors ...
4
votes
0
answers
181
views
Specify the embedding of special unitary group in a Spin group via their representation map
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
4
votes
0
answers
430
views
Comparison between spinor representations in $\operatorname{SL}(2,\mathbb C)=\operatorname{Spin}(1,3)$ and $\operatorname{Spin}(4)$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that
$$
\Spin(1,3)=\SL(2,\mathbb C)
$$
and
$$
\Spin(4)=\SU(2) \times \SU(2).
$$
The $\Spin(1,3)$ is the ...
- 2,147
4
votes
0
answers
367
views
Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
- 6,025
4
votes
0
answers
173
views
Ring of SO(n)-invariant differential operators on M_n,m
I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...
- 41
3
votes
1
answer
279
views
Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) ...
- 145
3
votes
1
answer
3k
views
Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $
My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...
user21574
3
votes
1
answer
160
views
Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
3
votes
2
answers
761
views
Representations of SU(2) and tensors on SU(2)
I have only recently started exploring this region of homogeneous spaces and its geometry and the question is born from that and given the beginner state of my exploration the questions might sound ...
- 3,541
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
503
views
The definition of a homogeneous vector bundle
For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...
- 157
3
votes
0
answers
230
views
Possible to express the coadjoint orbits in terms of Kahler reduction?
I have heard for many times that the coadjoint orbits of a compact semi-simple Lie group are Kahler. While I know that the symplectic structure on a coadjoint orbit can be given by the symplectic ...
- 461
2
votes
1
answer
325
views
Orbit space of $\mathrm{SO}(3)$ irreducible representations
$\DeclareMathOperator\SO{SO}$Consider the $7$-dimensional $\mathbb R^7$ real irreducible orthogonal representation of $\SO(3)$. I am seeking a description of the orbit space (when the action is ...
- 71
2
votes
1
answer
174
views
The sum of the weights of an irreducible simple Lie algebra module
Let $\frak{g}$ be a simple Lie algebra (over $\mathbb{R}$ or $\mathbb{C}$) and $V_{\lambda_i}$ a fundamental representation. What happens if I take the sum, in the dual of the/a Cartan subalgebra $\...
- 643
2
votes
1
answer
280
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
2
votes
1
answer
320
views
Gelfand pairs and (self)-dual representations
For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of ...
- 459
2
votes
0
answers
125
views
The double quotient of SU(N) by its diagonal maximal torus
$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space
$...
2
votes
0
answers
156
views
Symmetric spaces and spherical functions
Let $G$ be a connected compact Lie group, $\sigma$ an involutive automorphism and $K$ a subgroup such that $(G^\sigma)_0\subset K\subset G^\sigma$. Then $M=G/K$ is a Riemannian symmetric space.
...
- 31
2
votes
1
answer
529
views
Cotangent bundle of symmetric space is symmetric space?
Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...
user21574
1
vote
1
answer
141
views
$G/T$ has finitely many $G^\theta$ orbits
Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace $.
I'm looking for ...
- 139
1
vote
1
answer
161
views
A question about G-Manifolds
I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...
user21574
1
vote
1
answer
102
views
$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$
I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and $\...
- 25
1
vote
1
answer
204
views
Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
- 317
1
vote
1
answer
93
views
How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$?
Let $(\mathcal{G}, \mathcal{G}^*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^*$ and $\mathcal{G}$ are
\begin{align}
& [t^a, t^b]_* = C_c^{ab} t_c, \\
& [...
- 6,201
1
vote
1
answer
608
views
Para-Complexification of Lie Groups
Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...
user21574
1
vote
0
answers
70
views
Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
- 71
1
vote
0
answers
92
views
The $U({\frak g})v$-module generated by a single element of a $U({\frak g})v$-module
Let $\frak{g}$ be a finite dimensional complex semisimple Lie algebra and let $U(\frak{g})$ be its universal enveloping algebra. Take $V$ an infinite dimensional module over $U(\frak{g})$. Let $v \in ...
1
vote
2
answers
360
views
When representation of two different coadjoint orbits are equivalent?
Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where $\mathfrak{t}$...
user21574
0
votes
0
answers
128
views
How to build a representation of the diffeomorphism group of $U(n)$?
Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...
0
votes
0
answers
53
views
Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099