All Questions
Tagged with rt.representation-theory lie-groups
832 questions
4
votes
3
answers
2k
views
Lie subgroups of SU(4)
Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.
Additionally, which of these subgroups admit four dimensional ...
- 141
4
votes
1
answer
237
views
Aschbacher classes for compact simple group
Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO
Aschbacher's theorem says that every maximal subgroup of a finite simple classical ...
- 4,081
4
votes
1
answer
183
views
Multiplicities in Plancherel theorem for SL2(R)
The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. ...
user1688
4
votes
1
answer
256
views
Weyl group action on complexified Iwasawa decomposition
Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...
- 4,902
4
votes
1
answer
370
views
Cohomology of Projective Classical Lie Groups
Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...
- 4,920
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
4
votes
2
answers
369
views
Confusion over spin representation and coordinate ring of orthogonal Grassmannian
This is a copy from MSE where the question did not attract much attention.
I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic ...
- 24.2k
4
votes
2
answers
313
views
Do all unitary representations weakly converge to zero at infinity?
Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...
- 692
4
votes
1
answer
237
views
Average of product of matrix elements in irreducible representations of unitary groups
Let $\mathcal{U}(N)$ be the unitary group.
It is well known that
$$ \int_{\mathcal{U}(N)} U_{ij} U^\dagger_{nm} \,dU=\delta_{im}\delta_{jn}\frac{1}{N},$$
where $dU$ is the Haar measure.
More ...
- 2,552
4
votes
1
answer
1k
views
Complexification of compact Lie groups and complex algebraic linear reductive groups
I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker".
I found on internet that there is a bijection between complexifications of compact Lie ...
- 73
4
votes
1
answer
364
views
Are norm-continuous representations smooth?
Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x_i\to x\quad\Longrightarrow\quad ||\varphi(...
- 7,426
4
votes
1
answer
1k
views
When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?
This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected.
It seems that the ...
- 5,565
4
votes
1
answer
381
views
The existence of a finite dimensional Lie algebra with a given symmetric invariant metric
The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
- 755
4
votes
1
answer
715
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
2
answers
505
views
comprehensive presentation of the unitary dual of $SO_0(n,1)$
The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO_0(...
- 2,446
4
votes
1
answer
742
views
Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
- 2,160
4
votes
1
answer
133
views
Universal character ring for classical groups
The universal character ring for the general linear group is well understood but I want to ask about the universal character ring for the symplectic and orthogonal groups. For the general linear group,...
- 81
4
votes
2
answers
428
views
A morphism intertwining two induced representations
TL;DR:
Given representations $D,\Lambda$ of subgroups $K,Q$ of a Lie group $G$, is it true that every intertwining operator $T$ between the resulting induced representations of $G$ can be written
$$
(...
- 603
4
votes
1
answer
202
views
Branching to Levi subgroups in SAGE and the circle action
In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...
- 835
4
votes
1
answer
509
views
Lie Symmetries of the Bessel Differential Equation
The Bessel differential equation has an arbitrary looking form, but a lot is known about it.
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$
Is there a way to derive the Bessel ...
- 22.8k
4
votes
1
answer
516
views
Calculation with weights of $E_6$
Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...
- 4,729
4
votes
1
answer
303
views
The Gysin Sequence for an Associated Bundle over a Partial Flag Variety
Let $G$ be a connected, simply-connected complex semisimple Lie group, and let $P\subseteq G$ be a parabolic subgroup. Suppose that $V$ is a $1$-dimensional complex $P$-representation and consider the ...
- 4,920
4
votes
1
answer
923
views
About using the character formula for $SO(2n)$
I have known of the following equation for characters of a $SO(2n)$ representation with highest weights $(h_1,...,h_n)$ and for $(t_1,t_2,..,t_n,t_1^{-1},t_2^{-1},..,t_n^{-1})$ being the eigenvalues ...
- 1,893
4
votes
1
answer
346
views
Orbits of Root Vectors
Let $G$ be a connected, simply-connected complex semisimple Lie group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$, and let $$\frak{g}=\frak{t}\oplus\bigoplus_{\alpha\in\Delta}\frak{...
- 4,920
4
votes
1
answer
677
views
An identity for sheaf cohomology of flag varieties
Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection.
Given a representation ($\theta,V$) ...
user6311
4
votes
1
answer
101
views
K-types of a representation of the minimal Gelfand-Kirillov dimension
Let $G$ be a noncompact real simple Lie group not of Hermitian type, and $\mathfrak{g}_0$ its Lie algebra. Fix a maximal compact subgroup $K$ in $G$ with its Lie algebra $\mathfrak{k}_0$. Write $\...
- 951
4
votes
1
answer
348
views
Verma modules and Borel–Weil
Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
- 181
4
votes
1
answer
230
views
Are isotypic components of $S(\mathfrak{g})$ finite-dimensional?
Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic ...
- 443
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 ...
- 21.8k
4
votes
1
answer
907
views
Is there a generalization of Schur - Weyl duality and plethysm for direct product of special unitary groups?
Consider the semisimple compact group $K=SU(N_1)\times SU(N_2) \times \ldots \times SU(N_S)$ acting naturally on $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
- 965
4
votes
1
answer
321
views
Average of product of matrix elements in the special orthogonal group
Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed, in Integration with respect to the Haar measure on unitary, orthogonal and symplectic group (see ...
- 2,552
4
votes
1
answer
235
views
A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$
This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here.
$G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...
- 675
4
votes
1
answer
319
views
How to write down the connection morphism in the long exact sequence in Čech cohomology explicitly in this specific case?
Fix an integer $k$. Let $X=G/P$ be a complex rational homogeneous variety. I assume here $G$ is a simply connected semi simple complex Lie group and $P=P_k$ is a maximal parabolic subgroup defined by ...
- 279
4
votes
1
answer
633
views
Homomorphisms from binary polyhedral group to compact Lie groups
Are homomorphisms from binary polyhedral groups to (simple and simply connected) compact Lie groups classified?
For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie ...
- 6,123
4
votes
1
answer
1k
views
Reductive Lie Groups and Complexification
Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$...
- 4,920
4
votes
0
answers
87
views
Doubling constructions beyond classical groups: general principles?
The doubling method for constructing integral representations of L-functions has been successfully applied to classical groups, as demonstrated in this paper. However, extending this method to a wider ...
- 111
4
votes
0
answers
100
views
Embedding of a nilpotent algebraic group in upper triangular matrices
Suppose we have a polynomial group law on $G=\mathbb{R}^n$ which gives it a structure of a nilpotent algebraic group.
Is it true that there exists an embedding of $G$ into the group of upper-...
- 728
4
votes
0
answers
105
views
Irreducible representation of $\mathrm{GL}(2,\mathbb{R})$ that is not admissible
It is a basic theorem of Harish-Chandra that an irreducible unitary representation $\pi$ of a reductive group $G$ over $\mathbb{R}$ on a Hilbert space is admissible, meaning that every irreducible ...
- 413
4
votes
0
answers
143
views
Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
- 141
4
votes
0
answers
136
views
Two definitions of intertwining operators and Harish-Chandra's Plancherel measure
I guess this question is a well-known fact to experts, but I didn't find any explicit explanation in the literature.
So let $F$ be a $p$-adic field. (There're parallel definitions and results in the ...
- 709
4
votes
0
answers
120
views
Why are all "non-swinging" representations self-dual?
Let $\mathfrak{g}$ be a semisimple (say complex) Lie algebra, and $V$ an irreducible finite-dimensional representation of $\mathfrak{g}$. Denote by $w_0$ the longest element of the Weyl group, i.e. ...
- 1,574
4
votes
0
answers
111
views
How many diagrams interlace a given Young diagram?
For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff
$$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
4
votes
0
answers
128
views
Real Representation ring of $U(n)$ and the adjoint representation
I have two questions:
It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...
user340793
4
votes
0
answers
147
views
Is the homogeneous coordinate ring of a flag variety a UFD?
I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
- 201
4
votes
0
answers
108
views
Minimal dimension for $ \mathrm{PSU}_n $ as a matrix group
$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$
Here's the new question:
$ \SU_2 $ is a subgroup of $ \GL_2(\...
- 4,081
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
366
views
Derivative of a representation
I'm learning about Maass--Shimura operators, and there's a term that I'm not sure how to generalize nicely.
Let $\mathfrak{h}$ be the upper half-plane with parameter $z= x + iy$, and write $s = \frac{...
- 949
4
votes
0
answers
350
views
Interpolated simple integral fusion categories of Lie type
$\DeclareMathOperator\PSL{PSL} \DeclareMathOperator\Rep{Rep}$The idea motivating this post is that there should exist a global understanding of the unitary fusion categories $\Rep(G(q))$, with $G(q)$ ...
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
68
views
The weak restriction of the Jacquet module
Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
- 41