All Questions
Tagged with rt.representation-theory lie-groups
832 questions
1
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0
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65
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Classical groups generated by tensor products of subgroups
Let $ G $ denote a classical group.
Question:
Is it the case that
$$
\langle G_n \otimes G_m,G_m \otimes G_n\rangle=G_{nm}
$$
as long as $ n \neq m $?
For example, if $ G $ is the classical group $ GL(...
- 4,081
1
vote
0
answers
119
views
Question on Artin's Gamma function on $\operatorname{SO}(2,0)(\mathbb R)$
$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$.
Let $\pi$ be an ...
- 12.9k
5
votes
2
answers
669
views
classification for coadjoint orbits of lower or upper triangular matrices
Is there any classification for coadjoint orbits of lower or upper triangular matrices in general case $n\times n$. Is there any reference?
- 1,446
3
votes
0
answers
503
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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
6
votes
1
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303
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Compact Lie group has finitely many Lie primitive subgroups
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$A closed subgroup $ \Gamma $ of a Lie group $ G $ is called Lie primitive if it is not contained in any proper ...
- 63.9k
8
votes
1
answer
366
views
Is the GL(2,R)-representation of smooth, odd and 0-homogeneous functions on the punctured plane irreducible?
Let me preface this by saying that I have next to no background in representation theory. I come from geometry but the following representations showed up naturally in my work.
We let $ V = C^\infty \...
- 8,374
1
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0
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105
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Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
24
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2
answers
1k
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Lie groups generated by finitely many Lie algebra elements
Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x_1)\cdots\exp(x_n)$ ...
- 33.3k
2
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0
answers
67
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A branching law involving 2-power exterior representations
Let $K=SU(n)$.
We take a maximal torus $T$ in $K$ and fixed a simple root system with fundamental weights $\eta_1,\dots,\eta_{n-1}$.
For $\mu$ a dominant weight of $K$, we denote by $(\tau_\mu,V_\mu)$ ...
- 2,446
3
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0
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139
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Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
- 24.2k
4
votes
0
answers
135
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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
2
votes
0
answers
180
views
Howe duality vs first fundamental theorem in invariant theory
I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
- 63.9k
1
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0
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70
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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|...
- 63.9k
6
votes
1
answer
244
views
Certain isotypical component of the tensor product of irreducible representations of $\mathrm{U}(n)$
The following question is closely related to this one.
Let $\mathrm{U}(n)$ be the group of all (complex) unitary matrices $n\times n$. It is known that all irreducible representations of $\mathrm{U}(n)...
- 44.7k
21
votes
6
answers
2k
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How do I stop worrying about root systems and decomposition theorems (for reductive groups)?
I apologize for this being a very very vague question.
Just as personal experience, I never feel that I fully grasped the theory of root systems in Lie algebras and Lie/algebraic groups (I shall ...
- 1,446
1
vote
0
answers
59
views
K-finiteness of unitary representations of Poincaré-like groups?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\ISO{ISO}$I'd like to know if there are any papers that study the following problems:
Determine when decomposing the unitary irreps of $\ISO(d,1)$ into ...
- 63.9k
2
votes
3
answers
318
views
Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra
Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
- 44.7k
2
votes
1
answer
429
views
Representation ring of the general linear group
The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of ...
- 16.2k
3
votes
1
answer
140
views
Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
- 188k
2
votes
0
answers
159
views
The Cartan is a complex vector space but the root system is real?
Let $\frak{g}$ be a complex semisimple Lie algebra with some choice of Cartan subalgebra $\frak{h}$. The dual space $\frak{h}^* = \mathrm{Hom}_{\mathbb{C}}(\frak{h},\mathbb{C})$ is a complex vector ...
- 1,144
2
votes
1
answer
95
views
Image of minimal degree representation of quasisimple group unique up to conjugacy
Let $ G $ be a quasisimple finite group. Let $ d_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d_{min} $ irreps of ...
- 16.2k
3
votes
2
answers
492
views
Pairing a root with the half-sum of positive roots
Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
- 12.9k
2
votes
1
answer
337
views
Knapp's proof that the fundamental group of a compact semisimple Lie group is finite
In Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, he proves the following theorem of Weyl: If $G$ is a compact connected semisimple Lie group, then $\pi_1(G)$ ...
6
votes
0
answers
139
views
Why are all representations of split groups of real type?
(I am using a throwaway account because I plan on possibly pointing to this answer in a referee report I am writing, and using my main account would be a bit too obvious.)
Let $\mathfrak{g}$ be a ...
1
vote
1
answer
423
views
Quaternion representation and Haar measure of $SU(3)$ [closed]
Do we have easily and practically useful quaternion representation for $SU(3)$ group element and for Haar measure?
Also, is $SU(2)$ really simplified in the quaternion base?
- 188k
2
votes
0
answers
81
views
Complex semisimple Lie algebra modules with non-semisimple Cartan action
Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
- 257
4
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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
7
votes
1
answer
736
views
Infinite dimensional representations of $\frak{sl}_2$
The finite-dimensional representations of a complex semisimple Lie algebra $\frak{g}$ are well known to be classifiable by their highest weight vectors, giving a convenient countable indexing set. I ...
- 761
3
votes
1
answer
221
views
Cartan subspace of graded Lie algebras
Suppose $\mathfrak{g}$ is a complex reductive Lie algebra and $\theta$ is an automorphism of order $2$. Let $\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$ be the corresponding $\mathbb Z_2$-...
- 12.9k
2
votes
0
answers
185
views
Representation determined by traces
A discrete, faithful representation of a surface group $G:\pi_1(S_g) \to PSL_2(\mathbb{R})$ is determined, up to conjugacy by $PGL_2(\mathbb R)$, among such representations by the squares of traces of ...
- 133
6
votes
2
answers
685
views
When is the normalizer of the maximal torus maximal?
This is a cross post from MSE of
https://math.stackexchange.com/questions/4562196/normalizer-of-maximal-torus-is-maximal
Let $ T $ be a maximal torus in a compact connected simple Lie group $ K $. For ...
- 4,081
3
votes
1
answer
280
views
Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as
$$
V(\pi_1) \...
- 12.9k
2
votes
0
answers
48
views
Multiplicative invariants of non-reduced root systems
It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
- 553
1
vote
1
answer
189
views
Tensoring irreducible representations corresponding to root lattice elements
Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible ...
- 327
0
votes
1
answer
130
views
Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules
Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial?
...
- 24.2k
4
votes
1
answer
239
views
Number of representations of a semisimple Lie algebra of any given dimension
For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension.
More formally, let us define an ...
- 44.7k
3
votes
1
answer
451
views
Topological vector spaces in direct sum
A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow.
This question had emerged as an offshoot of a bigger ...
- 603
2
votes
2
answers
680
views
Complete representation theory of $\mathrm{SL}(2,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...
- 954
0
votes
1
answer
194
views
Sum of weights of an irreducible representation of $U(N)$
Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries.
Firstly, I would like to know ...
- 12.9k
2
votes
1
answer
94
views
Unitary dual of universal cover
The universal covering group $G$ of $\mathrm{SL}_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\...
- 63.9k
4
votes
1
answer
278
views
Complexification of a Lie subalgebra of a compact real form
I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made.
In this paper, $\mathfrak{g}$ is a ...
- 12.9k
3
votes
1
answer
147
views
Does every nilpotent orbit have an element supported on the simple root spaces?
Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...
- 12.9k
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 $G(q)$ ...
0
votes
1
answer
175
views
Centralizer of a reductive subgroup
Let $G$ be a reductive group over $\mathbb{C}$ and $H\subseteq G$ a reductive subgroup. Let $\rho$ be a faithful irreducible finite dimensional representation of $G$ over $\mathbb{C}$. Assume that $\...
- 12.9k
2
votes
1
answer
357
views
Tensor product of fundamental representations
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...
- 391
15
votes
1
answer
656
views
Linear embeddings of nilpotent pro-$p$ groups
Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
- 63.9k
14
votes
1
answer
360
views
Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups
If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here)...
- 954
3
votes
3
answers
581
views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
- 413
1
vote
0
answers
201
views
On the center of the universal enveloping algebra of a Lie algebra
Consider a finite-dimensional Lie algebra over C. Sometimes (e.g. for any semisimple Lie algebra) the center of its universal enveloping algebra is isomorphic to a freely generated polynomial ring.
...
- 63.9k
3
votes
1
answer
1k
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Highest weight orbit characterization (reformulated and extended)
Edit 1: I think that the question was not stated clearly enough so modified it a little.
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original ...
- 12.9k