All Questions
298 questions
5
votes
1
answer
339
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Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$
Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...
- 2,696
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?
user21574
5
votes
1
answer
283
views
Finite order automorpisms of affine Kac-Moody Lie algebras
It is known that for a finite order automorphism $\phi$ of a complex semisimple Lie algebra $L$, the fixed point subalgebra $L^{\phi}$ is a reductive Lie algebra and the centralizer of a Cartan ...
- 59
5
votes
1
answer
256
views
Definition of a Dirac operator
So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
- 327
5
votes
1
answer
368
views
Regular functions on nilpotent orbits and their covers
Let $G$ be a complex semisimple algebraic group with Lie algebra $\mathfrak{g}$.
In 1989 McGovern described the structure (as $G$-module) of the ring of regular functions on a finite cover of the ...
- 181
5
votes
1
answer
459
views
Some questions on analytic vectors and the integrability of Lie-algebra representations
I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
- 1,396
5
votes
0
answers
92
views
Canonical parabolics vs Levi subgroups
Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn ...
- 3,799
5
votes
0
answers
152
views
Explicit branching rules from $G(n+m)$ to $G(n) \times G(m)$ (where $G = \operatorname{SL}$, $\operatorname{SO}$ or $\operatorname{Sp}$)
Is there in the literature any explicit combinatorial description of the branching rules from $\operatorname{SL}(n+m)$ to $\operatorname{SL}(n) \times \operatorname{SL}(m)$, from $\operatorname{SO}(n+...
- 1,574
5
votes
0
answers
137
views
Differential operators on $G/K$
Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...
- 574
5
votes
0
answers
126
views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
5
votes
0
answers
167
views
Plancherel formula for $L^2(G/N)$
Let $G$ be a connected real semisimple or reductive Lie group. Let $TA$ be a Cartan subgroup, where $T$ is compact and $A$ is split. Let $MA$ be the centralizer of $A$ in $G$, and let $N$ be the ...
- 51
5
votes
0
answers
428
views
Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
- 181
5
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0
answers
281
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Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?
More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
- 118k
4
votes
4
answers
474
views
Algebra of regular functions on the quadratic cone and SU(2) representations
I was reading the paper "Short Star-Products for Filtered Quantizations" by Pavel Etingof and Douglas Stryker (MSN), where in the introduction they claim that the algebra of regular functions on the ...
- 175
4
votes
2
answers
4k
views
Nilpotent Lie algebras and unipotent Lie groups
$\mathbf{n}$ is nilpotent Lie algebra with $N$ being the corresponding algebraic Lie group. Now one neat feature of this setting is that you can take the exponential map to be identity. In other words ...
- 741
4
votes
1
answer
160
views
Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$
Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...
- 951
4
votes
3
answers
1k
views
Finite Order Automorphisms on Complex Simple Lie Algebras
Let $L$ be a finite dimensional complex simple Lie algebra, and
let $F(L)$ be the set of all finite order automorphisms on $L$.
Suppose that we declare $f,h \in F(L)$ to be equivalent if there exists
...
- 41
4
votes
1
answer
966
views
SU(6) -> SU(3) branching rule
I read in at least one paper and in the wiki below
http://en.wikipedia.org/wiki/Quark_model
that the 56 symmetric irrep of SU(6) breaks down into 10^{3/2} + 8^{1/2}
irreps of SU(3)xSU(2). Here the ...
- 199
4
votes
1
answer
2k
views
Reference request - localisation de g-modules
Does anyone have a link to a copy of Beilinson-Bernstein's "Localisation de g-modules", in which they prove the Beilinson-Bernstein theorem? I can't find it anywhere.
- 105
4
votes
2
answers
1k
views
Kostant's theorem on principal 3-dimensional subalgebras
I have a few questions concerning Kostant's work on principal three-dimensional subalgebras (TDS). Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and $\mathfrak{a}\subseteq\...
- 4,920
4
votes
2
answers
422
views
GAP versus SageMath for branching to Lie subgroups
Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...
- 835
4
votes
1
answer
288
views
The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators
Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$).
How is the embedding $\mathfrak{g}...
- 1,066
4
votes
1
answer
713
views
Criterion for nilradical of a maximal parabolic subalgebra to be abelian?
This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which ...
- 52.9k
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 ...
- 323
4
votes
1
answer
302
views
On maximal closed connected subgroups of a compact connected semisimple Lie group?
Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra.
Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...
- 2,446
4
votes
2
answers
393
views
Moving Between Weight Spaces in Highest-Weight Representations
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$ and let $\Delta\subseteq Hom(T,\mathbb{C}^*)$ be the ...
- 4,920
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 ...
- 327
4
votes
2
answers
350
views
Generating Irreducible representations of a simple lie algebra with Schur functors
Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
- 7,789
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
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
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
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
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
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
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
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
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
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
73
views
a property of the characters for center of universal enveloping algebra
Let $\mathfrak g$ be a complex simple Lie algebra. We fix Cartan subalgebra $\mathfrak h$ and a system of positive roots $\Psi$ for the root system of the pair $(\mathfrak g, \mathfrak h).$ For each $...
- 574
4
votes
0
answers
300
views
Number of connected components of the centre of a Levi subgroup
Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...
- 41
4
votes
0
answers
91
views
Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
- 951
4
votes
0
answers
174
views
Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal
irreducible representation. It can be shown that the number of
$\mathfrak{g}$-...
- 59
3
votes
1
answer
304
views
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
Consider the natural representations of $\operatorname{GL}(n,\mathbb{C})$ in the spaces
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$.
Is it known how to ...
- 21.8k
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 ...
- 327