All Questions
Tagged with rt.representation-theory lie-groups
832 questions
5
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1
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459
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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
86
views
Spherical functions in the space of functions on real Grassmannians
Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
- 21.8k
5
votes
0
answers
156
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When is a unitary group over a ring of integers dense?
Let $ SU_n(O_d) $ denote an integral unitary group of $ n \times n $ matrices over a totally real number field $ K_d:=\mathbb{Q}(\cos(\frac{2\pi }{d})) $ where $ O_d $ is the ring of integers of $ K_d ...
- 4,081
5
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0
answers
135
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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
5
votes
0
answers
92
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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
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 2,071
5
votes
0
answers
152
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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
373
views
Gelfand pairs in $SO(p,q)$
I am considering the groups $SO(p,q)$ over the reals. And inside it some parabolic subgroup, $P$. It can be the minimal parabolic but that is not the issue.
It is well known that $P$ has a Levi ...
- 51
5
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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
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0
answers
109
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Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
- 229
5
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0
answers
155
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Restriction of discrete series
QUESTION
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals the rank of its maximal compact subgroup. Suppose that $G'$ is a reductive subgroup of $G$ with equal rank. If $\...
- 951
5
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0
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126
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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
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0
answers
99
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Does there always exist an irreducible representation occurring with multiplicity one when inducing from $M=Z_K(A)$ to $K$?
This question is a more specific version of Does there always exist an irreducible representation occurring with multiplicity one when inducing from a closed subgroup to a compact Lie group? .
Since ...
- 1,942
5
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0
answers
136
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Modules of algebras with idempotents and the Stone-von Neumann theorem
The Stone-von Neumann theorem tells us that all unitary irreducible representations of the integrated/exponentiated/Weyl form of the canonical commutation relations (CCR) algebra in finite dimensions ...
- 653
5
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0
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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
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0
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171
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Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
- 1,719
5
votes
0
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304
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Decompositions of a compact Lie group into "fixed point set types"
Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...
- 1,942
5
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0
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214
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What do we know about isospectral Cayley graphs?
Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
- 1,893
5
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0
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400
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A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?
The following is my first question here on mathoverflow.
Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the ...
- 1,942
5
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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
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0
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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
5
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0
answers
519
views
Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups
In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
- 2,392
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
3
answers
681
views
Real points of reductive groups and connected components
Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...
- 6,180
4
votes
2
answers
578
views
Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 471
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
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2
answers
1k
views
Why limit of discrete series representation?
In what sense is the limit of discrete series representation of $SL(2, \mathbb{R})$ a limit of discrete series representations? Where does the name origin from?
- 11.2k
4
votes
3
answers
1k
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How to calculate symmetric tensor products of SO(10) representations?
I just want to consider the simplest case:
Let S=[0,0,0,0,1], how to derive the general formula for $Sym^k$ S?
My conjectured formula based on the results of LiE program for finite k values is:
$...
- 161
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
2
answers
3k
views
weyl group representations
I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the ...
- 188
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
2
answers
341
views
reference help indecomposable representations of SL(2,R)
Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible $(\mathfrak{g},K)$-...
- 2,709
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
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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
1
answer
511
views
Invariants of symmetric forms with respect to the symplectic group
Take a 6-dimensional vector space $V$ (for simplicity, over $\mathbb{C}$) and play the following game (for example, by employing the online Lie program): consider the 21-dimensional space $S^2V^*$ of ...
- 2,527
4
votes
1
answer
317
views
Orbits of tensor product $\operatorname{St}_2\otimes\operatorname{Sym}^2(\mathbb C ^3)$
Let $G_1=\operatorname{GL}_2(\mathbb C)$ act on $V_1=\mathbb C^2$ via the standard multiplication. Denote this representation by $\operatorname{St}_2$. Let $G_2=\operatorname{SL}_3(\mathbb C^3)$ act ...
- 960
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
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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
1k
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How to think about the simple reflection $s_0$ in the affine Weyl group?
Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
- 1,719
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
2
answers
363
views
Complexification or 'real'ization of Mapping Class group.
So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
- 327
4
votes
1
answer
976
views
Character determines the representation?
Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?
- 11.2k
4
votes
1
answer
278
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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
2
answers
272
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Schur positivity of a polynomial
Suppose a polynomial of the form
$$\prod_i^d \sum_j^p x_i^{f_j}$$
clearly symmetric, where $f_j\in \mathbb{N}$. There is a way to find the set of $f$ numbers such that this polynomial is Schur ...
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
163
views
Representations of Finite Subgroups on Homology
Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...
- 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