All Questions
Tagged with rt.representation-theory lie-groups
832 questions
6
votes
1
answer
319
views
Integral of Schur functions over $SU(N)$ instead of $U(N)$
Schur functions are irreducible characters of the unitary group $U(N)$. This implies
$$ \int_{{U}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\delta_{\lambda\mu},$$
where the overline means complex ...
- 1,549
6
votes
2
answers
788
views
Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations
I recently got interested in representation theory in quantum mechanics and I read the following theorem:
Let $G$ be a simply-connected Lie group with $H^2(\mathfrak{g},\mathbb{R})=0$ and let $\...
- 189
6
votes
1
answer
394
views
Branching rules for $SU(p,q)$
What is the branching rule for the subgroup $SU(p,q-1)\subset SU(p,q)$, i.e., the structure of the restriction of irreducible, finite-dimensional representations of $SU(p,q)$ to $SU(p,q-1)$? I would ...
- 395
6
votes
1
answer
390
views
Can these two irreducible $GL_n \mathbb Z$-representations be isomorphic?
Fix $n\in \mathbb N$ and a partition $\lambda$ with at most $n-1$ parts (of length at most $n-1$). Let $V$ be the irreducible $GL_n \mathbb R$-representation with highest weight $\lambda$ and $D$ the ...
- 236
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
1
answer
506
views
Lattice model for Affine Grassmannians of non type A
There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C}-...
- 1,719
6
votes
1
answer
1k
views
Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$
Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
- 755
6
votes
1
answer
444
views
Infinitesimal equivalence of admissible representations
Let $G_0$ be the $\mathbb{R}$-points of a real reductive group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. What is the precise relation between the category of ...
- 3,605
6
votes
1
answer
1k
views
Centralizers of nilpotent elements in semisimple Lie algebras
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $...
- 4,920
6
votes
2
answers
315
views
Covering relations in $K\backslash G/B$
Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\...
- 27.8k
6
votes
1
answer
317
views
Which Lie groups are covers of matrix groups?
I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...
- 3,115
6
votes
2
answers
280
views
How to check whether a given matrix is in the image of a representation?
Let $G$ be a compact simple Lie group, and let $\rho$ be a (faithful, unitary) irreducible representation thereof of $\mathbb K$-dimension $n$, where $\mathbb K=\mathbb C/\mathbb R/\mathbb H$ if $R$ ...
6
votes
1
answer
376
views
Does $SU(2)\cong Sp(1)\subset SO(5)$ factor through $Spin(5)\cong Sp(2)$ as the standard embedding $Sp(1) \to Sp(2)$?
$SU(2)$ can be seen as a subgroup of $SO(5)$ through the following chain of subgroups
$$
SU(2) \subset SO(4) \subset SO(5).
$$
If we identify $SU(2)\cong Sp(1)$, does the inclusion $Sp(1) \to SO(5)$ ...
- 225
6
votes
1
answer
445
views
references for faithful orthogonal (or unitary) representation of symmetric groups
Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).
(1). There does not exist any faithful orthogonal representation
$$
S_n\...
- 543
6
votes
1
answer
1k
views
Relation between Different Definitions of Induced Representation
I've seen two different ways to define induced representation.
One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of $...
- 1,040
6
votes
1
answer
573
views
Is there a "Cartan product" of Harish-Chandra modules?
If $\lambda,\mu$ are two dominant weights for a Lie group $G$, then
there is a canonical (up to scale, perhaps)
surjection $V_\lambda \otimes V_\mu \to V_{\lambda+\mu}$
of finite-dimensional ...
- 27.8k
6
votes
2
answers
5k
views
Representations of Lorentz group
Questions:
What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?
Why should one read Bargmann's paper on irred. ...
- 361
6
votes
2
answers
358
views
Duals of the spinor representations of $\frak{so}_{2n}$
For the $D_n$-series simple Lie algebra $\frak{so}_{2n}$
a curious phenomenon occurs for the fundamental representations corresponding to the spinor nodes of the Dynkin diagram, which is to say the ...
6
votes
1
answer
194
views
Involutions and Little Adjoint Representations of Simple Algebras
In what follows I'm going to use $V_{\theta_s}$ for the little adjoint representation af a Lie algebra i.e. the representation associated with the highest short rooth $\theta_s$.
Is easy to see that ...
- 233
6
votes
1
answer
190
views
Invariant regular cones in Lie group representation
I am following Analysis and Geometry on Complex Homogeneous Spaces by Faraut et al. I'll set up all of what I need and then ask my questions.
Let $G$ be a connected semi-simple non-compact real Lie ...
- 165
6
votes
3
answers
757
views
Decomposition of $L^2(\Gamma \backslash G)$
Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ (...
- 809
6
votes
4
answers
477
views
Topological properties of $K$ orbits in $G/B$
I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down ...
- 1,595
6
votes
1
answer
393
views
An alternative form of the Kazhdan-Lusztig conjecture
Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...
- 1,391
6
votes
1
answer
303
views
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 ...
- 4,081
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)...
- 379
6
votes
1
answer
551
views
Two definitions of automorphic forms on Lie groups
My question is the about the equivalence of two definitions of automorphic forms on a semisimple Lie group.
The most common definition of automorphic forms on a semisimple Lie group $G$ with respect ...
- 391
6
votes
1
answer
765
views
What is a map for the representation theory of reductive groups?
I have finished learning about linear algebraic groups (minus their representation theory) and the associated algebraic structures (root data, root systems, etc.), and will next attempt to summarize ...
- 2,071
6
votes
1
answer
538
views
Action of longest element of Weyl group on zero weight space
Let:
$G$ be a real semisimple Lie group;
$\rho$ be an irreducible representation of $G$ on a finite-dimensional real vector space;
$A$ be a "Cartan subspace" of $G$ (a Lie subalgebra which is abelian,...
- 1,574
6
votes
1
answer
749
views
Origin of symbols used for half-sum of positive roots in Lie theory?
The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...
- 52.9k
6
votes
3
answers
466
views
Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator
Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
- 3,244
6
votes
2
answers
194
views
Counting adjoints in the symmetric or antisymmetric square of a Lie group representation
EDIT (November 1, 2022): Over the weekend I think I found a technique to determine the exact multiplicities, according to how conjugation acts on the fundamental weights. While I haven't done the ...
- 630
6
votes
2
answers
334
views
Multiplication in universal enveloping algebra in terms of PBW isomorphism
Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...
- 7,377
6
votes
1
answer
2k
views
A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
6
votes
1
answer
2k
views
How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
- 3,541
6
votes
1
answer
255
views
Which Lie groups are a central extension of an algebraic group?
Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
- 161
6
votes
1
answer
169
views
Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$
Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected ...
user17697
6
votes
1
answer
321
views
Branching from $E(6)$ to $SO(10) \times U(1)$
In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...
- 835
6
votes
1
answer
403
views
Orbits for homogenous complex polynomials under unitary rotation of variables
Let's have two complex homogeneous polynomials of degree $k$: $f(z_1,\cdots,z_n)$ and $g(z_1,\cdots,z_n)$. We consider rotations of variables in the form of $\vec{z}' = U \vec{z}$, where $U\in SU(n)$.
...
- 1,612
6
votes
0
answers
149
views
Algorithmic representation of the Spin (and Pin) group [duplicate]
Performing algorithmic computations in $\mathit{SO}_n(\mathbb{R})$ or $\mathit{O}_n(\mathbb{R})$ is easy: its elements are represented by $n\times n$ orthogonal matrices of reals so, assuming we have ...
- 32.5k
6
votes
0
answers
139
views
Why are all representations of split groups of real type?
(I am using a throwaway account because I plan on possibly pointing to this answer in a referee report I am writing, and using my main account would be a bit too obvious.)
Let $\mathfrak{g}$ be a ...
6
votes
0
answers
200
views
Why should Serre's conjecture on congruence subgroup property hold?
There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$.
Whether a lattice in the group satisfies the congruence subgroup property,
...
- 1,024
6
votes
0
answers
201
views
Hall-Littlewood polynomials of non-dominant weights
$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
- 528
6
votes
0
answers
190
views
Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$
Lie algebraically, the eigenvalue of the spherical function
\begin{align*}
\phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*)
\end{align*}
...
- 83
6
votes
0
answers
273
views
Branching rules for E6 into SU(3)^3
I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
- 489
6
votes
0
answers
1k
views
Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
- 6,180
6
votes
0
answers
184
views
Reference request: fusion rules for unitary dual of SL(3,R)?
By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
- 25.8k
6
votes
0
answers
315
views
An analogue of Deligne--Lusztig theory for real groups?
I am considering the following analogue of Deligne--Lusztig theory:
Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have
$G^F=GL_n(\mathbb{R})$. Consider the ``Lang map''...
- 1,666
6
votes
0
answers
427
views
Non invertibility of certain integral arising from group action
Edit 1: According to the comment of Andreas Cap I revise the integral formula in the question.
Edit 2: I understand from the following post that some part of the previos version of my question has ...
- 356
6
votes
0
answers
455
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,719
6
votes
0
answers
244
views
Zariski closure of orbits of real groups on complex flag manifolds
Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
- 27.8k