All Questions
Tagged with rt.representation-theory lie-groups
832 questions
8
votes
2
answers
462
views
The action of $GL_{\infty}$ on the infinite wedge space
This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...
- 671
8
votes
1
answer
1k
views
Symmetric tensor product of bosonic/fermionic Hilbert space
Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, $Sym^2(\wedge^k\...
- 965
8
votes
1
answer
929
views
Raising and lowering operators for SL(n,K) on homogenous polynomials
The short version:
Can the theory of weights for SL(n,C) be explained concretely in terms of raising and lowering operators on spaces of polynomials?
A deleted question asked how to prove SL(3,C) ...
- 10.7k
8
votes
2
answers
1k
views
Killing form vs its counterpart in a given represenation
Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...
- 965
8
votes
2
answers
826
views
Weil's theorem about maps from a discrete group to a Lie group.
Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...
- 3,203
8
votes
1
answer
355
views
finite upper half-plane model for the $\text{GL}_2(\Bbb{F}_q)$ Weil representation
Let $\Bbb{F}_q$ be a finite field with $q$ elements, let $\Bbb{F}_{q^2}$
be its quadratic extension, and consider the finite "upper" half space
${\frak{H}}_q := \Bbb{F}_{q^2} - \Bbb{F}_q$. Apeing a ...
- 2,137
8
votes
2
answers
344
views
Integrals of representations over geodesics
Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the ...
- 8,086
8
votes
1
answer
584
views
Tensor products of unitary irreducible representations of $SU(2,2)$
What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater ...
- 1,488
8
votes
0
answers
267
views
A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
- 1,856
8
votes
0
answers
381
views
Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
- 2,751
8
votes
0
answers
411
views
Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
- 1,586
8
votes
0
answers
129
views
Is there a splitting rule for the restriction of a $GL(23, \mathbb{Q})$-representation to $O(23, \mathbb{Q})$?
I am interested in a $23$-dimensional $\mathbb{Q}$-vector space $V$ which I am viewing as a GL$_{23}(\mathbb{Q})$ representation. Schur functors can be defined over $\mathbb{Q}$, so we get ...
- 233
8
votes
0
answers
408
views
Connection between two theorems on character values?
In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
- 52.9k
8
votes
0
answers
388
views
Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
- 431
8
votes
1
answer
382
views
Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
- 4,902
7
votes
2
answers
1k
views
Representation ring of SU(n)?
What's the structure of representation ring of SU(n)?
What are the representations of generators?
- 3,804
7
votes
2
answers
418
views
About the map $S(\mathfrak{g}^ * )^G\rightarrow S(\mathfrak{h}^ * )^H$ for $H < G$
Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) $...
- 9,019
7
votes
3
answers
599
views
Root system of fixed point Lie sub-algebra
It is known that a non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can ...
- 103
7
votes
1
answer
2k
views
Automorphism group of the special unitary group $SU(N)$
Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)...
- 2,147
7
votes
3
answers
2k
views
Explicit isomorphism between distributions and universal enveloping algebra
The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the ...
- 741
7
votes
2
answers
775
views
Why is the generalized flag variety a “variety”?
In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\...
- 159
7
votes
1
answer
466
views
Stabilizer of Sp(n) and U(n) in GL(n)
I would be very grateful for a reference
to the following results (which are, I think, true,
though I never saw it in the literature).
Let $G\subset GL(n,{\Bbb C})$ be $U(n)$,
abd $A\in GL(2n,{\Bbb ...
- 9,237
7
votes
3
answers
2k
views
Characterising the adjoint representation of SU(N)
One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an $...
- 71
7
votes
2
answers
533
views
Schur polynomial, change of variable
Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...
- 3,804
7
votes
3
answers
3k
views
why are all characters of the maximal torus in a Lie group weights?
Let $G$ be a compact connected Lie group, $T$ maximal torus, identified with $\mathbb{R}^n/\mathbb{Z}^n$, $X^*(T)$
the set of characters of $T$, naturally identified with $\mathbb{Z}^n$. Let next $\...
- 109k
7
votes
1
answer
1k
views
Irreducible representation of the product of two groups and tensor product
Let $G_1, G_2$ be two lie groups, $V$ be a finite dimensional (continuous) irreducible complex representation of $G_1 \times G_2$, must $V \cong V_1 \otimes V_2$ for some irreducible representation $...
- 6,622
7
votes
1
answer
573
views
Faithful representation of the projective unitary group with the lowest dimension?
What is the lowest dimension of a faithful ordinary representation (as compared with projective representation) of the projective unitary group $\rm{PU}(d)$? Is it $d^2-1$?
- 241
7
votes
1
answer
561
views
How does the right regular of GL(n, R) and GL(n,Qp) decompose?
The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...
- 11.2k
7
votes
2
answers
1k
views
What is the theorem of the highest weight used for?
$\DeclareMathOperator\End{End}$Over the past few months, I have taught myself the classification of reductive groups, and continued to non-abelian (as well as a small venture to non-compact) Harmonic ...
- 2,071
7
votes
1
answer
1k
views
Elementary question about Langlands decomposition
Let $G$ be a a complex reductive algebraic group, together with an $\mathbb{R}$-form. Is it true that any continuous homomorphism $G(\mathbb{R}) \to \mathbb{R}^{\times}$ comes from an algebraic ...
- 5,562
7
votes
1
answer
378
views
Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?
In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...
- 1,509
7
votes
1
answer
426
views
Lie algebra "generated" by matrix-valued curve?
Let $A(t)$ be a $n\times n$-matrix-valued continuous (plus possibly other niceness conditions; see below) curve, with the matrix entries being complex in general. If I am not mistaken, $A(t)$ ...
- 784
7
votes
1
answer
364
views
Peter–Weyl theory for vector fields
Let $G$ be a compact Lie group. The classical Peter-Weyl theorem shows that $L^2(G)$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$. This is a ...
- 5,824
7
votes
1
answer
456
views
Can Galois conjugates of lattices in SL(2,R) be discrete?
Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...
- 454
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
7
votes
2
answers
314
views
Holomorphic discrete series vs. discrete series
(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete ...
- 709
7
votes
1
answer
606
views
Motivating the existence of Canonical Bases for Representations
In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the ...
- 1,073
7
votes
1
answer
743
views
schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups
I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...
- 4,466
7
votes
1
answer
1k
views
Morphisms of principal bundles with different structure groups and associated bundles
Consider a pair of principal bundles $P \to M$ and $P' \to M'$ with groups $G$ and $G'$, respectively. A morphism from $P$ to $P'$ is a pair $(\Phi, \phi)$ where $\phi: G \to G'$ is a Lie group ...
- 6,025
7
votes
1
answer
337
views
Decay of Fourier coefficients for compact Lie groups
Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...
- 5,759
7
votes
1
answer
1k
views
When is the Ad (Adjoint Representation) Morphism a Closed Map
Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
- 1,161
7
votes
1
answer
739
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 ...
- 257
7
votes
2
answers
656
views
Universal property of induced representation
Let $H$ be a closed subgroup of the compact Lie group $G$. Let $E$ be a continuous representation of $H$. In the book "Representations of compact Lie groups" by Bröcker and Dieck the induced ...
- 3,031
7
votes
1
answer
237
views
Finite subgroups of $PSU(3)$
I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
- 71
7
votes
2
answers
668
views
Branching laws for $SO(n)$
The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article:
https://en.wikipedia.org/wiki/Restricted_representation#...
- 71
7
votes
1
answer
381
views
Homology of symplectic groups in the unstable range
Let $Sp(2n,{\mathbb R})$ be the symplectic group and $H_3(Sp(2n,{\mathbb R});{\mathbb Z})$ its 3rd group homology (i.e., for the group with the discrete topology).
It is known that $$H_3(Sp(2n,{\...
- 10.4k
7
votes
1
answer
389
views
Lie group actions with only one orbit type, but not defining a principal bundle
Searched-for situation: A compact connected Lie group acts effectively on a closed Riemannian manifold by isometries, such that there is only one orbit type of dimension strictly less than that of the ...
- 1,942
7
votes
1
answer
426
views
lowest weight representation of loop groups
I am trying to understand lowest representations of loop groups as developed in Pressley and Segal's book. Specifically I want to be able to compute the weight spaces that appear in a lowest weight ...
- 3,968
7
votes
1
answer
323
views
Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?
In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
- 492
7
votes
1
answer
824
views
Infinite-dimensional admissible representations of SL(2,C)
I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
