All Questions
Tagged with rt.representation-theory lie-groups
832 questions
10
votes
1
answer
375
views
Invariants for $SO_n \backslash \mathfrak{gl}_n / SO_n$
Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines ...
- 4,991
10
votes
1
answer
659
views
how to view homology of affine Grassmannian as a subring of symmetric function
Let $G=SL_n$, it is proven that $R:=H_*(Gr_G)\cong \mathbb{C}[\sigma_1,...,\sigma_{n-1}]$ where $\sigma_i$ are of degree $2i$ as a polynomial ring generated by $n-1$ variables and the ring structure ...
- 849
10
votes
1
answer
323
views
Integral of product of Schur functions
Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae
$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...
- 2,552
10
votes
1
answer
292
views
Lagrangian Grassmannian from an Involution
I don't know if this is already answered somewhere in MO. The dynkin involution of $SL_{2n}$ that is $\alpha_i \mapsto \alpha_{2n-i}$ gives an outer automorphism of $SL_{2n}$ and then the maximal ...
- 185
10
votes
1
answer
354
views
Derivation of Blattner's conjecture in the Beilinson-Bernstein picture
On the last page of Schmid's article "Discrete Series", he says
"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb ...
- 27.8k
10
votes
0
answers
225
views
Third homology of simply connected Chevalley–Demazure group schemes
I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley ...
- 545
10
votes
0
answers
291
views
Each simple real Lie algebra as a representation of its maximal compact subalgebra
I am interested in a detailed description of the Cartan decomposition of each type of simple, real, finite-dimensional Lie algebra. (This is essentially a question about the classification of simple, ...
- 56.6k
10
votes
0
answers
392
views
Fake degrees: why coinvariant algebra and classical groups over finite fields?
Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.
...
- 24.2k
9
votes
3
answers
894
views
Representation rings of exceptional Lie groups
Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G_2$, $F_4$ or $E_8$, then it is known that $R(G)$ is a polynomial ring [F. ...
- 3,174
9
votes
2
answers
834
views
Lattices in $SL(n,\mathbb R)$
If $\Gamma\subseteq SL(n,\mathbb{R})$ is a lattice (i.e. discrete and finite covolume), does $\Gamma$ necessarily contain some $\mathbb{R}$-diagonalizable copy of $\mathbb{Z}^{n-1}$?
I know that the ...
- 93
9
votes
3
answers
576
views
Reference Request: Structure constants for G2
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
- 6,180
9
votes
2
answers
1k
views
Meaning of the coadjoint representation and its orbits
Given a Lie group $G$ there is a natural representation of $G$ on the dual of its Lie algebra $\mathfrak{g}^*$ given by the coadjoint representation. This representation is obtained by differentiating ...
- 1,474
9
votes
2
answers
2k
views
Fundamental representations and weight space dimension
For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all ...
- 643
9
votes
2
answers
870
views
$C^\infty$-vectors in general representations of Lie groups on locally convex spaces
This question is related to
this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...
- 3,380
9
votes
2
answers
522
views
Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup?
Let $ G $ be a simple linear algebraic group. Let $ G_\mathbb{R} $ be the real points of $ G $. Let $ G_\mathbb{Z} $ be the integer points of $ G $. Is $ G_\mathbb{Z} $ a maximal closed subgroup? In ...
- 4,081
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
9
votes
1
answer
543
views
Polynomial invariants for simple algebraic groups
Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible ...
- 2,751
9
votes
2
answers
634
views
Extension of the Weyl dimension formula
Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
- 2,446
9
votes
1
answer
987
views
Closures of torus orbits in flag varieties
Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$.
Now, as far as I ...
- 3,513
9
votes
1
answer
1k
views
Learning representation theory of real reductive lie groups
I am interested in any sources that can be helpful for learning the representation theory of real reductive groups.
I am currently reading Wallach book, but I feel that I don't understand the subject ...
9
votes
2
answers
831
views
Signed permutations in $ \operatorname{SO}(n) $ and normalizing an extraspecial group
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Lift{Lift}$The subgroup of $ \SO(n) $ of signed permutations has order $ n!2^{n-1} $. I will call ...
- 4,081
9
votes
2
answers
746
views
Characters of irreducible unitary representations of the Poincaré group
Consider Poincare group $\mathrm{ISO}(1,d-1)$, given by $\mathbb R^{1,d-1}\rtimes SO(1,d-1)$ in signature $(1,d-1)$, for some odd $d \geq 3$.
Denote the universal cover of the component connected to ...
- 533
9
votes
1
answer
277
views
Algorithmically handling the Spin groups in larg(ish) dimensions
Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...
- 32.5k
9
votes
1
answer
444
views
Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
- 835
9
votes
1
answer
335
views
Triple product formula on $K = \mathrm{SU}(2)$
Let $K = \mathrm{SU}(2) = \{ k[\alpha ,\beta] \mid \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with
$$ k [ \alpha , \beta ] =
\begin{pmatrix}
\alpha & \beta \\
- \...
- 185
9
votes
1
answer
414
views
Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces
If $G$ is a connected semisimple Lie group with finite center, Harish-Chandra defined a Schwartz space of rapidly decreasing functions on $G$ as the space of $\mathrm{C}^\infty$ functions defined by ...
- 3,816
9
votes
1
answer
732
views
Intertwiners and Clebsch-Gordan coefficients
Consider two unitary irreducible representations on vector spaces $V_1$ and $V_2$ of a Lie group $G$. For $G$ is compact and $V_1$ and $V_2$ finite dimensional there is a unique decomposition of $V_1 \...
- 131
9
votes
0
answers
366
views
Mappings of the sphere (to itself) defined by homogeneous polynomials
Preamble
$\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that
If $G$ is a subgroup of $\SO(m+1)$ ...
- 39k
9
votes
0
answers
234
views
Non-algebraic representations of $\text{SL}_n(\mathbb{R})$
My question is easily stated: are all continuous finite-dimensional real representations of $\text{SL}_n(\mathbb{R})$ algebraic representations?
This is false if you drop the word "continuous" (e.g. ...
- 383
9
votes
0
answers
161
views
Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit
$$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$
...
- 1,383
9
votes
0
answers
470
views
Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
8
votes
4
answers
3k
views
"Why" is every polynomial representation of SL(2) selfdual?
Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...
- 33.8k
8
votes
2
answers
2k
views
Lie algebras to classify Lie groups
What does the classification of Complex Semi-simple Lie algebras buy us in terms of classifying Lie groups? Certainly it classifies complex semi-simple lie groups but can we get any better? I know we ...
- 81
8
votes
1
answer
1k
views
Complex Lie group without faithful real representations?
Does there exist a complex analytic Lie group which doesn't have faithful representations in $GL(N,\mathbb R)$, viewed as a real Lie group?
There are examples of complex Lie groups which do not allow ...
- 1,050
8
votes
2
answers
572
views
reference containing the list of irreducible finite dimensional representation of real general linear group
It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\...
- 2,709
8
votes
2
answers
2k
views
Does $SU(N)$ have pseudo-real representation?
For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).
A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\...
- 169
8
votes
1
answer
534
views
Representation theory of $\mathrm{GL}_n(\mathbb{Z})$
I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
- 81
8
votes
2
answers
1k
views
number of irreducible representations over general fields
For a finite group, there are finitely many irreducible representations of complex numbers.
What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In ...
- 1,503
8
votes
1
answer
1k
views
Multiplicity one theorem
I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) (http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/automorphic-forms-and-l-functions-...
8
votes
2
answers
617
views
Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field
Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality.
Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$?
I understand ...
- 960
8
votes
3
answers
2k
views
Splitting of a Short-Exact Sequence of Lie Groups
Let $G$ be a Lie group (possibly disconnected). Consider the natural short-exact sequence $$1\rightarrow G_0\rightarrow G\rightarrow\pi_0(G)\rightarrow 1,$$ where $G_0$ is the identity component of $G$...
- 4,920
8
votes
2
answers
1k
views
Which Lie groups have adjoint representations that are bounded away from zero?
Studying stability of certain non-autonomous dynamical systems on Lie groups I have come across the following question: Exactly which finite-dimensional, real Lie groups have adjoint representations ...
- 187
8
votes
1
answer
374
views
Invariant ring of $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ under $\textrm{SO}(4)$
Consider the representation of $\textrm{SO}(4)$ on $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ induced by the standard representation of $\textrm{SO}(4)$ on $\mathbb{R}^4$. I am interested in the ring of ...
- 3,031
8
votes
2
answers
436
views
Infinite Krull-Schmidt categories?
In a Krull--Schmidt category, if
$$
X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s},
$$
where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...
- 643
8
votes
1
answer
3k
views
The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$
The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia article. Am I correct in reading from this that any given ...
- 449
8
votes
1
answer
1k
views
Weingarten function for unitary group
Studying integration over unitary group I came across this function, the Weingarten function Wg, such that
$$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k}
U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in S_n}...
- 1,549
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 \...
- 183
8
votes
2
answers
482
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 14.2k
8
votes
2
answers
619
views
Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity?
$\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, Lusztig's $q$-analog of weight multiplicty $K_{\lambda,...
- 24.2k
8
votes
1
answer
702
views
Central extensions of loop groups
Let $LG=\operatorname{Maps}(S^1,G)$ be the loop group of a compact Lie group $G$. I should add some adjectives to $G$, but for sake of simplicity let's just take $G=SU(2)$.
There is a central ...
- 18.7k