All Questions
298 questions
2
votes
2
answers
353
views
Particular reduced expression of the longest element of Weyl group
Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it. Let $w_0^K$ be the longest word of the Dynkin subdiagram $K$, which might be a disjoint union of connected Dynkin ...
- 201
2
votes
2
answers
336
views
Orthosymplectic superalgebra
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 673
2
votes
1
answer
174
views
The sum of the weights of an irreducible simple Lie algebra module
Let $\frak{g}$ be a simple Lie algebra (over $\mathbb{R}$ or $\mathbb{C}$) and $V_{\lambda_i}$ a fundamental representation. What happens if I take the sum, in the dual of the/a Cartan subalgebra $\...
- 643
2
votes
1
answer
640
views
Simple modules for direct sum of simple Lie algebras
I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...
- 951
2
votes
2
answers
1k
views
Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
- 671
2
votes
1
answer
244
views
Decomposition of an $\text{SL}_n(\mathbb{C})$ representation
Let $W = V \oplus V^*$, where $V$ is the standard $\text{SL}_n(\mathbb{C})$ rep and $V^*$ is its dual. I'm ultimately trying to decompose the space $(W \otimes \bigwedge^2 W) / {\bigwedge^3 W}$.
This ...
- 181
2
votes
1
answer
165
views
Trivial representation of a maximal torus
Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
- 128
2
votes
1
answer
160
views
Is the restriction of the Cartan 3-form on conjugacy classes exact?
Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
...
- 443
2
votes
1
answer
206
views
Extending representations of Lie subalgebras to the whole Lie algebra
Let $\frak{g}$ be a complex simple Lie algebra and let $\frak{k}$ be a non-zero semisimple Lie subalgebra of $\frak{g}$. Is it possible to realize every simple $\frak{k}$-module $W$ as a $\frak{k}$-...
- 435
2
votes
1
answer
229
views
Action of the negative Cartan-Weyl generators on a highest weight element
Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha_1, \cdots, \alpha_r$ be the corresponding choice of simple roots, $X_{\alpha_i}, ...
- 123
2
votes
1
answer
144
views
Characterisation of even nilpotent elements in $\mathfrak{sl}_n$
Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...
- 1,677
2
votes
3
answers
367
views
Invariant subbundles of tangent bundle of flag variety
Suppose that $G$ is a complex semisimple Lie group, $P$ a parabolic subgroup of $G$. What are all of the $P$-invariant subspaces of $\mathfrak{g}/\mathfrak{p}$? In various low dimensional examples, I ...
- 26.3k
2
votes
1
answer
107
views
Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces
For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
- 157
2
votes
1
answer
191
views
Normalizer of SU$(2)$ in SU$(6)$
Consider the $\mathfrak{su}(2)$ subalgebra of $\mathfrak{su}(6)$ embedded as
$$\mathfrak{su(2)}=\text{Span}\{\mathbb{1}_3 \times \sigma^i\}, \quad i=1,2,3$$
with $\sigma^i$ the Pauli matrices and $\...
- 155
2
votes
1
answer
196
views
Weight space dimension of the fundamental representation $\pi_n$ for type $C_n$
Will the fundamental representation $\pi_n$ of type $C_n$, for $n > 3$, have weight spaces of dimension greater than $1$? Is there some online resource where weight space multiplicities can be ...
2
votes
1
answer
261
views
Representation of a Lie algebra from a representation of Lie group
Let $\pi$ be a unitary representation of a Lie group $G$ on a Hilbert space $H_{\pi}$. Thus $\pi:G \to B(H_{\pi})$. One can extend $\pi$ to a representation of $L^1(G)$ via the formula
$\pi(f)=\int_{...
- 9,330
2
votes
1
answer
358
views
Is the Adjoint Action self dual over finite fields?
Given a finite group $G$, and representation $\rho: G \to H(\mathbb{F}_q)$ where $H$ is some classical algebraic group ($Gl$, $Sl$, $O$, $SO$, $SP$, $GSP$, $U$, etc), is the induced Adjoint ...
- 1,629
2
votes
1
answer
320
views
Gelfand pairs and (self)-dual representations
For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of ...
- 459
2
votes
1
answer
223
views
The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular
This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We ...
- 9,019
2
votes
1
answer
359
views
Characterization of the weight orbit in the projective space via second order Casimir.
This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
- 965
2
votes
1
answer
144
views
Paper request: Graev's classification of SU(2,2) irreducible unitary representations
I am interested in Graev's paper in "M. L. Graev:Dokl. Akad. Nauk SSSR,98, 517 (1954); Amer. Math. Soc. Transl.,66, 1 (1968)." in which the irreducible unitary representations of SU(2,2) are ...
- 679
2
votes
1
answer
90
views
Non-isomorphic direct products of a solvable and a semisimple Lie algebra
Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \...
- 393
2
votes
1
answer
261
views
Semisimple Lie algebra modules with $1$-dimensional weight spaces
Given a semisimple complex Lie algebra $\frak{g}$ of rank $r$, with Chevally generators $E_i,F_i,K_i$. Let $V$ be a finite dimensional representation of $\mathfrak{g}$ such that each weight space of $...
- 643
2
votes
1
answer
271
views
Discrete decomposability of unitary representation
[INTRODUCTION]
Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of ...
- 951
2
votes
1
answer
143
views
Trivial representation of $SL_n$ in $V(\alpha_1)\otimes \cdots \otimes V(\alpha_m)$
Suppose $\alpha_1,\cdots, \alpha_m$ are partitions of lenght at most $n$ with $c = \frac{1}{n}\sum_s\sum_i \alpha_s(i)$ an integer. If the representation of $SL_n$ $V(\alpha_1)\otimes \cdots \otimes V(...
- 291
2
votes
1
answer
251
views
Characterization of restricted weights of representations of real semisimple Lie groups
I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of restricted roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of ...
- 1,574
2
votes
1
answer
159
views
Adjoint action on orthogonal complement
Consider a compact Lie algebra $\mathfrak{g} \subset \mathfrak{u}(n)$ and its associated connected, compact Lie group $G$. Let $\mathfrak{g}^{\perp}$ denote $\mathfrak{g}$'s orthogonal complement (as ...
- 179
2
votes
1
answer
338
views
Knapp's proof that the fundamental group of a compact semisimple Lie group is finite
In Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, he proves the following theorem of Weyl: If $G$ is a compact connected semisimple Lie group, then $\pi_1(G)$ ...
- 241
2
votes
1
answer
357
views
Tensor product of fundamental representations
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...
- 391
2
votes
1
answer
219
views
Are finite-dimensional real representations of semisimple real Lie algebras completely reducible?
Suppose $\mathfrak{g}$ is a real form of a semisimple Lie algebra $\mathfrak{g}_\mathbb{C} = \mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$. Then we have the following:
There is an equivalence of ...
- 618
2
votes
1
answer
93
views
Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight
Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.
It is an exercise of Bröcker's book on Representations of Compact Lie ...
- 169
2
votes
1
answer
90
views
Plancherel expansion for Spin(n-1,1)
I am interested in the principal series (unitary irreducible) representations of $Spin(n-1,1)$, and in the generalized Pancherel's formula for the delta function on the group in terms of a sum (and an ...
- 129
2
votes
1
answer
256
views
The real group orbits on the flag variety always contains the holomorphic directions?
Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\...
- 9,019
2
votes
1
answer
197
views
Polytopes related to the conjugation action of a Lie group on multiple copies of itself?
Let G be a finite dimensional real Lie group. As I understand it, the quotient space of G acting on itself by conjugation is a well studied polytope which can be identified with the fundamental alcove ...
- 27.5k
2
votes
0
answers
67
views
A branching law involving 2-power exterior representations
Let $K=SU(n)$.
We take a maximal torus $T$ in $K$ and fixed a simple root system with fundamental weights $\eta_1,\dots,\eta_{n-1}$.
For $\mu$ a dominant weight of $K$, we denote by $(\tau_\mu,V_\mu)$ ...
- 2,446
2
votes
0
answers
180
views
Howe duality vs first fundamental theorem in invariant theory
I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
2
votes
0
answers
159
views
The Cartan is a complex vector space but the root system is real?
Let $\frak{g}$ be a complex semisimple Lie algebra with some choice of Cartan subalgebra $\frak{h}$. The dual space $\frak{h}^* = \mathrm{Hom}_{\mathbb{C}}(\frak{h},\mathbb{C})$ is a complex vector ...
- 1,144
2
votes
0
answers
81
views
Complex semisimple Lie algebra modules with non-semisimple Cartan action
Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
- 257
2
votes
0
answers
82
views
Question on a remark in Speh's paper
I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
- 1,121
2
votes
0
answers
135
views
Fusion rules for the Lie algebra $\frak{so}_{2n+1}$
For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...
- 393
2
votes
0
answers
69
views
Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group
Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
- 1,077
2
votes
0
answers
81
views
The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 10.5k
2
votes
0
answers
59
views
Finite-dimensional graded Lie algebras with $2$ generators
Does anyone know of a classification of those (complex) Lie algebras which are:
generated by two elements
$\mathbb{Z}$-graded Lie algebras
finite dimensional
- 643
2
votes
0
answers
573
views
Clebsch–Gordan(CG) coefficients for SO(N) and Sp(N) group
I know how to calculate the CG coefficients for $SU(N)$, but there are other simple Lie group like $SO(N)$ and $Sp(N)$. But up to now I can't find any textbook tells me how to calculate these and I ...
- 249
2
votes
0
answers
808
views
Casimir operators of a given Lie Algebra
I am a Physicist, so let me apologize in advance for some possible imprecisions.
I'm working on a 10-dimensional Lie Algebra. Each element of the algebra represents a quantum mechanical operator, and ...
- 255
2
votes
0
answers
135
views
Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations
I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
$U_iU_{i+1}U_i=...
- 71
2
votes
0
answers
128
views
On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?
Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...
- 327
2
votes
0
answers
115
views
The condition of maximality in branching rules of $SO$ group representations
Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
- 1,893
2
votes
0
answers
562
views
Complex Finite Dimensional Representation of GL(N,C)
What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?
We already know all the complex finite dimensional linear representation of SU(N).
- 3,804
2
votes
0
answers
224
views
$(\mathfrak{g},K)$-modules and parabolic category $\mathcal{O}$
I am trying to get acquainted with various infinite dimensional representations of Lie groups. So a general reference would be appreciated. Right now I am trying to figure out the following question.
...
- 8,597