All Questions
298 questions
3
votes
1
answer
231
views
Does a representation of the universal cover of a Lie group induce a projective representation of the group itself?
Suppose that $G$ is a connected Lie group, $\tilde{G}$ its universal cover, $p:\tilde{G}\to G$ the covering map. Does a representation $\rho$ of $\tilde{G}$ on a finite-dimensional vector space $V$ ...
- 3,115
3
votes
1
answer
462
views
R-linear representations of sl(2,C)
Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$?
Equivalently, what ...
- 10.4k
3
votes
2
answers
493
views
Pairing a root with the half-sum of positive roots
Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
3
votes
1
answer
484
views
Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More ...
- 393
3
votes
1
answer
279
views
Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) ...
- 145
3
votes
2
answers
978
views
The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
- 131
3
votes
1
answer
162
views
Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
3
votes
1
answer
280
views
Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as
$$
V(\pi_1) \...
- 257
3
votes
1
answer
147
views
Does every nilpotent orbit have an element supported on the simple root spaces?
Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...
- 57
3
votes
1
answer
294
views
Lie group GL(4) representation decomposition
Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$.
Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation.
My question: How does $$V\otimes ...
- 131
3
votes
1
answer
3k
views
Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $
My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...
user21574
3
votes
1
answer
129
views
Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group
When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.
In the paper, we assume that $\...
- 137
3
votes
1
answer
195
views
A representation of $\frak{sl}_n$ as partial derivatives on polynomials
As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
- 1,144
3
votes
1
answer
572
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
- 31
3
votes
2
answers
704
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
- 1,719
3
votes
1
answer
160
views
Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
3
votes
1
answer
221
views
Cartan subspace of graded Lie algebras
Suppose $\mathfrak{g}$ is a complex reductive Lie algebra and $\theta$ is an automorphism of order $2$. Let $\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$ be the corresponding $\mathbb Z_2$-...
- 673
3
votes
2
answers
686
views
Real Adjoint representations of complex type
Let $G$ be a semi-simple compact Lie group. Let $V$ be a real vector space and let:
$\rho : G \to Aut_{\mathbb{R}}(V)$
be an irreducible real representation of $G$ on $V$. We say that $\rho$ is a ...
- 2,818
3
votes
2
answers
236
views
Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors
Dear all,
I have some difficulties with the following assertion in the book of Kirillov.
Let $G$ be a connected Lie group, and T a given (!) representation of G on a Banach space V.
Let $V^\omega$ ...
- 399
3
votes
1
answer
112
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
3
votes
1
answer
140
views
Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
- 179
3
votes
1
answer
209
views
The limit of a deformation of the ring structure on $\mathbb{C}[G]$
Let $G$ be a complex semisimple Lie group. Let $\Lambda^+$ denote its dominant Weyl chamber (by fixing a Cartan and Borel) and $V_{\lambda}$ the irreducible representation of $G$ with highest weight $\...
- 461
3
votes
1
answer
121
views
Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra
Background:
Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ ...
- 3,637
3
votes
1
answer
1k
views
Highest weight orbit characterization (reformulated and extended)
Edit 1: I think that the question was not stated clearly enough so modified it a little.
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original ...
3
votes
1
answer
176
views
Example of a supersolvable Lie group/algebra whose nilradical does not have a complement
What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is ...
- 33
3
votes
1
answer
242
views
Notions of integrability for affine Lie algebras and positive energy representations
Let $\mathfrak{g}$ be a simple (complex) Lie algebra. Given an invariant bilinear form $\kappa : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C}$, we can form the central extension $\hat{\mathfrak{g}}...
- 3,019
3
votes
1
answer
246
views
Distinguished dominant integral weight related to a branching problem
Let $G$ be a simple compact connected Lie group and let $K$ be a connected closed subgroup of $G$.
Let $\widehat G$ and $\widehat K$ denote the corresponding unitary duals, that is, the (equivalence ...
- 2,446
3
votes
1
answer
211
views
Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we ...
- 55
3
votes
1
answer
190
views
Critical points of characters on semisimple groups
Let $G$ be a semisimple connected complex Lie group, compact real Lie group or linear algebraic group. Let $\chi$ be the character of a finite dimensional irreducible representation of $G$ (I am ...
- 32.5k
3
votes
1
answer
281
views
Prescribed spherical representations, symplectic group $Sp(n)$
An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by $K$...
- 2,446
3
votes
2
answers
1k
views
Representations of reductive Lie group
Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie ...
- 671
3
votes
0
answers
195
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 673
3
votes
0
answers
200
views
Theorem of highest weight of semisimple Lie algebras: what fails precisely for reductive case
Let $\mathfrak{g}$ a complex semisimple Lie algebra. It is well known that by the theorem of the highest weight, all finite-dimensional complex irreps of $\mathfrak{g}$ are (up to iso) classified by ...
- 6,038
3
votes
0
answers
139
views
Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
3
votes
0
answers
167
views
The subalgebras of $\mathfrak{su}(2^n)$
$\DeclareMathOperator\su{\mathfrak{su}}$I want to find out all the subalgebras of $\su(N)$, in particular, $N=2^n$, which is the Lie algebra of $n$-qubits.
I don't know whether this is a hard question ...
- 89
3
votes
0
answers
101
views
Character formula for real representations
For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. The real representations of a real semisimple Lie algebra are classified using their ...
- 103
3
votes
0
answers
123
views
Decomposition of Schur modules over the orthogonal group
Let $V=\mathbb{R}^n$ and $O(n)$ the orthogonal group acting with its standard action on $V$. Now for any partition $\lambda$ we have the Schur module $S_\lambda V$ which is a representation of $O(n)$. ...
- 3,031
3
votes
0
answers
106
views
Induced $(\mathfrak{g},K)$-modules
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
- 951
3
votes
0
answers
111
views
Simple $\mathfrak{g}$-modules preserved by twisting
Let $G$ be a semi-simple lie group (simply connected for simplicity), $\mathfrak{g}$ its lie algebra. Write $\overline{G}=Inn(\mathfrak{g})$ for the adjoint form of $G$ which we identify here with ...
- 1,077
3
votes
0
answers
116
views
Extension of representations of certain compact Lie groups
Let $G$ be a real, connected, non-compact, semisimple Lie group with finite center and real rank $1$. Let $G=KAN$ be an Iwasawa decomposition, then $K\subset G$ is a maximal compact subgroup, and $\...
- 1,942
3
votes
0
answers
130
views
About the purpose of introducing '"groups of Heisenberg type"
I would like to know, can we say that the "groups of Heisenberg type" where introduced by A. Kaplan in "Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by ...
- 650
3
votes
0
answers
274
views
Invariant functions on the dual Lie algebra
Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.
...
- 95
3
votes
0
answers
126
views
Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra
Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...
- 159
3
votes
0
answers
236
views
Deligne-Simpson problem for classical groups
Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...
- 181
3
votes
0
answers
170
views
The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$
It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...
- 2,091
3
votes
0
answers
235
views
The fundamental in the tensor square of a complex representation of $SO(N)$
I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental ...
- 173
3
votes
0
answers
359
views
Does Branching in the Weight Diagram affect an embedding?
All groups here are compact semisimple Lie groups. Out of laziness I will use $B_7$ to mean $Spin(15)$.
Suppose that one has a group $H$ and a subgroup $G$. The embedding determines the decomposition ...
- 5,191
3
votes
1
answer
129
views
Exhaustion of restrictions of holomorphic / antiholomorphic representations
Let $G$ be a simple Lie group of Hermitian type, and $G'$ be a reductive subgroup of $G$. Suppose that $G'$ is also of Hermitian type and contains the center of the maximal compact subgroup of $G$. ...
- 951
2
votes
2
answers
237
views
Tensoring $\frak{g}$-modules by fundamental representations
Given a fundamental representation $V(\nu_k)$ of a semisimple Lie algebra $\frak{g}$, and a general irreducible finite-dimensional representation $V$, is it ever possible that the tensor product $V \...
2
votes
3
answers
318
views
Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra
Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
- 159