All Questions
14 questions with no upvoted or accepted answers
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
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
5
votes
0
answers
126
views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
4
votes
0
answers
143
views
Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
- 141
4
votes
0
answers
141
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Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
4
votes
0
answers
304
views
What is known about the projective representations of $\mathrm{SO}(n_1,n_2)$?
He${}$llo MO.
Let $\mathrm{O}(n_1,n_2)$ be the pseudo-orthogonal group. I am interested in its (continuous, not necessarily unitary, finite-dimensional) irreducible projective representations, for ...
3
votes
1
answer
310
views
Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?
Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
- 965
2
votes
0
answers
53
views
Integral of product of characters of GL(N, C)
We know that
$\int_{U(N)} \chi_{R}(A\Omega B\Omega^{\dagger}) [d\Omega ] = \chi_R(A)\chi_R(B)/\chi_R(1)$
where $[d\Omega ]$ is the Haar measure on the unitary group $U(N)$, $A, B$ are hermitian $N\...
- 21
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
95
views
what kind of Gaussian matrix models are these?
In a physics paper I found a very complicated Gaussian matrix model:
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n}
\frac{
\prod_{i < j}\left[2 \sinh \frac{\mu_i - \mu_j}{2} \right]^2 \...
- 22.8k
2
votes
0
answers
115
views
The condition of maximality in branching rules of $SO$ group representations
Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
- 1,893
1
vote
0
answers
59
views
K-finiteness of unitary representations of Poincaré-like groups?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\ISO{ISO}$I'd like to know if there are any papers that study the following problems:
Determine when decomposing the unitary irreps of $\ISO(d,1)$ into ...
- 144
1
vote
0
answers
103
views
Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
- 1,155
0
votes
0
answers
66
views
How to Evaluate the ABJM partition function for N=2
This is the ABJM partition function on the 3-sphere,
$$ Z(2) = \int \frac{d^2\mu}{(2\pi)^2} \frac{d^2\nu}{(2\pi)^2}
\frac{\left[ 2 \sinh \frac{\mu_1 - \mu_2}{2}\right]^2\left[ 2 \sinh \frac{\nu_1 - \...
- 22.8k