All Questions
9 questions
23
votes
2
answers
611
views
Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?
Consider the compact Lie group $E_8$. Its second-smallest fundamental representation is $3875$-dimensional and admits a symmetric invariant form, and so is real: $E_8 \curvearrowright \mathbb{R}^{3875}...
- 54.6k
19
votes
5
answers
4k
views
Matrix representation for $F_4$
Has anyone ever bothered to write down the 26-dimensional fundamental representation of $F_4$? I wouldn't mind looking at it. Is it in $\mathfrak{so}(26)$?
I'm familiar with the construction of the ...
- 2,123
15
votes
2
answers
416
views
Constructing $E_8$ from its branching to $A_8$
Background/motivation: One of the usual constructions of [the adjoint representation of] the $E_8$ exceptional Lie group (found, e.g., in J. F. Adams's, "Lectures on Exceptional Lie Groups", esp. chap....
- 32.5k
14
votes
1
answer
360
views
Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups
If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here)...
- 32.5k
6
votes
2
answers
280
views
How to check whether a given matrix is in the image of a representation?
Let $G$ be a compact simple Lie group, and let $\rho$ be a (faithful, unitary) irreducible representation thereof of $\mathbb K$-dimension $n$, where $\mathbb K=\mathbb C/\mathbb R/\mathbb H$ if $R$ ...
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
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
2
votes
1
answer
304
views
Decomposition into irreducible components of a representation of $Spin(9)$
It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey).
Consider the induced representation of $Spin(9)...
- 21.8k
1
vote
1
answer
303
views
A representation of Spin(9,1)
Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ).
Consider the induced ...
- 21.8k