ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 25 Oct 2021 12:20:33 +0200Simple roots from weights in type Ahttps://ask.sagemath.org/question/59474/simple-roots-from-weights-in-type-a/Hi everyone,
I ran into the following strange issue. Running
for a in RootSystem('A3').root_lattice().simple_roots():
a.to_ambient()
gives the expected formulas for the simple roots in type A.
However, if you run instead
for a in RootSystem('A3').weight_lattice().simple_roots():
a.to_ambient()
then the last root looks like
(1,1,2,0)
which differs from the first output by the "determinant weight" (1,1,1,1).
Anyone came across this before? This behaviour can easily cause some trouble...
Best,
Maxim
Maxim SmirnovMon, 25 Oct 2021 12:20:33 +0200https://ask.sagemath.org/question/59474/Find sphere points in a latticehttps://ask.sagemath.org/question/49784/find-sphere-points-in-a-lattice/Hello, say I have a lattice as in
Q = RootSystem('E8').weight_lattice()
with it's canonical bilinear form. How do I find the vectors $v$ such that $(v,v)=n$ for a positive integer $n$?
Edit: it would be even better if there's a way to get the points inside a cone, say the principal chamber in the above example.heluaniTue, 04 Feb 2020 20:01:35 +0100https://ask.sagemath.org/question/49784/Branching to Levi Subgroups in Sagehttps://ask.sagemath.org/question/45691/branching-to-levi-subgroups-in-sage/In the Sage computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup. See for example the root system $branching Rules \subseteq $ combinatorics in the Sage manual
Explicitly, one is branching to subgroup corresponding to a Dynkin sub-diagram, obtained by removing a single node.
For example, we can branch from $\operatorname{SL}(n)$ to the subgroup $\operatorname{SL}(n-1)$.
However, $\operatorname{SL}(n-1)$ can be considered as "living" in the larger subgroup
$\operatorname{SL}(n-1) \times \operatorname{U}(1)$. This is true for every subgroup coming from a deleted node, i.e. one can always take the product of the subgroup with $\operatorname{U}(1)$, to obtain a larger subgroup.
How does one branch to this subgroup in Sage. For example, it is done in the LieArt program for mathematica: see A3 of the ArXiv version of Lie Art.
Is this also possible in Sage?nadiasusyWed, 06 Mar 2019 22:10:48 +0100https://ask.sagemath.org/question/45691/Rename Lambda in weight spacehttps://ask.sagemath.org/question/44280/rename-lambda-in-weight-space/ It's a basic question, but after looking at the manual for a while I couldn't find the answer, so here goes.
R = RootSystem(["A",7])
print R.weight_lattice().fundamental_weight(1)
returns `Lambda[1]`. Is it possible to rename Lambda here? I could hack this by overriding the `.__str__` method, but is there a cleaner method?pbelmansWed, 14 Nov 2018 15:53:24 +0100https://ask.sagemath.org/question/44280/