A project for computing lower bounds for the Lichnerowicz Laplacian on homogeneous spaces G/H with G simple and Einstein standard metric. This code was used to generate data for the article arXiv:2304.10607.
Running the code requires an installation of both Sage and LiE.
Open sage in a terminal and type:
load("Sym2Bounds.sage")
Then enjoy the implemented functionalities, or navigate to the subfolders you are interested in and load the .sage files there.
The Sage function WeylCharacterRing is the standard to handle compact semisimple Lie groups. Typing
G = WeylCharacterRing("A2xB3",style="coroots")
initializes G as the compact Lie group A2xB3. The optional parameter refers to the basis in which highest weights are given (in this case the simple coroots, i.e. fundamental weights). For example G(1,0,1,0,0) will correspond to the tensor product of standard representations of A2 and B3.
Most of the implemented functions require a branching rule as input. This can be obtained using the Sage function branching_rule.
LiE instead uses a restriction matrix for branching. Since LiE works with row vectors rather than column vectors, this matrix is the transpose of the one defined in the article.
Sym2LowerBounds(G,H,b,startscan=0,endscan=-1) takes as input the WeylCharacterRings G and H corresponding to a simple Lie group G and a semisimple subgroup H, a branching rule b (alternatively the restriction matrix as used in LiE, given as a list of lists), and the two optional arguments startscan and endscan which refer to the range of Casimir eigenvalues of G for which Fourier modes should be scanned. If left blank, the appropriate upper bounds is automatically determined using the crude estimate.
The output consists of fibrewise estimates for A*A and q(R) on the standard homogeneous space G/H as well as Fourier-mode-wise lower bounds for the Lichnerowicz Laplacian (if q(R)>E does not follow from the first step). Additionally, the program computes the dimension of the space of tt-matrix coefficients and tries to find Killing tensors in each Fourier mode by comparing Hom(.,Sym^2_0m) with Hom(.,m) and Hom(.,Sym^3m).
This is printed to the console and also to a .txt file, named by the Cartan types of G and H.
Sym2LowerBoundsWithTorus(G,Hss,rm,startscan=0,endscan=-1) takes as input the WeylCharacterRings G and Hss corresponding to a simple Lie group G and the semisimple part of a non-semisimple subgroup H, plus a LiE restriction matrix rm (given as a list of lists) for the embedding of H in G. The optional arguments and the output are as above.
Sym2LowerBoundsFullFlag(cartantype,startscan=0,endscan=-1) takes as input just the Dynkin type string cartantype of a simple Lie group G. The optional arguments and the output are as above, except that the fibrewise estimates are not printed for each H-isotype to avoid cluttering, and the program does not search for Killing tensors.
There is a subfolder containing a .sage file for each family and for the two classes of exceptions. Make sure to comment/uncomment or alter the for loops at the end of each file as is needed.
This involves an alternative, faster branching routine for large rank(G), similar to what one would do by hand. Requires loading
load("Sym2BoundsBigBranch.sage")
inside the subfolder BigBranch.
The branching routine does currently not work if there are too few right padding zeroes in the occurring highest weight vectors. This happens if rank(G) is too small, for spin modules, and always if G=SU(n). Again, there is a subfolder containing a .sage file for all the families where the procedure does work.
Sym2LowerBoundsBig(Gstr,Hstr,o1,endscan=-1,verbose=False) takes as input the Dynkin type string Gstr of a simple Lie group G, the Dynkin type string Hstr of a semisimple subgroup H (in Sage notation, e.g. "A12xC7"), a string o1 containing the branched standard representation of G, denoted as a LiE polynomial. The optional argument endscan is as above. verbose is a debug option that prints what happens during the Fourier scan phase.
Sym2LowerBoundsBigWithTorus(Gstr,Hssstr,o1,rm,endscan=-1,verbose=False) works similarly. The additional input rm (LiE restriction matrix, given as a list of lists) is only needed to compute Casimir eigenvalues of H.
Here are some of the most common errors that may occur.
This can be remedied by raising the LiE parameter maxobjects (standard 99999). Type, for example:
lie.eval("maxobjects 999999")
Sometimes an even higher value is needed. However, this may lead to longer computation times.
There seems to be no way to manipulate the hash table used in the LiE interface. We tried to be parsimonious with the amount of LiE objects in use, but it will exceed the system capabilities at some point. We recommend dividing up the Fourier modes by use of the startscan and endscan parameters.
This seems to be a problem within LiE itself. A reproducible example is the branching of the D124-module [2,2,0,0,0,...] to E8. We recommend using the BigBranch procedure instead.
This may happen sometimes, but not deterministically so. Unfortunately the Sage-LiE interface does not recover automatically from this, so you need to restart Sage itself.
Also a problem of LiE. It occurs when taking the symmetric power of a G-module if rank(G) is too large.