Mod-2 Cohomology of crystallographic space groups
Author
Chunxiao Liu, Weicheng Ye
Title
Mod-2 Cohomology of crystallographic space groups
Description
Mod-2 cohomology of crystallographic space groups: ring structure, Poincaré series, and standard cocycle functions. Article reference: arXiv:2410.03607.
Category
Academic Articles & Supplements
Keywords
group cohomology, crystallography, Poincaré series, Lieb-Schultz-Mattis theorem
URL
http://www.notebookarchive.org/2024-11-a6i9y5n/
DOI
https://notebookarchive.org/2024-11-a6i9y5n
Date Added
2024-11-22
Date Last Modified
2024-11-22
File Size
57.35 megabytes
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
Mod-2 Cohomology of crystallographic space groups
Mod-2 Cohomology of crystallographic space groups
Chunxiao Liu, Weicheng Ye
This notebook (the “Notebook”) contains data for the mod-2 cohomology ring of the 3D crystallographic space groups. It is one of the auxiliary files for • [LY24a] Liu, Ye, Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints, arXiv:2410.03607.For other auxiliary files, see • [LY24b] https://github.com/liuchx1993/Space-Group-Cohomology-and-LSM.
It contains two chapters:• Collection of Cohomology Rings It collects the mod-2 cohomology rings for 230 space groups, including generators, relations, as well as Z2 ranks at each degree.• Group Structure and Cocycles It collects the group structure (i.e. representation of elements in the group and multiplication rules) and cocycles written as functions of elements of the group. A number of auxiliary codes to check the consistency of these information is also included.
Collection of Cohomology Rings
The main piece of information is collected in the two arrays Rings and PoincareSeries, which collects the information of:
• Rings: mod-2 cohomology ring written as generators and relations, for each of the 230 space groups
For example, we have (after evaluation of the code)
For example, we have (after evaluation of the code)
Rings[[1]](*ThisisforspacegroupNo.1*)
Out[]=
{{Ax,Ay,Az},{{,,}}}
2
Ax
2
Ay
2
Az
Rings[[1]][[1]] are the generators of the mod-2 cohomology ring for space group No.1, Rings[[1]][[2]] are the relations. In particular, Rings[[1]][[2]][[i]] are relations at degree i+1.
• PoincareSeries: PoincareSeries representing the Z2 rank of each degree, for each of the 230 space groups
For example, we have (after evaluation of the code)
For example, we have (after evaluation of the code)
In[]:=
PoincareSeries[[1]]
Out[]=
1+3x+3+
2
x
3
x
This means that for space group No. 1, the Z2 ranks of each degree (starting from degree 0) are 1, 3, 3, 1, ... This is calculated from the ResolutionSpaceGroup in GAP
This is compared with the codes to obtain the Z2 ranks of each degree from the mod-2 cohomology rings, GetAllElementsYGetAllElementsY[symbols_, degrees_, relations_, target_]For a mod-2 cohomology ring symbols, degrees: generators of the cohomology ring and their corresponding degrees relations: the relations of the cohomology ring (generators of a sub - ring that should be mod out) target: the degree which we look at Output: the Z2 rank in each degree up to target (*Caveat: The codes to get the basis may be inefficient*) For example, we have
In[]:=
GetAllElementsY[Rings[[1]][[1]],ToDegrees[Rings[[1]][[1]]],Join[Flatten[Rings[[1]][[2]]]],15]
Out[]=
{1,3,3,1,0,0,0,0,0,0,0,0,0,0,0,0}
This means that from the mod-2 cohomology rings of space group No. 1, the Z2 ranks of each degree (up to degree 15) exactly match with the results from Poincare Series.
Collection of cohomology rings and Z2 ranks
Collection of cohomology rings and Z2 ranks
Codes for checking Z2 ranks from cohomology rings
Codes for checking Z2 ranks from cohomology rings
Check
Check
Group Structure and Cocycles
Cite this as: Chunxiao Liu, Weicheng Ye, "Mod-2 Cohomology of crystallographic space groups" from the Notebook Archive (2024), https://notebookarchive.org/2024-11-a6i9y5n
Download
