The Seitz notation is widely used in crystallographic symmetry analysis based on group theory. In nature, as described here and here, all the operations represented by Seitz notation form a Euclidean group.

The following are some relevant backgrounds that I raised this question and hope to tackle it:

Recently, my friend Dr. Gui-Bin Liu [1] has developed the SpaceGroupIrep [2-3] which is a Mathematica package for irreducible representations of space group. In this package, the two kernel Seitz notation based calculation, i.e., inverse of a Seitz notation and multiplication between two Seitz notation are implemented but only work numerically, as shown in the attached screenshot. Therefore, I want to expand this feature so that it can provide some convenience to the relevant symbolic derivation and theoretical research.

It's well known that the Seitz (Koster-Seitz) notation is a widely used Euclidean transformations tool in three-dimensional crystallography and solid state physics, just as described in Koster's book [4]. But insofar, I can't find any tools/packages which can do the Seitz (Koster-Seitz) notation based derivation and operation.

In my opinion, if the operation based on Seitz symbol can be realized, it will bring great convenience to the study of quantum mechanics, solid state physics, crystallography, and other related disciplines.

In summary, I want to know if it's possible to do/implement the Seitz notation based symbol algebra with Mathematica.

Any hints/comments will be highly appreciated.

Regards,
HZ