Link to EStA documentation: https://skphy.github.io/aug

More to add about the theory:

Lattice transformation

point group symmetry

space group symmetry :

The geometric meaning of (W, w): How can one find the geometric meaning of a matrix–column pair ? (see Matrices, Mappings and Crystallographic Symmetry by Hans Wondratschek; page 64 of 82; good-sk!!)

Direct sum of matrix representatives: the matrix representative of operator A in space S is the direct sum of the matrix representatives of A in the component subspaces(say S1, S2,and S3 in 3D and so on). The space S must be decomposed into a sum of non-overlapping sub-spaces SI, S2.. .. under the A operator. [GROUP THEORY IN SOLID STATE PHYSICS BY D. F. JOHKSTON]

• 3D matrix as direct sum of 1D and 2D matrices. Find the example; you know it!! water example: space group operations ...confirm it!!

Direct product of matrix representatives: If S’ x S“ is the product space constructed from spaces S‘, basis {u_p,} and S”, basis {u_r}, each invariant under an operator A. The matrix representative of A in S’ x S” is the direct product of the matrix representatives of A in S’ and S”.

Decomposition of an invariant space: The concept of irreducibility at once raises the question: can a space S, invariant under a set of operators {A}, be decomposed into a sum of non-overlapping sub-spaces S', S", , , ,, each invariant and irreducible under {A} ? T h e answer is yes-if certain conditions, which we now consider, are satisfied. It is important to note that we shall only give sufficient conditions ; a discussion of necessary conditions would be too long and difficult here. We prove that if ( A } is a set of unitary operators, then complete decomposition of a finite space, invariant under (A}, is always possible.