In K space, the basis of T is P times the basis of S. The covolume of T is the determinant of its basis, which is the determinant of P times the covolume of S.IndexThe index of a sublattice within a larger lattice is the quotient of the two covolumes, which is the determinant of the transforming matrix P, as described above. This is an element of R. In real space, it represents the number of cells of S that fit into a cell of T, although the cells may not pack perfectly. Let’s consider an example.Let S be the unit lattice in the plane, and let T be generated by [2,0] and [1,1]. This is a parallelogram of area 2. You can’t place two whole squares inside this parallelogram, but you can if you cut them up and rearrange the pieces.When S = TAssume S and T are the same lattice. A transformation P represents the basis of T in terms of the basis of S, while another transformation Q represents the basis of S in terms of the basis of T. The product PQ represents the basis of S in terms of the basis of S. This has to be the identity matrix. Therefore P and Q are inverses, and their determinants are units in R.

Source: Sublattice, Covolume, Discriminant