# Generate symmetry-adapted polynomial basis functions (1st-3rd order)

crystod-group --generate-basis --point-group m-3m

* Point group *
m-3m

* Input basis functions: 1st order (linear) *
 x, y, z

* Closed basis-function space: 1st order (linear) *
 dimension: 3
 basis: [x, y, z]

* Reducible characters: 1st order (linear) *
  E: 3
  C3: 0
  C2: -1
  C4: 1
  C4^2: -1
  i: -3
  S4: -1
  S6: 0
  sgh: 1
  sgd: 1

* Decomposition: 1st order (linear) *
 1.0 [T1u]

* Irreducible representations for basis functions: 1st order (linear) *
  T1u: [x, y, z]

* Input basis functions: 2nd order (quadratic) *
 x^2, y^2, z^2, xy, yz, xz

* Closed basis-function space: 2nd order (quadratic) *
 dimension: 6
 basis: [x^2, y^2, z^2, xy, yz, xz]

* Reducible characters: 2nd order (quadratic) *
  E: 6
  C3: 0
  C2: 2
  C4: 0
  C4^2: 2
  i: 6
  S4: 0
  S6: 0
  sgh: 2
  sgd: 2

* Decomposition: 2nd order (quadratic) *
 1.0 [A1g] + 1.0 [Eg] + 1.0 [T2g]

* Irreducible representations for basis functions: 2nd order (quadratic) *
  A1g: [x^2 + y^2 + z^2]
  Eg: [-2 x^2 + y^2 + z^2, x^2 - 2 y^2 + z^2]
  T2g: [xy, yz, xz]

* Input basis functions: 3rd order (cubic) *
 x^3, y^3, z^3, x^2 y, x^2 z, xy^2, y^2 z, xz^2, yz^2, xyz

* Closed basis-function space: 3rd order (cubic) *
 dimension: 10
 basis: [x^3, y^3, z^3, x^2 y, x^2 z, xy^2, y^2 z, xz^2, yz^2, xyz]

* Reducible characters: 3rd order (cubic) *
  E: 10
  C3: 1
  C2: -2
  C4: 0
  C4^2: -2
  i: -10
  S4: 0
  S6: -1
  sgh: 2
  sgd: 2

* Decomposition: 3rd order (cubic) *
 1.0 [A2u] + 2.0 [T1u] + 1.0 [T2u]

* Irreducible representations for basis functions: 3rd order (cubic) *
  A2u: [xyz]
  T1u: [x^3, y^3, z^3, x^2 y + yz^2, x^2 z + y^2 z, xy^2 + xz^2]
  T2u: [-x^2 y + yz^2, -x^2 z + y^2 z, -xy^2 + xz^2]



# Space-group mode at a k point; --order restricts to a single order
crystod-group --generate-basis --space-group Pm-3m --kpoint 0 0 0 --order 2

* Space group *
Pm-3m (221)

* Little group of k *
Pm-3m (221)

* k-point (primitive) *
 GM [0.0, 0.0, 0.0]

* Input basis functions: 2nd order (quadratic) *
 x^2, y^2, z^2, xy, yz, xz

* Closed basis-function space: 2nd order (quadratic) *
 dimension: 6
 basis: [x^2, y^2, z^2, xy, yz, xz]

* Little-group characters: 2nd order (quadratic) *
 (48 operations, omitted here -- run the command to see them)

* Decomposition: 2nd order (quadratic) *
 1.0 [GM1+(1)] + 1.0 [GM3+(2)] + 1.0 [GM5+(3)]

* Irreducible representations for basis functions: 2nd order (quadratic) *
  GM1+(1): [x^2 + y^2 + z^2]
  GM3+(2): [-2 x^2 + y^2 + z^2, x^2 - 2 y^2 + z^2]
  GM5+(3): [xy, yz, xz]
