Example 6.4: D6 acting on R^3 

Linear Action & SAB 

> Rep := GroupDihedral[linearAction](3, 6, type = nonreflexion); -1; Generators = Rep[2], Rep[7]; 1; Irr := GroupDihedral[irredRepresentation](6); -1; `Dimensions of Irreducible Representations` = map(p...
Rep := GroupDihedral[linearAction](3, 6, type = nonreflexion); -1; Generators = Rep[2], Rep[7]; 1; Irr := GroupDihedral[irredRepresentation](6); -1; `Dimensions of Irreducible Representations` = map(p...
Rep := GroupDihedral[linearAction](3, 6, type = nonreflexion); -1; Generators = Rep[2], Rep[7]; 1; Irr := GroupDihedral[irredRepresentation](6); -1; `Dimensions of Irreducible Representations` = map(p...
Rep := GroupDihedral[linearAction](3, 6, type = nonreflexion); -1; Generators = Rep[2], Rep[7]; 1; Irr := GroupDihedral[irredRepresentation](6); -1; `Dimensions of Irreducible Representations` = map(p...
 

 

Typesetting:-mprintslash([Generators = Matrix([[Typesetting:-mfrac(Typesetting:-mn(
`Dimensions of Irreducible Representations` = [[1, 1, 1, 1, 2, 2], []] (3.1.1.1)
 

> SymmetryAdaptedPolynomialList := [seq(SAPolynomialBasis(Rep, Irr, i .. i)[1], i = 0 .. 9)]; -1; `Multiplicities within a degree` = [seq(map(proc (p) options operator, arrow; nops(p[1]) end proc, Symme...
SymmetryAdaptedPolynomialList := [seq(SAPolynomialBasis(Rep, Irr, i .. i)[1], i = 0 .. 9)]; -1; `Multiplicities within a degree` = [seq(map(proc (p) options operator, arrow; nops(p[1]) end proc, Symme...
SymmetryAdaptedPolynomialList := [seq(SAPolynomialBasis(Rep, Irr, i .. i)[1], i = 0 .. 9)]; -1; `Multiplicities within a degree` = [seq(map(proc (p) options operator, arrow; nops(p[1]) end proc, Symme...
 

`Multiplicities within a degree` = [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0], [2, 0, 0, 0, 1, 1], [0, 2, 1, 1, 2, 1], [3, 0, 1, 1, 2, 3], [0, 3, 2, 2, 4, 3], [5, 1, 2, 2, 4, 5], [1, 5, 3, 3, 7, 5], [7, ...
`Multiplicities within a degree` = [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0], [2, 0, 0, 0, 1, 1], [0, 2, 1, 1, 2, 1], [3, 0, 1, 1, 2, 3], [0, 3, 2, 2, 4, 3], [5, 1, 2, 2, 4, 5], [1, 5, 3, 3, 7, 5], [7, ...		 (3.1.1.2)
 

Invariants and Equivariants 

> Inv, Equiv := Equivariants(Rep, Irr, SymmetryAdaptedPolynomialList); -1
 

> map(primpart, Inv); 1
 

[`*`(`^`(x[3], 2)), `+`(`*`(`^`(x[1], 2)), `*`(`^`(x[2], 2))), `+`(`-`(`*`(`^`(x[1], 6))), `*`(15, `*`(`^`(x[1], 4), `*`(`^`(x[2], 2)))), `-`(`*`(15, `*`(`^`(x[1], 2), `*`(`^`(x[2], 4))))), `*`(`^`(x[... (3.1.2.1)
 

> map(reorganize, Equiv); 1
 

[[[1]], [[x[3]], [`+`(`*`(`/`(1, 240), `*`(`^`(10, `/`(1, 2)), `*`(`+`(`*`(3, `*`(`^`(x[1], 2))), `-`(`*`(`^`(x[2], 2)))), `*`(`+`(`*`(`^`(x[1], 2)), `-`(`*`(3, `*`(`^`(x[2], 2))))), `*`(x[1], `*`(x[2...
[[[1]], [[x[3]], [`+`(`*`(`/`(1, 240), `*`(`^`(10, `/`(1, 2)), `*`(`+`(`*`(3, `*`(`^`(x[1], 2))), `-`(`*`(`^`(x[2], 2)))), `*`(`+`(`*`(`^`(x[1], 2)), `-`(`*`(3, `*`(`^`(x[2], 2))))), `*`(x[1], `*`(x[2...
[[[1]], [[x[3]], [`+`(`*`(`/`(1, 240), `*`(`^`(10, `/`(1, 2)), `*`(`+`(`*`(3, `*`(`^`(x[1], 2))), `-`(`*`(`^`(x[2], 2)))), `*`(`+`(`*`(`^`(x[1], 2)), `-`(`*`(3, `*`(`^`(x[2], 2))))), `*`(x[1], `*`(x[2...
[[[1]], [[x[3]], [`+`(`*`(`/`(1, 240), `*`(`^`(10, `/`(1, 2)), `*`(`+`(`*`(3, `*`(`^`(x[1], 2))), `-`(`*`(`^`(x[2], 2)))), `*`(`+`(`*`(`^`(x[1], 2)), `-`(`*`(3, `*`(`^`(x[2], 2))))), `*`(x[1], `*`(x[2...
[[[1]], [[x[3]], [`+`(`*`(`/`(1, 240), `*`(`^`(10, `/`(1, 2)), `*`(`+`(`*`(3, `*`(`^`(x[1], 2))), `-`(`*`(`^`(x[2], 2)))), `*`(`+`(`*`(`^`(x[1], 2)), `-`(`*`(3, `*`(`^`(x[2], 2))))), `*`(x[1], `*`(x[2...		 (3.1.2.2)