58.6 Operations for Special Kinds of Bases

If the underlying free left module V of the basis B supports a canonical basis (see CanonicalBasis) then IsCanonicalBasis returns true if B is equal to the canonical basis of V, and false otherwise.

Let B be an S-basis of a field F, say, and let R and M be the rings of algebraic integers in S and F, respectively. IsIntegralBasis returns true if B is also an R-basis of M, and false otherwise.

Let B be an S-basis of a field F, say. IsNormalBasis returns true if B is invariant under the Galois group (see GaloisGroup.field) of the field extension F / S, and false otherwise.

gap> B:= CanonicalBasis( GaussianRationals );
gap> IsIntegralBasis( B );  IsNormalBasis( B );
true
false

Let B be a basis of a free left module R, say, that is also a ring. In this case StructureConstantsTable returns a structure constants table T in sparse representation, as used for structure constants algebras (see Section Algebras of the GAP User's Tutorial).

The coefficients w.r.t. B of the product of the i-th and j-th basis vector of B are stored in T[i][j] as a list of length 2; its first entry is the list of positions of nonzero coefficients, the second entry is the list of these coefficients themselves.

The multiplication in an algebra A with vector space basis B with basis vectors [ v₁, ¼, v_n ] is determined by the so-called structure matrices M_k = [ m_ijk ]_ij, 1 £ k £ n. The M_k are defined by v_i v_j = å_k m_i,j,k v_k. Let a = [ a₁, ¼, a_n ] and b = [ b₁, ¼, b_n ]. Then

gap> A:= QuaternionAlgebra( Rationals );;
gap> StructureConstantsTable( Basis( A ) );
[ [ [ [ 1 ], [ 1 ] ], [ [ 2 ], [ 1 ] ], [ [ 3 ], [ 1 ] ], [ [ 4 ], [ 1 ] ] ], 
  [ [ [ 2 ], [ 1 ] ], [ [ 1 ], [ -1 ] ], [ [ 4 ], [ 1 ] ], [ [ 3 ], [ -1 ] ] ], 
  [ [ [ 3 ], [ 1 ] ], [ [ 4 ], [ -1 ] ], [ [ 1 ], [ -1 ] ], [ [ 2 ], [ 1 ] ] ], 
  [ [ [ 4 ], [ 1 ] ], [ [ 3 ], [ 1 ] ], [ [ 2 ], [ -1 ] ], [ [ 1 ], [ -1 ] ] ]
    , 0, 0 ]

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