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Norm Spaces and Basis Reduction





NormSpace(A) : AlgQuat -> ModTupFld


NormSpace(S) : AlgQuatOrd -> ModTupRng


NormModule(S) : AlgQuatOrd -> ModTupRng



Given an algebra A or an order or ideal S, returns the underlying 
space or module over its base ring, with inner product respect to the 
norm, followed by the map into the structure.



GramMatrix(S) : AlgQuatOrd -> AlgMat



The Gram matrix with respect to the norm on the basis for S.



ReducedBasis(S) : AlgQuatOrd -> SeqEnum



Given an order or ideal S over Z in a definite quaternion algebra, 
returns a Minkowski-reduced basis for S.



ReducedGramMatrix(S) : AlgQuatOrd -> AlgMat



The unique Minkowski-reduced Gram matrix of a reduced basis for the 
definite quaternion order or ideal S.  





Example AlgQuat_Basis_Reduction (H68E11)

The quaternion ideal machinery makes use of a Minkowski basis reduction 
algorithm which returns a uniquely normalized reduced Gram matrix for 
any definite quaternion ideal.  This forms the core of the isomorphism 
testing for quaternion ideals. 





> A := QuaternionOrder(19,2);
> ideals := LeftIdealClasses(A);
> #ideals;
5
> [ (1/Norm(I))*ReducedGramMatrix(I) : I in ideals ];
[
    [ 2  0  1  1]
    [ 0  2  1  1]
    [ 1  1 20  1]
    [ 1  1  1 20],


    [6 0 1 3]
    [0 6 3 1]
    [1 3 8 1]
    [3 1 1 8],


    [6 0 1 3]
    [0 6 3 1]
    [1 3 8 1]
    [3 1 1 8],


    [ 4  0  1 -1]
    [ 0  4  1  1]
    [ 1  1 10  0]
    [-1  1  0 10],


    [ 4  0  1 -1]
    [ 0  4  1  1]
    [ 1  1 10  0]
    [-1  1  0 10]
]






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