Invariant Forms

Let \(G\) be a finite matrix group \(G < {\operatorname{GL}}_n({\mathbb{Q}})\). A matrix \(F \in M_n({\mathbb{Q}})\) is \(G\)-invariant if \(gFg^{tr} = F\) for all \(g \in G\).

PositiveDefiniteForm(G): GrpMat -> Mtrx

For a finite integral or rational matrix group \(G\), return a positive definite symmetric \(G\)-invariant form.

InvariantForms(G): GrpMat -> [ AlgMatElt ]

SymmetricForms(G): GrpMat -> [ AlgMatElt ]

AntisymmetricForms(G): GrpMat -> [ AlgMatElt ]

For an integral or rational matrix group \(G\), return a basis for the space of \(G\)-linear forms or for the subspace of (anti-) symmetric forms respectively.

The first form returned by InvariantForms and SymmetricForms will be positive definite.

InvariantForms(G, n): GrpMat, RngIntElt -> [ AlgMatElt ]

SymmetricForms(G, n): GrpMat, RngIntElt -> [ AlgMatElt ]

AntisymmetricForms(G, n): GrpMat, RngIntElt -> [ AlgMatElt ]

For an integral or rational matrix group \(G\), return a sequence consisting of \(n\geq 0\) \(G\)-invariant (symmetric or antisymmetric) bilinear forms for \(G\).

NumberOfInvariantForms(G): GrpMat -> RngIntElt, RngIntElt

NumberOfSymmetricForms(G): GrpMat -> RngIntElt

NumberOfAntisymmetricForms(G): GrpMat -> RngIntElt

For an integral or rational matrix group \(G\) or a \(G\)-lattice \(L\), return the dimension of the space of (symmetric or anti-symmetric) invariant bilinear forms for \(G\).

The algorithm uses a modular method which is much faster than the actual computation of the forms.