Wall Forms

Given an isometry f of a quadratic, symplectic or unitary space V with bilinear or sesquilinear form β, the Wall form of f is the form θ defined on the image I of 1 - f by θ(u, v) = β(w, v), where u = w(1 - f). In general, the Wall form is not reflexive.

WallForm(V, f) : ModTupFld, Mtrx -> ModTupFld, Map

The space of the Wall form of the isometry f and its embedding in V.

WallIsometry(V, I, mu) : ModTupFld, ModTupFld, Map -> Mtrx

The inverse of WallForm. This is an isometry corresponding to the embedding μ : I to V, where V is a quadratic, symplectic or unitary space.

WallDecomposition(V, f) : ModTupFld, Mtrx -> Mtrx, Mtrx

An isometry f of a quadratic or symplectic space V is Wall-regularif the restriction of 1 - f to the image of 1 - f is invertible. If f is any isometry of V this function returns a Wall-regular element f_r and a unipotent element f_u such that f = f_rf_u = f_u f_r.

SemiOrthogonalBasis(V) : ModTupFld -> SeqEnum

If V is a vector space with a bilinear form β, a basis e₁, e₂, ..., e_n for V is semi-orthogonal if β(e_i, e_j) = 0 for i < j. This function returns a semi-orthogonal basis with respect to the non-degenerate, non-alternating form attached to V. If the base field is GF(2), the form should be symmetric.

GeneralisedWallForm(V, f) : ModTupFld, Mtrx -> ModTupFld, Map

This function returns the space of the generalised Wall form of the similarity f and its embedding in the quadratic space V. Suppose that the quadratic form Q of V is nondegenerate and let β be its polar form. Then Q(vf) = η Q(v) for some η. Suppose that η = ζ² and let V(f, ζ) denote the ζ-eigenspace of f. The generalised Wall form θ of f is defined on the orthogonal complement of V(f, ζ) by θ(u, v) = β(w, v), where u = ζ w - wf.