Wall Forms

Given an isometry \(f\) of a quadratic, symplectic or unitary space \(V\) with bilinear or sesquilinear form \(\beta\), the Wall form of \(f\) is the form \(\theta\) defined on the image \(I\) of \(1-f\) by \(\theta(u,v) = \beta(w,v)\), 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 \(\mu : I \to V\), 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-regular if 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 \(\beta\), a basis \(e_1\), \(e_2\), …, \(e_n\) for \(V\) is semi-orthogonal if \(\beta(e_i,e_j) = 0\) 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 \({\bf F}_{2}\), 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 \(\beta\) be its polar form. Then \(Q(vf) = \eta Q(v)\) for some \(\eta\). Suppose that \(\eta = \zeta^2\) and let \(V(f,\zeta)\) denote the \(\zeta\)-eigenspace of \(f\). The generalised Wall form \(\theta\) of \(f\) is defined on the orthogonal complement of \(V(f,\zeta)\) by \(\theta(u,v) = \beta(w,v)\), where \(u = \zeta w - wf\).