# 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 $u = \zeta w - wf$.