FactorWeylSpinor - Maple Help

Tensor[FactorWeylSpinor] - factorize a rank 4 symmetric spinor

Calling Sequences

FactorWeylSpinor( W, PT)

Parameters

W- a symmetric rank 4 covariant spinor

PT    - the Petrov type of the spinor $W$

Description

 • A rank 4 symmetric spinor ${W}_{\mathrm{ABCD}}$ can always be factorized as the symmetric product of rank 1 spinors,

.

The (non-unique) spinors are called the principal spinors of and the corresponding null vectors are called the principal null directions. The Petrov type of the spinor ${W}_{\mathrm{ABCD}}$ (see AdaptedSpinorDyad or PetrovType) determines the multiplicities of the principal spinors.

TYPE I. The principal spinors are all distinct, that is, non -proportional and .

TYPE II. Two of the principal spinors are proportional, where  are non-proportional.

TYPE III. Three of principal spinors are proportional, where are non-proportional.

TYPE D. Two pairs of the principal spinors are proportional, where  are non-proportional.  Equivalently, there is a spinor dyad () and a complex number such that .

TYPE N. The principal spinors are all proportional, .

 • The command FactorWeylSpinor returns a list of 4 spinors] and a scaling factor such that .
 • The command FactorWeylSpinor is part of the DifferentialGeometry:-Tensor package. It can be used in the form FactorWeylSpinor(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-FactorWeylSpinor(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

We calculate a factorization of Weyl spinors of each Petrov type and we use the command SymmetrizeIndices to verify that the factorization is correct.

We first create a spinor bundle over a 4-dimensional spacetime.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],\mathrm{Spin}\right)$
 ${\mathrm{frame name: Spin}}$ (2.1)

In order to construct the Weyl spinors for our examples, we need a basis for the vector space of symmetric rank 4 spinors. This we obtain from the GenerateSymmetricTensors command.

 Spin > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dz1},\mathrm{dz2}\right],4\right)$
 ${S}{:=}\left[{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{,}\frac{{1}}{{4}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{4}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{4}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{4}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{,}\frac{{1}}{{6}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{6}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{6}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{6}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{6}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{6}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{,}\frac{{1}}{{4}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{4}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{4}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{4}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{,}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}\right]$ (2.2)

Set the global environment variable _EnvExplicit to true to insure that our factorizations are free of expressions.

 Spin > $\mathrm{_EnvExplicit}≔\mathrm{true}:$

Example 1. Type I

Define a rank 4 spinor ${W}_{1}.$

 Spin > $\mathrm{W1}≔\mathrm{DGzip}\left(\left[6,12,30,24,6\right],S,"plus"\right)$
 ${\mathrm{W1}}{:=}{6}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{5}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{5}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{5}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{5}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{5}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{5}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.3)

Calculate the Newman-Penrose coefficients for ${W}_{1}$ with respect to the given dyad basis .

 Spin > $\mathrm{NP1}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W1},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP1}}{:=}{\mathrm{table}}\left(\left[{"Psi4"}{=}{6}{,}{"Psi1"}{=}{-}{6}{,}{"Psi2"}{=}{5}{,}{"Psi3"}{=}{-}{3}{,}{"Psi0"}{=}{6}\right]\right)$ (2.4)

Use the Newman-Penrose coefficients to find the Petrov type of ${W}_{1}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP1}\right)$
 ${"I"}$ (2.5)

Factor ${W}_{1}.$

 Spin > $\mathrm{PS1},\mathrm{η1}≔\mathrm{FactorWeylSpinor}\left(\mathrm{W1},"I"\right)$
 ${\mathrm{PS1}}{,}{\mathrm{η1}}{:=}\left[\left(\frac{{1}}{{2}}{+}{I}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{dz1}}{+}{I}{}{\mathrm{dz2}}{,}\left(\frac{{1}}{{2}}{-}{I}{-}\frac{{1}}{{2}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{,}\left(\frac{{1}}{{2}}{+}{I}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{dz1}}{+}{I}{}{\mathrm{dz2}}{,}\left(\frac{{1}}{{2}}{-}{I}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}\right]{,}{6}$ (2.6)

We check that this answer is correct by computing the symmetric tensor product of the 4 spinors ${\mathrm{PS}}_{1}$.

 Spin > $\mathrm{W1Check}≔\mathrm{η1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{SymmetrizeIndices}\left(\left(\left(\mathrm{PS1}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS1}\left[2\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS1}\left[3\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS1}\left[4\right],\left[1,2,3,4\right],"Symmetric"\right)$
 ${\mathrm{W1Check}}{:=}{6}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{5}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{5}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{5}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{5}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{5}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{5}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{6}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.7)
 Spin > $\mathrm{W1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{W1Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.8)

Example 2. Type II

Define a rank 4 spinor ${W}_{2}.$

 Spin > $\mathrm{W2}≔\mathrm{DGzip}\left(\left[4,4,6,16,10\right],S,"plus"\right)$
 ${\mathrm{W2}}{:=}{4}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{10}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.9)

Calculate the Newman-Penrose coefficients for ${W}_{2}$ with respect to the given dyad basis .

 Spin > $\mathrm{NP2}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W2},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP2}}{:=}{\mathrm{table}}\left(\left[{"Psi4"}{=}{4}{,}{"Psi1"}{=}{-}{4}{,}{"Psi2"}{=}{1}{,}{"Psi3"}{=}{-}{1}{,}{"Psi0"}{=}{10}\right]\right)$ (2.10)

Find the Petrov type of ${W}_{2}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP2}\right)$
 ${"II"}$ (2.11)

Factor ${W}_{2}.$

 Spin > $\mathrm{PS2},\mathrm{η2}≔\mathrm{FactorWeylSpinor}\left(\mathrm{W2},"II"\right)$
 ${\mathrm{PS2}}{,}{\mathrm{η2}}{:=}\left[\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz1}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz2}}{,}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz1}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}{}{\mathrm{dz2}}{,}\left(\frac{{1}}{{3}}{}{I}{}\sqrt{{3}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}\right){}{\mathrm{dz1}}{+}\left({-}\frac{{2}}{{3}}{}{I}{}\sqrt{{3}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}\right){}{\mathrm{dz2}}{,}\left({-}\frac{{1}}{{3}}{}{I}{}\sqrt{{3}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}\right){}{\mathrm{dz1}}{+}\left(\frac{{2}}{{3}}{}{I}{}\sqrt{{3}}{+}\frac{{1}}{{3}}{}\sqrt{{3}}\right){}{\mathrm{dz2}}\right]{,}{18}$ (2.12)

Note that the first two factors are identical.

 Spin > $\mathrm{PS2}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS2}\left[2\right]$
 ${0}{}{\mathrm{dt}}$ (2.13)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${\mathrm{PS}}_{2}.$

 Spin > $\mathrm{W2Check}≔\mathrm{η2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{SymmetrizeIndices}\left(\left(\left(\mathrm{PS2}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS2}\left[2\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS2}\left[3\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS2}\left[4\right],\left[1,2,3,4\right],"Symmetric"\right)$
 ${\mathrm{W2Check}}{:=}{4}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{10}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.14)
 Spin > $\mathrm{W2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{W2Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.15)

Example 3. Type III

Define a rank 4 spinor ${W}_{3}.$

 Spin > $\mathrm{W3}≔\mathrm{DGzip}\left(\left[-8,-20I,12,-4I,4\right],S,"plus"\right)$
 ${\mathrm{W3}}{:=}{-}{8}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.16)

Calculate the Newman-Penrose coefficients for ${W}_{3}$ with respect to the given dyad basis .

 Spin > $\mathrm{NP3}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W3},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP3}}{:=}{\mathrm{table}}\left(\left[{"Psi4"}{=}{-}{8}{,}{"Psi1"}{=}{I}{,}{"Psi2"}{=}{2}{,}{"Psi3"}{=}{5}{}{I}{,}{"Psi0"}{=}{4}\right]\right)$ (2.17)

Find the Petrov type of ${W}_{3}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP3}\right)$
 ${"III"}$ (2.18)

Factor ${W}_{3.}$

 Spin > $\mathrm{PS3},\mathrm{η3}≔\mathrm{FactorWeylSpinor}\left(\mathrm{W3},"III"\right)$
 ${\mathrm{PS3}}{,}{\mathrm{η3}}{:=}\left[\left({-}\frac{{1}}{{2}}{}\sqrt{{6}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{6}}\right){}{\mathrm{dz1}}{+}\left({-}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz2}}{,}\left({-}\frac{{1}}{{2}}{}\sqrt{{6}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{6}}\right){}{\mathrm{dz1}}{+}\left({-}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz2}}{,}\left({-}\frac{{1}}{{2}}{}\sqrt{{6}}{+}\frac{{1}}{{2}}{}{I}{}\sqrt{{6}}\right){}{\mathrm{dz1}}{+}\left({-}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz2}}{,}\left(\frac{{1}}{{9}}{-}\frac{{1}}{{9}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz1}}{+}\left({-}\frac{{1}}{{18}}{-}\frac{{1}}{{18}}{}{I}\right){}\sqrt{{6}}{}{\mathrm{dz2}}\right]{,}{-}{4}$ (2.19)

Note that the first three factors are identical.

 Spin > $\mathrm{PS3}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS3}\left[2\right],\mathrm{PS3}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS3}\left[3\right]$
 ${0}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}$ (2.20)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${\mathrm{PS}}_{3}.$

 Spin > $\mathrm{W3Check}≔\mathrm{η3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{SymmetrizeIndices}\left(\left(\left(\mathrm{PS3}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS3}\left[2\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS3}\left[3\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS3}\left[4\right],\left[1,2,3,4\right],"Symmetric"\right)$
 ${\mathrm{W3Check}}{:=}{-}{8}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}{5}{}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{5}{}{I}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{2}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{I}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{4}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.21)
 Spin > $\mathrm{W3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{W3Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.22)

Example 4. Type D

Define a rank 4 spinor ${W}_{4}.$

 Spin > $\mathrm{W4}≔\mathrm{DGzip}\left(\left[3,-18,3,72,48\right],S,"plus"\right)$
 ${\mathrm{W4}}{:=}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{48}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.23)

Calculate the Newman-Penrose coefficients for ${W}_{4}$ with respect to the given dyad basis .

 Spin > $\mathrm{NP4}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W4},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP4}}{:=}{\mathrm{table}}\left(\left[{"Psi4"}{=}{3}{,}{"Psi1"}{=}{-}{18}{,}{"Psi2"}{=}\frac{{1}}{{2}}{,}{"Psi3"}{=}\frac{{9}}{{2}}{,}{"Psi0"}{=}{48}\right]\right)$ (2.24)

Find the Petrov type of ${W}_{4}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP4}\right)$
 ${"D"}$ (2.25)

Factor ${W}_{4.}$

 Spin > $\mathrm{PS4},\mathrm{η4}≔\mathrm{FactorWeylSpinor}\left(\mathrm{W4},"D"\right)$
 ${\mathrm{PS4}}{,}{\mathrm{η4}}{:=}\left[{\mathrm{dz1}}{+}{\mathrm{dz2}}{,}{\mathrm{dz1}}{+}{\mathrm{dz2}}{,}{-}\frac{{1}}{{5}}{}{\mathrm{dz1}}{+}\frac{{4}}{{5}}{}{\mathrm{dz2}}{,}{-}\frac{{1}}{{5}}{}{\mathrm{dz1}}{+}\frac{{4}}{{5}}{}{\mathrm{dz2}}\right]{,}{75}$ (2.26)

Note that the first two factors and last two factors are identical.

 Spin > $\mathrm{PS4}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS4}\left[2\right],\mathrm{PS4}\left[3\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS4}\left[4\right]$
 ${0}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}$ (2.27)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${\mathrm{PS}}_{4}.$

 Spin > $\mathrm{W4Check}≔\mathrm{η4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{SymmetrizeIndices}\left(\left(\left(\mathrm{PS4}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS4}\left[2\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS4}\left[3\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS4}\left[4\right],\left[1,2,3,4\right],"Symmetric"\right)$
 ${\mathrm{W4Check}}{:=}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{-}\frac{{9}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{18}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{48}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.28)
 Spin > $\mathrm{W4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{W4Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.29)

Example 5. Type N

Define a rank 4 spinor ${W}_{5}.$

 Spin > $\mathrm{W5}≔\mathrm{DGzip}\left(\left[1,12,54,108,81\right],S,"plus"\right)$
 ${\mathrm{W5}}{:=}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{81}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.30)

Calculate the Newman-Penrose coefficients for ${W}_{5}$ with respect to the given dyad basis .

 Spin > $\mathrm{NP5}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W5},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP5}}{:=}{\mathrm{table}}\left(\left[{"Psi4"}{=}{1}{,}{"Psi1"}{=}{-}{27}{,}{"Psi2"}{=}{9}{,}{"Psi3"}{=}{-}{3}{,}{"Psi0"}{=}{81}\right]\right)$ (2.31)

Find the Petrov type of ${W}_{5}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP5}\right)$
 ${"N"}$ (2.32)

Factor ${W}_{5.}$

 Spin > $\mathrm{PS5},\mathrm{η5}≔\mathrm{FactorWeylSpinor}\left(\mathrm{W5},"N"\right)$
 ${\mathrm{PS5}}{,}{\mathrm{η5}}{:=}\left[{\mathrm{dz1}}{+}{3}{}{\mathrm{dz2}}{,}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz2}}{,}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz2}}{,}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz2}}\right]{,}{1}$ (2.33)

Note that all four factors are identical.

 Spin > $\mathrm{PS5}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS5}\left[2\right],\mathrm{PS5}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS5}\left[3\right],\mathrm{PS5}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS5}\left[4\right]$
 ${0}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}{,}{0}{}{\mathrm{dt}}$ (2.34)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${\mathrm{PS}}_{5}.$

 Spin > $\mathrm{W5Check}≔\mathrm{η5}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&mult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{SymmetrizeIndices}\left(\left(\left(\mathrm{PS5}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS5}\left[2\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS5}\left[3\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{PS5}\left[4\right],\left[1,2,3,4\right],"Symmetric"\right)$
 ${\mathrm{W5Check}}{:=}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{3}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{9}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{27}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{81}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}$ (2.35)
 Spin > $\mathrm{W5}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{W5Check}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.36)