N=4.nb

$RecursionLimit=Infinity

Import Adinkra package

<<Adinkra`

BuildDate[Adinkra]

Function Lists

FunctionList[SpaceTime]

FunctionList[GenerateLandR]

FunctionList[AdinkraEssentials]

FunctionList[BasisDecomposition]

FunctionList[BC4Tools]

FunctionList[GraphingTools]

FunctionList[Adinkra]

Introduction to some of these functions

Within FunctionList[SpaceTime]

η[1,1]

Cmetric

Lap[SomeField[t,x,y,z]]

Lap[A[2][t,x,y,z]]

Within FunctionList[AdinkraEssentials]

IndexRange[AdinkraEssentials][ReportLevel]

FunctionList[AdinkraEssentials]

N=4, Various Representations

ONshellCM,  and a non-adinkraic representation to test catches

L[OnShellCM] ={(({{1, 0, 0, 0}, {0, 0, 0, -1}})),(({{0, 1, 0, 0}, {0, 0, 1, 0}})),(({{0, 0, 1, 0}, {0, -1, 0, 0}})),(({{0, 0, 0, 1}, {1, 0, 0, 0}}))}

R[OnShellCM] = Table[Transpose[L[OnShellCM][[Ii]]],{Ii,1,4}]

L[ID] = L[OnShellCM];
R[ID] = L[Q];

GenerateAdinkraData[OnShellCM]

GenerateCoeffs[OnShellCM]

PrintAllGARL[OnShellCM]

PrintGARLBasis[OnShellCM][1,1]

PrintGARLBasis[OnShellCM][3,3]

PrintAllVtilde[OnShellCM]

PrintVtildeBasis[OnShellCM][3,4]

PrintRBasis[OnShellCM][1]

GALR[OnShellCM][[1,1]]

PrintGALR[OnShellCM][1,1]

PrintAllGALR[OnShellCM]

PrintAllGARL[OnShellCM]

Q: a base representation built in to Adinkra.m
((Subscript[((Subscript[Q, I])), i])^(OverHat[j]))= {(), ((OverBar[13]))(12)(34), ((OverBar[14]))(13)(24), ((OverBar[12]))(14)(23)} 
            = {I, -i (α^2), -i (α^3), -i (α^1)} = {I, i, j, k}

PrintAllL[Q]

MatrixPower[L[Q][[2]],2]//MatrixForm

L[Q][[3]].(-Transpose[L[Q][[3]]])//MatrixForm

L[Q][[3]].L[Q][[2]]== -L[Q][[4]]

PrintAllL[Q]

GraphAdinkra[Q]

GraphAdinkra[Q,buildrules[{{8},{1,2,3,4},{5,6,7}}]]

GraphAdinkra[Q,buildrules[{{5,6},{1,2,3,4},{7,8}}]]

GenerateAdinkraData[Q,Orthogonal]

PrintAllGALR[Q]

PrintAllGARL[Q]

GenerateCoeffs[Q]

PrintGALRBasis[Q][1,1]

PrintVtildeBasis[Q][1,2]

PrintGALRBasis[Q][1,2]

(OverTilde[Q]): a base representation built in to Adinkra.m 
((Subscript[((Subscript[(OverTilde[Q]), I])), i])^(OverHat[j]))= {(), ((OverBar[23]))(12)(34), ((OverBar[12]))(13)(24), ((OverBar[13]))(14)(23)}
            = {I, i (β^3), -i (β^2), -i (β^1)}  = {I, (OverTilde[i]), (OverTilde[j]), (OverTilde[k])}

PrintAllL[Qtilde]

GraphAdinkra[Qtilde]

CM

L[CM]={ (({{1, 0, 0, 0}, {0, 0, 0, -1}, {0, 1, 0, 0}, {0, 0, -1, 0}})),(({{0, 1, 0, 0}, {0, 0, 1, 0}, {-1, 0, 0, 0}, {0, 0, 0, -1}})),(({{0, 0, 1, 0}, {0, -1, 0, 0}, {0, 0, 0, -1}, {1, 0, 0, 0}})),(({{0, 0, 0, 1}, {1, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 0}}))};

L[CM] == L[362511]

GraphAdinkra[CM]

TM

L[TM]={ (({{1, 0, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}, {0, -1, 0, 0}})),(({{0, 1, 0, 0}, {0, 0, 0, 1}, {0, 0, -1, 0}, {1, 0, 0, 0}})),(({{0, 0, 1, 0}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, -1}})),(({{0, 0, 0, 1}, {0, -1, 0, 0}, {1, 0, 0, 0}, {0, 0, 1, 0}}))};

L[TM] == L[553212]

GraphAdinkra[TM]

VM

L[VM]={ (({{0, 1, 0, 0}, {0, 0, 0, -1}, {1, 0, 0, 0}, {0, 0, -1, 0}})),(({{1, 0, 0, 0}, {0, 0, 1, 0}, {0, -1, 0, 0}, {0, 0, 0, -1}})),(({{0, 0, 0, 1}, {0, 1, 0, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}})),(({{0, 0, 1, 0}, {-1, 0, 0, 0}, {0, 0, 0, -1}, {0, 1, 0, 0}}))};

L[VM] == L[-342412]

GraphAdinkra[VM]

VM1

L[CM]={ (({{1, 0, 0, 0}, {0, 0, 0, -1}, {0, 1, 0, 0}, {0, 0, -1, 0}})),(({{0, 1, 0, 0}, {0, 0, 1, 0}, {-1, 0, 0, 0}, {0, 0, 0, -1}})),(({{0, 0, 1, 0}, {0, -1, 0, 0}, {0, 0, 0, -1}, {1, 0, 0, 0}})),(({{0, 0, 0, 1}, {1, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 0}}))};

L[CM] == L[362511]

GraphAdinkra[CM]

cis adinkra

L[cis] = L[111712];

PrintAllL[cis]

GraphAdinkra[cis]

trans- adinkra

L[trans] = L[111112];

PrintAllL[trans]

GraphAdinkra[trans]

L[-734262]

L[-734262]

GraphAdinkra[-734262]

GenerateAdinkraData[-734262,Orthogonal]

PrintAllGALR[-734262]

PrintAllGARL[-734262]

AdinkraPreliminaryReport[Rep] prints preliminary data to see if we want to proceed with analysis

GraphAdinkra[CM]

GenerateAdinkraData[CM,Orthogonal]

PrintAllGARL[CM]

AdinkraReport[CM,1]

AdinkraPreliminaryReport[OnShellCM]

AdinkraReport[trans,0]

AdinkraPreliminaryReportO[cis]

AdinkraPreliminaryReport[-734262]

Need to use AdinkraPreliminaryReportO[-734262] as never stored the R matrices for this Rep

AdinkraPreliminaryReportO[-734262]

GenerateData

GenerateAdinkraData[Q,Orthogonal]

GenerateCoeffs[Q]

GenerateAdinkraData[Qtilde,Orthogonal]

GenerateCoeffs[Qtilde]

GenerateAdinkraData[CM,Orthogonal]

GenerateCoeffs[CM]

GenerateAdinkraData[TM,Orthogonal]

GenerateCoeffs[TM]

GenerateAdinkraData[VM,Orthogonal]

GenerateCoeffs[VM]

GenerateAdinkraData[ID]

GenerateCoeffs[ID] (* Need to fix this, shouldn't generate anything *)

GenerateAdinkraData[OnShellCM]

GenerateAdinkraData[cis,Orthogonal]

GenerateCoeffs[cis]

GenerateAdinkraData[trans,Orthogonal]

GenerateCoeffs[trans]

GenerateAdinkraData[-734262,Orthogonal]

GenerateCoeffs[-734262]

Reports

AdinkraReport[ID,3]

AdinkraReport[Q,3]

AdinkraReport[Q,3]

AdinkraReport[Q,8]

AdinkraReport[OnShellCM]

AdinkraReport[OnShellCM]

AdinkraFullReport[OnShellCM]

AdinkraFullReport[TM]

V[VM][[1,2]]==V[VM][[3,4]]

V[VM][[1,2]] == 2VsoN[VM][[1,2]]

FunctionList[AdinkraEssentials]

When the LinearlyIndependent function does not give True, the output below is read as follows: 
VPM[-1][CM][[2,3]] + VPM[-1][CM][[1,4]]   =  0   =>   VPM[-1][CM][[2,3]] = - VPM[-1][CM][[1,4]] 
VPM[-1][CM][[2,4]] -  VPM[-1][CM][[1,3]]  =  0   =>     VPM[-1][CM][[2,4]] = VPM[-1][CM][[1,3]]
VPM[-1][CM][[3,4]] + VPM[-1][CM][[1,2]]  =  0   =>     VPM[-1][CM][[3,4]] = - VPM[-1][CM][[1,2]]

LinearlyIndependent[VPM[-1][CM]]

PrintAllVPM[-1][CM]

Define
BC4ColorPerm[n,a,μ,A]:=(((-1))^n)((Subscript[((H^a)), I])^K)((Subscript[((Subscript[S, 3]^μ)), K])^L)((Subscript[((𝒱^A)), L])^J)
(H^a)= { (),((OverBar[12])),((OverBar[13])),((OverBar[23])) ,((OverBar[1])),((OverBar[2])),((OverBar[3])),((OverBar[123]))} ,  sign flips on three elements
(S^μ) = { (), (12), (13), (23), (123), (132) }:,  (Subscript[S, 3]) permutation group:
(𝒱^A)= {(), (12)(34), (13)(24), (14)(23) }, the vierergruppe:

Here we are using the BC4Tools package, so lets display the function list to see what we have to use

FunctionList[BC4Tools]

Use Index Range to display the valid ranges for a given type of index listed in the FunctionList

IndexRange[BC4Tools][n]

IndexRange[BC4Tools][a]

IndexRange[BC4Tools][μ]

IndexRange[BC4Tools][A]

Some Permutation Matrices

S3Perm[123].VierPerm[1423]//MatrixForm

BC4Perm[0,12,123,1423]//MatrixForm

BC4Perm[0,12,123,1423][[1,2]]L[Q][[2]]

S3Perm[132]//MatrixForm

We are using BC4ColorPerm, rather than BC4Color so use the appropriate indices

L[NewQ]=BC4ColorPerm[-1,12,123,1423][L[Q]];

PrintAllL[NewQ]

GraphAdinkra[NewQ]

Compare to Q, the above is a permutation of the colors as encoded in PrintBC4ColorPerm below read right to left:
then flopping (swapping) colors 1 and 4 and 2 and 3:  (14)(23)
then flopping color 1 to 2 to 3 to 1:  (123)
--->    (142)
1 to 4 (green to red, as in what was green in Q is now red in NewQ),
4 to 2 (red to becomes purple)
2 to 1  (purple becomes green)
orange stays orange
the flipping (switching the sign of) colors 3 and 4 (orange and red):  ((OverBar[34]))

PrintBC4ColorPerm[-1,12,123,1423]

GraphAdinkra[Q]

GraphAdinkra[NewQ]

In terms of the Table indices:
BC4Color[n,a,μ,A][L[Rep]] := ((Subscript[((Subscript[BC, 4]^(mbνB))), I])^J)((Subscript[((Subscript[L, J]^((Rep)))), i])^(OverHat[j]))= (((-1))^m)((Subscript[((H^b)), I])^K)((Subscript[((Subscript[S, 3]^ν)), K])^L)((Subscript[((𝒱^B)), L])^J)((Subscript[((Subscript[L, J]^((Rep)))), i])^(OverHat[j]))
BC4Boson[n,a,μ,A][L[Rep]] := ((Subscript[((Subscript[BC, 4]^(mbνB))), i])^j)((Subscript[((Subscript[L, I]^((Rep)))), j])^(OverHat[j]))= (((-1))^m)((Subscript[((H^b)), i])^k)((Subscript[((Subscript[S, 3]^ν)), k])^p)((Subscript[((𝒱^B)), p])^j)((Subscript[((Subscript[L, I]^((Rep)))), j])^(OverHat[j]))
for n = 1 we have the minus sign
b=2 is the sign flip  ((OverBar[12])), so put these together we have  -((OverBar[12]))=((OverBar[34]))
μ=5 is the Permutation (123)
A=4 is the Vierergruppe element (14)(23)
so we should use BC4Color[1,2,5,4]

IndexRange[BC4Tools][n]

IndexRange[BC4Tools][a]

IndexRange[BC4Tools][μ]

IndexRange[BC4Tools][A]

L[NewQTable]=BC4Color[1,2,5,4][L[Q]];

PrintAllL[NewQTable]

L[NewQTable]==L[NewQ]

GraphAdinkra[NewQTable]

GraphAdinkra[Q]

GraphAdinkra[NewQTable]
GraphAdinkra[NewQ]

Isometry BC4Color and BC4Boson transformations on Q

(Subscript[Q, I])

LBP = {0,0,0,0};
LCP = {0,0,0,0};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]

BC4Boson: ((OverBar[1234])); BC4Color: ((OverBar[1234]))

LBP = {-1,0,0,0};
LCP = {-1,0,0,0};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[13]))(12)(34); BC4Color: ((OverBar[13]))(12)(34)

LBP = {0,13,0,1234};
LCP = {0,13,0,1234};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[24]))(12)(34); BC4Color: ((OverBar[24]))(12)(34)

LBP = {-1,13,0,1234};
LCP = {-1,13,0,1234};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[23]))(13)(24); BC4Color: ((OverBar[23]))(13)(24)

LBP = {0,23,0,1324};
LCP = {0,23,0,1324};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[14]))(13)(24); BC4Color: ((OverBar[14]))(13)(24)

LBP = {-1,23,0,1324};
LCP = {-1,23,0,1324};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[12]))(14)(23); BC4Color: ((OverBar[12]))(14)(23)

LBP = {0,12,0,1423};
LCP = {0,12,0,1423};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[34]))(14)(23); BC4Color: ((OverBar[34]))(14)(23)

LBP = {-1,12,0,1423};
LCP = {-1,12,0,1423};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Q]]];
GraphAdinkra[Local]
L[Local] == L[Q]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

Isometry BC4Color and BC4Boson transformations on (OverTilde[Q])

(Subscript[(OverTilde[Q]), I])

LBP = {0,0,0,0};
LCP = {0,0,0,0};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[1234])); BC4Color: ((OverBar[1234]))

LBP = {-1,0,0,0};
LCP = {-1,0,0,0};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[23]))(12)(34); BC4Color: ((OverBar[13]))(12)(34)

LBP = {0,23,0,1234};
LCP = {0,13,0,1234};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[14]))(12)(34); BC4Color: ((OverBar[24]))(12)(34)

LBP = {-1,23,0,1234};
LCP = {-1,13,0,1234};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[12]))(13)(24); BC4Color: ((OverBar[14]))(13)(24)

LBP = {0,12,0,1324};
LCP = {-1,23,0,1324};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[34]))(13)(24); BC4Color: ((OverBar[23]))(13)(24)

LBP = {-1,12,0,1324};
LCP = {0,23,0,1324};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[13]))(14)(23); BC4Color: ((OverBar[12]))(14)(23)

LBP = {0,13,0,1423};
LCP = {0,12,0,1423};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

BC4Boson: ((OverBar[24]))(14)(23); BC4Color: ((OverBar[34]))(14)(23)

LBP = {-1,13,0,1423};
LCP = {-1,12,0,1423};
L[Local] = BC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]][BC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]][L[Qtilde]]];
GraphAdinkra[Local]
L[Local] == L[Qtilde]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]
PrintBC4BosonPerm[LBP[[1]],LBP[[2]],LBP[[3]],LBP[[4]]]PrintBC4ColorPerm[LCP[[1]],LCP[[2]],LCP[[3]],LCP[[4]]]//TeXForm

96*384

48*8

All 36864 N=4, d=4 valise adinkras
((Subscript[((Subscript[L, I]^(±aμAbν))), i])^(OverHat[j])):=((Subscript[((Subscript[BC, 4]^(±aμA))), i])^j)((Subscript[((Subscript[L, I]^(bν))), j])^(OverHat[j]))=((Subscript[((Subscript[BC, 4]^(±aμA))), i])^j)((Subscript[((Subscript[HS, 3]^(bν))), I])^J) ((Subscript[((Subscript[Q, J])), j])^(OverHat[j]))  ,  (Subscript[(OverTilde[V]), IJ])~α  (ℓ-equivalence classes)
((Subscript[((Subscript[(OverTilde[L]), I]^(±aμAbν))), i])^(OverHat[j])):= ((Subscript[((Subscript[BC, 4]^(±aμA))), i])^j)((Subscript[((Subscript[(OverTilde[L]), I]^(bν))), j])^(OverHat[j]))=((Subscript[((Subscript[BC, 4]^(±aμA))), i])^j)((Subscript[((Subscript[HS, 3]^(bν))), I])^J) ((Subscript[((Subscript[(OverTilde[Q]), J])), j])^(OverHat[j])) ,  (Subscript[(OverTilde[V]), IJ])~β  ((OverTilde[ℓ])-equivalence classes)
L[RepCode] := ((Subscript[((Subscript[L, I]^(±aμAbν))), i])^(OverHat[j])) for RepCode = ±aμAbν1, ((Subscript[((Subscript[(OverTilde[L]), I]^(±aμAbν))), i])^(OverHat[j]))for RepCode= ±aμAbν2

PrintAllL[CM]

L[CM] == L[362511]

GraphAdinkra[CM]

Gadgets

Gadget[Q,Qtilde]

Gadget[Q,CM]

Gadget[CM,TM]

Gadget[TM,VM]

Gadget[VM,cis]

Gadget[cis,trans]

PrintAllL[cis]

Gadget[Qtilde,trans]

Gadget[Qtilde,cis]

Print Functions: choose Rep to be anything from the previous sections

GraphAdinkra[cis]
PrintAllL[cis]

GraphAdinkra[trans]
PrintAllL[trans]

Rep = trans;

PrintL[Rep][1]

PrintAllL[Rep]

PrintR[Rep][2]

PrintAllR[Rep]

PrintAllV[Rep]

Table[PrintV[Rep][Ii,Ji],{Ii,1,4},{Ji,1,4}] ==Table[V[Rep][[Ii,Ji]]//MatrixForm,{Ii,1,4},{Ji,1,4}]

PrintAllVtildePM[1][Rep]

PrintAllHoloraumytilde[Rep][{1,1,1,1}]

Holoraumytilde[Rep][[15]]//MatrixForm

PrintZetaGen[Rep][4]

PrintAllZetaGen[Rep]

PrintAllHoloraumy[Rep][{1,1,1,1}]==PrintAllHoloraumy[Rep][[15]]==(-ZetaGen[Rep][[2]].ZetaGen[Rep][[3]].ZetaGen[Rep][[4]]//MatrixForm)

WordW[{0,0,0,1}]

PrintAllHoloraumytilde[Rep]

PrintAllMonodromytilde[Rep]

N=4 dWvH 20x20 multiplet: time consuming

Matrices

L[Test20] = {(({{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, (1/2)}, {0, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, (1/2)}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2), 0, 0}, {0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, (1/2), 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -(1/2), 0}, {-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, (1/2)}, {2, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -(1/2), 0}, {0, 1, 0, 0, 0, 0, 0, -1, 2, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, (1/2), 0}, {0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, (1/2)}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}})),(({{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -(1/2), 0}, {0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -(1/2), 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, (1/2), 0}, {0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -(1/2), 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, (1/2)}, {0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, (1/2)}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, (1/2), 0}, {0, 2, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {-1, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, (1/2), 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}})),(({{0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, (1/2), 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, (1/2)}, {0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, (1/2)}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, (1/2), 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -(1/2), 0}, {0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, (1/2), 0}, {1, 0, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, -(1/2), 0, 0}, {0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, (1/2)}})),(({{0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, (1/2)}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -(1/2), 0, 0, 0}, {1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -(1/2), 0}, {0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, (1/2), 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2), 0, 0}, {0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, (1/2), 0, 0, 0}, {0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, -(1/2)}, {0, -1, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 1, 0, 0, 0, 0, -(1/2), 0, 0}, {0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, -(1/2), 0}, {0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, (1/2), 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -(1/2), 0, 0, 0}, {0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -(1/2), 0}}))};

R[Test20] = Table[Inverse[L[Test20][[Ii]]],{Ii,1,4}];

Generating Adinkra Data

GenerateAdinkraData[Test20]

GenerateCoeffs[Test20]

Reports

AdinkraReport[Test20,0]//AbsoluteTiming

AdinkraReport[Test20,1]//AbsoluteTiming

AdinkraReport[Test20,2]//AbsoluteTiming

AdinkraReport[Test20,3]//AbsoluteTiming

AdinkraReport[Test20,4]//AbsoluteTiming

AdinkraReport[Test20,5]//AbsoluteTiming

AdinkraReport[Test20,6]//AbsoluteTiming

AdinkraReport[Test20,7]//AbsoluteTiming

AdinkraReport[Test20,8]//AbsoluteTiming

AdinkraHoloMonoReport[Test20]

AdinkraReport[Test20]

AdinkraFullReport[Test20]

PrintVtildeBasis[Test20][1,2]/.nz[5]->2

Testing BC4Tools

FunctionList[BC4Tools]

Rep

PrintIell[Q]

PrintIell[CM]

PrintIell[VM]

PrintIell[TM]

ell[Q][1,1][1,2]

IellABCode[Q]

Vtilde[Q]