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Obs(4) is a 16 dimensional Frobenius Algrebra

fricas
)set output abbreviate on
V := OrderedVariableList [p,q,r,s]

\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q , r , s ])(1)
Type: TYPE
fricas
M := FreeMonoid V

\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r , s ]))(2)
Type: TYPE
fricas
gens:List M := enumerate()$V

\label{eq3}\left[ p , \: q , \: r , \: s \right](3)
Type: LIST(FMONOID(OVAR([p,q,r,s])))
fricas
divisible := Record(lm: M,rm: M)

\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r , s ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r , s ])))}(4)
Type: TYPE
fricas
leftDiv(k:Union(divisible,"failed")):M == (k::divisible).lm
Function declaration leftDiv : Union(Record(lm: FMONOID(OVAR([p,q,r, s])),rm: FMONOID(OVAR([p,q,r,s]))),"failed") -> FMONOID(OVAR([p,q ,r,s])) has been added to workspace.
Type: VOID
fricas
rightDiv(k:Union(divisible,"failed")):M == (k::divisible).rm
Function declaration rightDiv : Union(Record(lm: FMONOID(OVAR([p,q,r ,s])),rm: FMONOID(OVAR([p,q,r,s]))),"failed") -> FMONOID(OVAR([p, q,r,s])) has been added to workspace.
Type: VOID
fricas
K := FRAC POLY INT

\label{eq5}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ }))(5)
Type: TYPE
fricas
MK := FreeModule(K,M)

\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r , s ])))(6)
Type: TYPE
fricas
coeff(x:MK):K == leadingCoefficient(x)
Function declaration coeff : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r, s]))) -> FRAC(POLY(INT)) has been added to workspace.
Type: VOID
fricas
monomial(x:MK):M == leadingMonomial(x)
Function declaration monomial : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q ,r,s]))) -> FMONOID(OVAR([p,q,r,s])) has been added to workspace.
Type: VOID
fricas
m(x:M):K == subscript('m,[retract(x)::Symbol])
Function declaration m : FMONOID(OVAR([p,q,r,s])) -> FRAC(POLY(INT)) has been added to workspace.
Type: VOID
fricas
γ(x:M,y:M):K == subscript('γ,[concat(string retract x, string retract y)::Symbol])
Function declaration γ : (FMONOID(OVAR([p,q,r,s])),FMONOID(OVAR([p,q ,r,s]))) -> FRAC(POLY(INT)) has been added to workspace.
Type: VOID

Basis

fricas
basis := concat(gens,concat [[j*i for i in gens | i~=j] for j in gens])

\label{eq7}\left[ p , \: q , \: r , \: s , \:{p \  q}, \:{p \  r}, \:{p \  s}, \:{q \  p}, \:{q \  r}, \:{q \  s}, \:{r \  p}, \:{r \  q}, \:{r \  s}, \:{s \  p}, \:{s \  q}, \:{s \  r}\right](7)
Type: LIST(FMONOID(OVAR([p,q,r,s])))

Idempotent

fricas
rule1(ij:MK):MK ==
  for k in gens repeat
    kk := divide(monomial(ij),k*k)
    if kk case divisible then
      ij:=(coeff(ij) * m(k)*γ(k,k)) * (leftDiv(kk) * k * rightDiv(kk))
  return(ij)
Function declaration rule1 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r, s]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s]))) has been added to workspace.
Type: VOID

Reduction

fricas
rule2(ij:MK):MK ==
  for i in gens repeat
    for j in gens | j ~= i repeat
      for k in gens | k ~= j repeat
        ijk:=divide(leadingMonomial(ij),i*j*k)
        if ijk case divisible then
          --if i=k then
          --  ij := (coeff(ij)*m(i)*m(j)*γ(i,j)*γ(j,i) ) * _
          --        (leftDiv(ijk)*i*rightDiv(ijk))
          --else
            ij := (coeff(ij)*m(j)*γ(i,j)*γ(j,k) / γ(i,k) ) * _
                  (leftDiv(ijk)*i*k*rightDiv(ijk))
  return(ij)
Function declaration rule2 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r, s]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s]))) has been added to workspace.
Type: VOID

Modulo fixed point of applied rules

fricas
mod(ij:MK):MK ==
  ijFix:MK := 1
  while ijFix~=ij repeat
    ijFix := ij
    ij := rule1(ij)
    ij := rule2(ij)
  return(ij)
Function declaration mod : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s] ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s]))) has been added to workspace.
Type: VOID

Matrix

Multiplication is monoidal concatenation modulo the fixed point

fricas
MT := [[mod(i*j) for j in basis] for i in basis]
fricas
Compiling function monomial with type FM(FRAC(POLY(INT)),FMONOID(
      OVAR([p,q,r,s]))) -> FMONOID(OVAR([p,q,r,s]))
fricas
Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r,s]))) -> FRAC(POLY(INT))
fricas
Compiling function m with type FMONOID(OVAR([p,q,r,s])) -> FRAC(POLY
      (INT))
fricas
Compiling function γ with type (FMONOID(OVAR([p,q,r,s])),FMONOID(
      OVAR([p,q,r,s]))) -> FRAC(POLY(INT))
fricas
Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
      p,q,r,s])),rm: FMONOID(OVAR([p,q,r,s]))),"failed") -> FMONOID(
      OVAR([p,q,r,s]))
fricas
Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
      [p,q,r,s])),rm: FMONOID(OVAR([p,q,r,s]))),"failed") -> FMONOID(
      OVAR([p,q,r,s]))
fricas
Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r,s]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s])))
fricas
Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r,s]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s])))
fricas
Compiling function mod with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q,r,s]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s])))

\label{eq8}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{{m_{p}}\ {��_{pp}}\  p}, \:{p \  q}, \:{p \  r}, \:{p \  s}, \:{{m_{p}}\ {��_{pp}}\  p \  q}, \:{{m_{p}}\ {��_{pp}}\  p \  r}, \:{{m_{p}}\ {��_{pp}}\  p \  s}, \: \right.
\
\
\displaystyle
\left.{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\  p}, \:{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\  p \  r}, \:{{{{m_{q}}\ {��_{pq}}\ {��_{qs}}}\over{��_{ps}}}\  p \  s}, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}\  p}, \right.
\
\
\displaystyle
\left.\:{{{{m_{r}}\ {��_{pr}}\ {��_{rq}}}\over{��_{pq}}}\  p \  q}, \:{{{{m_{r}}\ {��_{pr}}\ {��_{rs}}}\over{��_{ps}}}\  p \  s}, \:{{m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}\  p}, \:{{{{m_{s}}\ {��_{ps}}\ {��_{sq}}}\over{��_{pq}}}\  p \  q}, \: \right.
\
\
\displaystyle
\left.{{{{m_{s}}\ {��_{ps}}\ {��_{sr}}}\over{��_{pr}}}\  p \  r}\right] 
(8)
Type: LIST(LIST(FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r,s])))))

Structure Constants

fricas
R:=FRAC DMP(concat [[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)

\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ <em> 01 mp , </em> 01 mq , <em> 01 mr , </em> 01 ms , * 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� ps , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� qs , * 01 �� rp , * 01 �� rq , * 01 �� rr , * 01 �� rs , * 01 �� sp , * 01 �� sq , * 01 �� sr , * 0
1 �� ss ] , \hbox{\axiomType{INT}\ }))(9)
Type: TYPE
fricas
mat3(y:M):List List R == map(z+->map(x+->coefficient(x,y)::FRAC POLY INT,z),MT)
Function declaration mat3 : FMONOID(OVAR([p,q,r,s])) -> LIST(LIST( FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps, *01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp, *01γsq,*01γsr,*01γss],INT)))) has been added to workspace.
Type: VOID
fricas
ss:=map(mat3, basis)
fricas
Compiling function mat3 with type FMONOID(OVAR([p,q,r,s])) -> LIST(
      LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γ
      ps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp
      ,*01γsq,*01γsr,*01γss],INT))))

\label{eq10}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{{m_{p}}\ {��_{pp}}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: 0, \: 0, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: 0, \: 0, \: \right.
\
\
\displaystyle
\left.{{m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \: 0, \: 0 \right] 
(10)
Type: LIST(LIST(LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)))))

Algebra

fricas
cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol
Function declaration cats : FMONOID(OVAR([p,q,r,s])) -> SYMBOL has been added to workspace.
Type: VOID
fricas
A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI,map(cats,basis),ss::Vector(Matrix R))
fricas
Compiling function cats with type FMONOID(OVAR([p,q,r,s])) -> SYMBOL

\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ <em> 01 mp , </em> 01 mq , <em> 01 mr , </em> 01 ms , * 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� ps , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� qs , * 01 �� rp , * 01 �� rq , * 0
1 �� rr , * 01 �� rs , * 01 �� sp , * 01 �� sq , * 01 �� sr , * 01 �� ss ] , \hbox{\axiomType{INT}\ })) , 16, [ p , q , r , s , pq , pr , ps , qp , qr , qs , rp , rq , rs , sp , sq , sr ]<a class=? , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ]?)" title=" \label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 ms , * 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� ps , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� qs , * 01 �� rp , * 01 �� rq , * 0 1 �� rr , * 01 �� rs , * 01 �� sp , * 01 �� sq , * 01 �� sr , * 01 �� ss ] , \hbox{\axiomType{INT}\ })) , 16, [ p , q , r , s , pq , pr , ps , qp , qr , qs , rp , rq , rs , sp , sq , sr ]? , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ]?)" class="equation" src="images/9151146123860165062-16.0px.png" align="bottom" Style="vertical-align:text-bottom" width="794" height="18"/>(11)
Type: TYPE
fricas
alternative?()$A
algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

\label{eq12} \mbox{\rm true} (12)
Type: BOOLEAN
fricas
antiAssociative?()$A
algebra is not anti-associative

\label{eq13} \mbox{\rm false} (13)
Type: BOOLEAN
fricas
antiCommutative?()$A
algebra is not anti-commutative

\label{eq14} \mbox{\rm false} (14)
Type: BOOLEAN
fricas
associative?()$A
algebra is associative

\label{eq15} \mbox{\rm true} (15)
Type: BOOLEAN
fricas
commutative?()$A
algebra is not commutative

\label{eq16} \mbox{\rm false} (16)
Type: BOOLEAN
fricas
flexible?()$A
algebra is flexible

\label{eq17} \mbox{\rm true} (17)
Type: BOOLEAN
fricas
jacobiIdentity?()$A
Jacobi identity does not hold

\label{eq18} \mbox{\rm false} (18)
Type: BOOLEAN
fricas
jordanAdmissible?()$A
algebra is not Jordan admissible

\label{eq19} \mbox{\rm false} (19)
Type: BOOLEAN
fricas
jordanAlgebra?()$A
algebra is not commutative this is not a Jordan algebra

\label{eq20} \mbox{\rm false} (20)
Type: BOOLEAN
fricas
leftAlternative?()$A
algebra is left alternative

\label{eq21} \mbox{\rm true} (21)
Type: BOOLEAN
fricas
lieAdmissible?()$A
algebra is Lie admissible

\label{eq22} \mbox{\rm true} (22)
Type: BOOLEAN
fricas
lieAlgebra?()$A
algebra is not anti-commutative this is not a Lie algebra

\label{eq23} \mbox{\rm false} (23)
Type: BOOLEAN
fricas
powerAssociative?()$A
Internal Error The function powerAssociative? with signature hashcode is missing from domain AlgebraGivenByStructuralConstants (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01m r) (*01m s) (*01γ pp) (*01γ pq) (*01γ pr) (*01γ ps) (*01γ qp) (*01γ qq) (*01γ qr) (*01γ qs) (*01γ rp) (*01γ rq) (*01γ rr) (*01γ rs) (*01γ sp) (*01γ sq) (*01γ sr) (*01γ ss)) (Integer))) 16(p q r s pq pr ps qp qr qs rp rq rs sp sq sr)UNPRINTABLE

Check Multiplication

fricas
AB := entries basis()$A

\label{eq24}\left[ p , \: q , \: r , \: s , \: pq , \: pr , \: ps , \: qp , \: qr , \: qs , \: rp , \: rq , \: rs , \: sp , \: sq , \: sr \right](24)
Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16,[p,q,r,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]))
fricas
A2MK(z:A):MK==reduce(+,map((x:R,y:M):MK+->(x::K)*y,coordinates(z),basis))
Function declaration A2MK : ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01ms, *01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp, *01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16,[p,q,r ,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,MATRIX, MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX, MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR( [p,q,r,s]))) has been added to workspace.
Type: VOID
fricas
test(MT=map(x+->map(A2MK,x),[[i*j for j in AB] for i in AB]))
fricas
Compiling function A2MK with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,
      *01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,
      *01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16
      ,[p,q,r,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),
      FMONOID(OVAR([p,q,r,s])))

\label{eq25} \mbox{\rm true} (25)
Type: BOOLEAN

Trace

fricas
[rightTrace(i)$A for i in AB]

\label{eq26}\begin{array}{@{}l}
\displaystyle
\left[{4 \ {m_{p}}\ {��_{pp}}}, \:{4 \ {m_{q}}\ {��_{qq}}}, \:{4 \ {m_{r}}\ {��_{rr}}}, \:{4 \ {m_{s}}\ {��_{ss}}}, \:{4 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \:{4 \ {m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \:{4 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{4 \ {m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}, \:{4 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{4 \ {m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}, \:{4 \ {m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}, \:{4 \ {m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}\right] 
(26)
Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)))
fricas
[leftTrace(i)$A for i in AB]

\label{eq27}\begin{array}{@{}l}
\displaystyle
\left[{4 \ {m_{p}}\ {��_{pp}}}, \:{4 \ {m_{q}}\ {��_{qq}}}, \:{4 \ {m_{r}}\ {��_{rr}}}, \:{4 \ {m_{s}}\ {��_{ss}}}, \:{4 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \:{4 \ {m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \:{4 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{4 \ {m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}, \:{4 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{4 \ {m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}, \:{4 \ {m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \: \right.
\
\
\displaystyle
\left.{4 \ {m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}, \:{4 \ {m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}\right] 
(27)
Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)))
fricas
trace(i)==rightTrace(i) / #gens
Type: VOID
fricas
[trace(i) for i in AB]
fricas
Compiling function trace with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr
      ,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,
      *01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16
      ,[p,q,r,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FRAC(DMP([*01mp,*01mq,
      *01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γ
      qs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT))

\label{eq28}\begin{array}{@{}l}
\displaystyle
\left[{{m_{p}}\ {��_{pp}}}, \:{{m_{q}}\ {��_{qq}}}, \:{{m_{r}}\ {��_{rr}}}, \:{{m_{s}}\ {��_{ss}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right.
\
\
\displaystyle
\left.{{m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{{m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}, \: \right.
\
\
\displaystyle
\left.{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \:{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{{m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}, \:{{m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}, \: \right.
\
\
\displaystyle
\left.{{m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}, \:{{m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}\right] 
(28)
Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)))

Center

fricas
C:=basisOfCenter()$AlgebraPackage(R,A); # C

\label{eq29}1(29)
Type: PI
fricas
c:=C(1)

\label{eq30}\begin{array}{@{}l}
\displaystyle
sr +{{{\left(
\begin{array}{@{}l}
\displaystyle
-{{m_{r}}\ {��_{pp}}\ {��_{rq}}\ {{��_{sr}}^{2}}}+{{m_{r}}\ {��_{pp}}\ {��_{rr}}\ {��_{sq}}\ {��_{sr}}}+ 
\
\
\displaystyle
{{m_{r}}\ {��_{pq}}\ {��_{rp}}\ {{��_{sr}}^{2}}}-{{m_{r}}\ {��_{pq}}\ {��_{rr}}\ {��_{sp}}\ {��_{sr}}}- 
\
\
\displaystyle
{{m_{r}}\ {��_{pr}}\ {��_{rp}}\ {��_{sq}}\ {��_{sr}}}+{{m_{r}}\ {��_{pr}}\ {��_{rq}}\ {��_{sp}}\ {��_{sr}}}
(30)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16,[p,q,r,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX])
fricas
[c*i-i*c for i in AB]

\label{eq31}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right](31)
Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16,[p,q,r,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]))
fricas
test(c*c=c)

\label{eq32} \mbox{\rm false} (32)
Type: BOOLEAN

Unit

fricas
rightTrace(c)

\label{eq33}{\left(
\begin{array}{@{}l}
\displaystyle
-{{16}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{qq}}\ {��_{rr}}\ {��_{sr}}\ {��_{ss}}}+ 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{qq}}\ {��_{rs}}\ {{��_{sr}}^{2}}}+{{16}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{qr}}\ {��_{rq}}\ {��_{sr}}\ {��_{ss}}}- 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}\ {��_{sr}}}-{{16}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{qs}}\ {��_{rq}}\ {{��_{sr}}^{2}}}+ 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{qs}}\ {��_{rr}}\ {��_{sq}}\ {��_{sr}}}+{{16}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qp}}\ {��_{rr}}\ {��_{sr}}\ {��_{ss}}}- 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qp}}\ {��_{rs}}\ {{��_{sr}}^{2}}}-{{16}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{sr}}\ {��_{ss}}}+ 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sp}}\ {��_{sr}}}+{{16}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{rp}}\ {{��_{sr}}^{2}}}- 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{rr}}\ {��_{sp}}\ {��_{sr}}}-{{16}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{sr}}\ {��_{ss}}}+ 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qp}}\ {��_{rs}}\ {��_{sq}}\ {��_{sr}}}+{{16}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qq}}\ {��_{rp}}\ {��_{sr}}\ {��_{ss}}}- 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qq}}\ {��_{rs}}\ {��_{sp}}\ {��_{sr}}}-{{16}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qs}}\ {��_{rp}}\ {��_{sq}}\ {��_{sr}}}+ 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sp}}\ {��_{sr}}}+{{16}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{rq}}\ {{��_{sr}}^{2}}}- 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{rr}}\ {��_{sq}}\ {��_{sr}}}-{{16}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qq}}\ {��_{rp}}\ {{��_{sr}}^{2}}}+ 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qq}}\ {��_{rr}}\ {��_{sp}}\ {��_{sr}}}+{{16}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qr}}\ {��_{rp}}\ {��_{sq}}\ {��_{sr}}}- 
\
\
\displaystyle
{{16}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qr}}\ {��_{rq}}\ {��_{sp}}\ {��_{sr}}}
(33)
Type: FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT))
fricas
n := #basis / rightTrace(c) * c

\label{eq34}\begin{array}{@{}l}
\displaystyle
{{{\left(
\begin{array}{@{}l}
\displaystyle
-{{��_{pp}}\ {��_{qq}}\ {��_{sr}}}+{{��_{pp}}\ {��_{qr}}\ {��_{sq}}}+{{��_{pq}}\ {��_{qp}}\ {��_{sr}}}-{{��_{pq}}\ {��_{qr}}\ {��_{sp}}}- 
\
\
\displaystyle
{{��_{pr}}\ {��_{qp}}\ {��_{sq}}}+{{��_{pr}}\ {��_{qq}}\ {��_{sp}}}
(34)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)),16,[p,q,r,s,pq,pr,ps,qp,qr,qs,rp,rq,rs,sp,sq,sr],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX])
fricas
trace(n)

\label{eq35}4(35)
Type: FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT))
fricas
test(n*n=n)

\label{eq36} \mbox{\rm true} (36)
Type: BOOLEAN

fricas
test(n=unit()$A)

\label{eq37} \mbox{\rm true} (37)
Type: BOOLEAN
fricas
f:=gcd map(x+->denom x,coordinates(n))

\label{eq38}\begin{array}{@{}l}
\displaystyle
{{��_{pp}}\ {��_{qq}}\ {��_{rr}}\ {��_{ss}}}-{{��_{pp}}\ {��_{qq}}\ {��_{rs}}\ {��_{sr}}}-{{��_{pp}}\ {��_{qr}}\ {��_{rq}}\ {��_{ss}}}+{{��_{pp}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}+ 
\
\
\displaystyle
{{��_{pp}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}-{{��_{pp}}\ {��_{qs}}\ {��_{rr}}\ {��_{sq}}}-{{��_{pq}}\ {��_{qp}}\ {��_{rr}}\ {��_{ss}}}+{{��_{pq}}\ {��_{qp}}\ {��_{rs}}\ {��_{sr}}}+ 
\
\
\displaystyle
{{��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{ss}}}-{{��_{pq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sp}}}-{{��_{pq}}\ {��_{qs}}\ {��_{rp}}\ {��_{sr}}}+{{��_{pq}}\ {��_{qs}}\ {��_{rr}}\ {��_{sp}}}+ 
\
\
\displaystyle
{{��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{ss}}}-{{��_{pr}}\ {��_{qp}}\ {��_{rs}}\ {��_{sq}}}-{{��_{pr}}\ {��_{qq}}\ {��_{rp}}\ {��_{ss}}}+{{��_{pr}}\ {��_{qq}}\ {��_{rs}}\ {��_{sp}}}+ 
\
\
\displaystyle
{{��_{pr}}\ {��_{qs}}\ {��_{rp}}\ {��_{sq}}}-{{��_{pr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sp}}}-{{��_{ps}}\ {��_{qp}}\ {��_{rq}}\ {��_{sr}}}+{{��_{ps}}\ {��_{qp}}\ {��_{rr}}\ {��_{sq}}}+ 
\
\
\displaystyle
{{��_{ps}}\ {��_{qq}}\ {��_{rp}}\ {��_{sr}}}-{{��_{ps}}\ {��_{qq}}\ {��_{rr}}\ {��_{sp}}}-{{��_{ps}}\ {��_{qr}}\ {��_{rp}}\ {��_{sq}}}+{{��_{ps}}\ {��_{qr}}\ {��_{rq}}\ {��_{sp}}}
(38)
Type: DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)
fricas
ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]

\label{eq39}\left[ 
\begin{array}{cccc}
{��_{pp}}&{��_{pq}}&{��_{pr}}&{��_{ps}}
\
{��_{qp}}&{��_{qq}}&{��_{qr}}&{��_{qs}}
\
{��_{rp}}&{��_{rq}}&{��_{rr}}&{��_{rs}}
\
{��_{sp}}&{��_{sq}}&{��_{sr}}&{��_{ss}}
(39)
Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)))
fricas
test(f = determinant(ff))

\label{eq40} \mbox{\rm true} (40)
Type: BOOLEAN
fricas
(f*n)::OutputForm / f::OutputForm

\label{eq41}{\left(
\begin{array}{@{}l}
\displaystyle
{{{\left(
\begin{array}{@{}l}
\displaystyle
-{{��_{pp}}\ {��_{qq}}\ {��_{sr}}}+{{��_{pp}}\ {��_{qr}}\ {��_{sq}}}+{{��_{pq}}\ {��_{qp}}\ {��_{sr}}}- 
\
\
\displaystyle
{{��_{pq}}\ {��_{qr}}\ {��_{sp}}}-{{��_{pr}}\ {��_{qp}}\ {��_{sq}}}+{{��_{pr}}\ {��_{qq}}\ {��_{sp}}}
(41)
Type: OUTFORM

Antisymmetric γ

fricas
eqAnti:List Equation EXPR INT := concat [[(i>j => γ(i,j)=γ(i,j);i=j =>γ(i,j)=0;γ(i,j)=-γ(j,i)) for j in gens] for i in gens]

\label{eq42}\begin{array}{@{}l}
\displaystyle
\left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{pr}}={��_{pr}}}, \:{{��_{ps}}={��_{ps}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}, \: \right.
\
\
\displaystyle
\left.{{��_{qr}}={��_{qr}}}, \:{{��_{qs}}={��_{qs}}}, \:{{��_{rp}}= -{��_{pr}}}, \:{{��_{rq}}= -{��_{qr}}}, \:{{��_{rr}}= 0}, \:{{��_{rs}}={��_{rs}}}, \: \right.
\
\
\displaystyle
\left.{{��_{sp}}= -{��_{ps}}}, \:{{��_{sq}}= -{��_{qs}}}, \:{{��_{sr}}= -{��_{rs}}}, \:{{��_{ss}}= 0}\right] 
(42)
Type: LIST(EQ(EXPR(INT)))
fricas
anti(x:R):R == subst(x::EXPR INT, eqAnti)::FRAC POLY INT
Function declaration anti : FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp ,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq, *01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)) -> FRAC(DMP([ *01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq ,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr, *01γss],INT)) has been added to workspace.
Type: VOID
fricas
(anti(f)*map(anti ,coordinates(n))::A)::OutputForm / anti(f)::OutputForm
fricas
Compiling function anti with type FRAC(DMP([*01mp,*01mq,*01mr,*01ms,
      *01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,
      *01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)) -> FRAC(
      DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,
      *01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,
      *01γsr,*01γss],INT))

\label{eq43}{\left(
\begin{array}{@{}l}
\displaystyle
{{{-{{{��_{pq}}^{2}}\ {��_{rs}}}+{{��_{pq}}\ {��_{pr}}\ {��_{qs}}}-{{��_{pq}}\ {��_{ps}}\ {��_{qr}}}}\over{{m_{r}}\ {m_{s}}\ {��_{rs}}}}\  sr}+ 
\
\
\displaystyle
{{{{{��_{pq}}\ {��_{pr}}\ {��_{rs}}}-{{{��_{pr}}^{2}}\ {��_{qs}}}+{{��_{pr}}\ {��_{ps}}\ {��_{qr}}}}\over{{m_{q}}\ {m_{s}}\ {��_{qs}}}}\  sq}+ 
\
\
\displaystyle
{{{-{{��_{pq}}\ {��_{qr}}\ {��_{rs}}}+{{��_{pr}}\ {��_{qr}}\ {��_{qs}}}-{{��_{ps}}\ {{��_{qr}}^{2}}}}\over{{m_{p}}\ {m_{s}}\ {��_{ps}}}}\  sp}+ 
\
\
\displaystyle
{{{-{{{��_{pq}}^{2}}\ {��_{rs}}}+{{��_{pq}}\ {��_{pr}}\ {��_{qs}}}-{{��_{pq}}\ {��_{ps}}\ {��_{qr}}}}\over{{m_{r}}\ {m_{s}}\ {��_{rs}}}}\  rs}+ 
\
\
\displaystyle
{{{-{{��_{pq}}\ {��_{ps}}\ {��_{rs}}}+{{��_{pr}}\ {��_{ps}}\ {��_{qs}}}-{{{��_{ps}}^{2}}\ {��_{qr}}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\  rq}+ 
\
\
\displaystyle
{{{{{��_{pq}}\ {��_{qs}}\ {��_{rs}}}-{{��_{pr}}\ {{��_{qs}}^{2}}}+{{��_{ps}}\ {��_{qr}}\ {��_{qs}}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\  rp}+ 
\
\
\displaystyle
{{{{{��_{pq}}\ {��_{pr}}\ {��_{rs}}}-{{{��_{pr}}^{2}}\ {��_{qs}}}+{{��_{pr}}\ {��_{ps}}\ {��_{qr}}}}\over{{m_{q}}\ {m_{s}}\ {��_{qs}}}}\  qs}+ 
\
\
\displaystyle
{{{-{{��_{pq}}\ {��_{ps}}\ {��_{rs}}}+{{��_{pr}}\ {��_{ps}}\ {��_{qs}}}-{{{��_{ps}}^{2}}\ {��_{qr}}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\  qr}+ 
\
\
\displaystyle
{{{-{{��_{pq}}\ {{��_{rs}}^{2}}}+{{��_{pr}}\ {��_{qs}}\ {��_{rs}}}-{{��_{ps}}\ {��_{qr}}\ {��_{rs}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\  qp}+ 
\
\
\displaystyle
{{{-{{��_{pq}}\ {��_{qr}}\ {��_{rs}}}+{{��_{pr}}\ {��_{qr}}\ {��_{qs}}}-{{��_{ps}}\ {{��_{qr}}^{2}}}}\over{{m_{p}}\ {m_{s}}\ {��_{ps}}}}\  ps}+ 
\
\
\displaystyle
{{{{{��_{pq}}\ {��_{qs}}\ {��_{rs}}}-{{��_{pr}}\ {{��_{qs}}^{2}}}+{{��_{ps}}\ {��_{qr}}\ {��_{qs}}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\  pr}+ 
\
\
\displaystyle
{{{-{{��_{pq}}\ {{��_{rs}}^{2}}}+{{��_{pr}}\ {��_{qs}}\ {��_{rs}}}-{{��_{ps}}\ {��_{qr}}\ {��_{rs}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\  pq}
(43)
Type: OUTFORM

Scalar Product

fricas
S := matrix [[trace(x*y) for y in AB] for x in AB]

\label{eq44}\left[ 
\begin{array}{cccccccccccccccc}
{{{m_{p}}^{2}}\ {{��_{pp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}
\
{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{q}}^{2}}\ {{��_{qq}}^{2}}}&{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {m_{s}}\ {��_{qq}}\ {��_{qs}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{{m_{q}}^{2}}\ {m_{s}}\ {��_{qq}}\ {��_{qs}}\ {��_{sq}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}
\
{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}&{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{r}}^{2}}\ {{��_{rr}}^{2}}}&{{m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{{m_{r}}^{2}}\ {m_{s}}\ {��_{rr}}\ {��_{rs}}\ {��_{sr}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}&{{{m_{r}}^{2}}\ {m_{s}}\ {��_{rr}}\ {��_{rs}}\ {��_{sr}}}
\
{{m_{p}}\ {m_{s}}\ {��_{ps}}\ {��_{sp}}}&{{m_{q}}\ {m_{s}}\ {��_{qs}}\ {��_{sq}}}&{{m_{r}}\ {m_{s}}\ {��_{rs}}\ {��_{sr}}}&{{{m_{s}}^{2}}\ {{��_{ss}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{p}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{sp}}\ {��_{ss}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}&{{m_{q}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{sq}}\ {��_{ss}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{m_{r}}\ {{m_{s}}^{2}}\ {��_{rs}}\ {��_{sr}}\ {��_{ss}}}&{{m_{p}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{sp}}\ {��_{ss}}}&{{m_{q}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{sq}}\ {��_{ss}}}&{{m_{r}}\ {{m_{s}}^{2}}\ {��_{rs}}\ {��_{sr}}\ {��_{ss}}}
\
{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{ps}}\ {��_{qp}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{s}}\ {��_{pq}}\ {��_{qq}}\ {��_{qs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pp}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{s}}\ {��_{pq}}\ {��_{qp}}\ {��_{qs}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{rp}}\ {��_{sr}}}
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{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{rp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{ps}}\ {��_{rp}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{pr}}\ {��_{rr}}\ {��_{rs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qp}}\ {��_{rs}}\ {��_{sq}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{pr}}\ {��_{rp}}\ {��_{rs}}\ {��_{sr}}}
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{{{m_{p}}^{2}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{m_{p}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{sp}}\ {��_{ss}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{ps}}\ {��_{qp}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{ps}}\ {��_{rp}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {{m_{s}}^{2}}\ {{��_{ps}}^{2}}\ {{��_{sp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qr}}\ {��_{rp}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{qs}}\ {��_{sp}}\ {��_{sq}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{rq}}\ {��_{sr}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{rs}}\ {��_{sp}}\ {��_{sr}}}&{{{m_{p}}^{2}}\ {{m_{s}}^{2}}\ {��_{pp}}\ {��_{ps}}\ {��_{sp}}\ {��_{ss}}}&{{m_{p}}\ {m_{q}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}\ {��_{ss}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}\ {��_{ss}}}
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{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{qp}}\ {��_{sq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{s}}\ {��_{pq}}\ {��_{qp}}\ {��_{qs}}\ {��_{sq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qp}}\ {��_{rs}}\ {��_{sq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{ps}}\ {��_{qp}}\ {��_{sp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{qq}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{rq}}\ {��_{sr}}}
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{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qr}}\ {��_{rp}}\ {��_{sq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rq}}^{2}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qr}}\ {��_{rr}}\ {��_{rs}}\ {��_{sq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{qq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qr}}\ {��_{rq}}\ {��_{rs}}\ {��_{sr}}}
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{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{{m_{q}}^{2}}\ {m_{s}}\ {��_{qq}}\ {��_{qs}}\ {��_{sq}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{m_{q}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{sq}}\ {��_{ss}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{s}}\ {��_{pq}}\ {��_{qq}}\ {��_{qs}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{qs}}\ {��_{sp}}\ {��_{sq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{s}}\ {��_{pq}}\ {��_{qp}}\ {��_{qs}}\ {��_{sq}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sq}}}&{{{m_{q}}^{2}}\ {{m_{s}}^{2}}\ {{��_{qs}}^{2}}\ {{��_{sq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{rp}}\ {��_{sr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{qq}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{m_{q}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{rs}}\ {��_{sq}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {{m_{s}}^{2}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}\ {��_{ss}}}&{{{m_{q}}^{2}}\ {{m_{s}}^{2}}\ {��_{qq}}\ {��_{qs}}\ {��_{sq}}\ {��_{ss}}}&{{m_{q}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}\ {��_{ss}}}
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{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{rp}}\ {��_{sr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{rp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{pr}}\ {��_{rp}}\ {��_{rs}}\ {��_{sr}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{ps}}\ {��_{rp}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qr}}\ {��_{rp}}\ {��_{sq}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{rr}}\ {��_{sr}}}
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{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{rq}}\ {��_{sr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{qq}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rq}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qr}}\ {��_{rq}}\ {��_{rs}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{rr}}\ {��_{sr}}}
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{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}}&{{{m_{r}}^{2}}\ {m_{s}}\ {��_{rr}}\ {��_{rs}}\ {��_{sr}}}&{{m_{r}}\ {{m_{s}}^{2}}\ {��_{rs}}\ {��_{sr}}\ {��_{ss}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{pr}}\ {��_{rr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{rs}}\ {��_{sp}}\ {��_{sr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qp}}\ {��_{rs}}\ {��_{sq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qr}}\ {��_{rr}}\ {��_{rs}}\ {��_{sq}}}&{{m_{q}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{rs}}\ {��_{sq}}\ {��_{sr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{pr}}\ {��_{rp}}\ {��_{rs}}\ {��_{sr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qr}}\ {��_{rq}}\ {��_{rs}}\ {��_{sr}}}&{{{m_{r}}^{2}}\ {{m_{s}}^{2}}\ {{��_{rs}}^{2}}\ {{��_{sr}}^{2}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}\ {��_{ss}}}&{{m_{q}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{qr}}\ {��_{rs}}\ {��_{sq}}\ {��_{ss}}}&{{{m_{r}}^{2}}\ {{m_{s}}^{2}}\ {��_{rr}}\ {��_{rs}}\ {��_{sr}}\ {��_{ss}}}
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{{{m_{p}}^{2}}\ {m_{s}}\ {��_{pp}}\ {��_{ps}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{p}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{sp}}\ {��_{ss}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pp}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pp}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}}&{{{m_{p}}^{2}}\ {{m_{s}}^{2}}\ {��_{pp}}\ {��_{ps}}\ {��_{sp}}\ {��_{ss}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{s}}\ {��_{pq}}\ {��_{ps}}\ {��_{qp}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qr}}\ {��_{rs}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {{m_{s}}^{2}}\ {��_{pq}}\ {��_{qs}}\ {��_{sp}}\ {��_{ss}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{ps}}\ {��_{rp}}\ {��_{sp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pr}}\ {��_{qs}}\ {��_{rq}}\ {��_{sp}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{pr}}\ {��_{rs}}\ {��_{sp}}\ {��_{ss}}}&{{{m_{p}}^{2}}\ {{m_{s}}^{2}}\ {{��_{ps}}^{2}}\ {{��_{sp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{qs}}\ {��_{sp}}\ {��_{sq}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{rs}}\ {��_{sp}}\ {��_{sr}}}
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{{m_{p}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}}&{{m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}}&{{{m_{r}}^{2}}\ {m_{s}}\ {��_{rr}}\ {��_{rs}}\ {��_{sr}}}&{{m_{r}}\ {{m_{s}}^{2}}\ {��_{rs}}\ {��_{sr}}\ {��_{ss}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{pq}}\ {��_{qs}}\ {��_{rp}}\ {��_{sr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{pr}}\ {��_{rp}}\ {��_{rs}}\ {��_{sr}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{rp}}\ {��_{sr}}\ {��_{ss}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {m_{s}}\ {��_{ps}}\ {��_{qp}}\ {��_{rq}}\ {��_{sr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qr}}\ {��_{rq}}\ {��_{rs}}\ {��_{sr}}}&{{m_{q}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{rq}}\ {��_{sr}}\ {��_{ss}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{ps}}\ {��_{rp}}\ {��_{rr}}\ {��_{sr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {m_{s}}\ {��_{qs}}\ {��_{rq}}\ {��_{rr}}\ {��_{sr}}}&{{{m_{r}}^{2}}\ {{m_{s}}^{2}}\ {��_{rr}}\ {��_{rs}}\ {��_{sr}}\ {��_{ss}}}&{{m_{p}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{ps}}\ {��_{rs}}\ {��_{sp}}\ {��_{sr}}}&{{m_{q}}\ {m_{r}}\ {{m_{s}}^{2}}\ {��_{qs}}\ {��_{rs}}\ {��_{sq}}\ {��_{sr}}}&{{{m_{r}}^{2}}\ {{m_{s}}^{2}}\ {{��_{rs}}^{2}}\ {{��_{sr}}^{2}}}
(44)
Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01ms,*01γpp,*01γpq,*01γpr,*01γps,*01γqp,*01γqq,*01γqr,*01γqs,*01γrp,*01γrq,*01γrr,*01γrs,*01γsp,*01γsq,*01γsr,*01γss],INT)))




  Subject:   Be Bold !!
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