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SandBoxObserverAsIdempotent

Edit detail for SandBoxObserverAsIdempotent revision 3 of 21

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Editor: Bill Page Time: 2014/01/24 17:34:41 GMT+0
Note:

added:
-- idempotent
--MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3)),m(gens(1))=1,m(gens(2))=1,m(gens(3))=1]),monomial(mod(i*j)))$MK for j in basis] for i in basis]
-- nilpotent
--MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,m(gens(1))=1,m(gens(2))=1,m(gens(3))=1]),monomial(mod(i*j)))$MK for j in basis] for i in basis]

Obs(3) is a 9 dimensional Frobenius Algrebra

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)set output abbreviate on

V := OrderedVariableList [p,q,r]

$\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q , r ])$

(1)

Type: TYPE

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M := FreeMonoid V

$\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ]))$

(2)

Type: TYPE

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gens:List M := enumerate()$V

$\label{eq3}\left[ p , \: q , \: r \right]$

(3)

Type: LIST(FMONOID(OVAR([p,q,r])))

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divisible := Record(lm: M,rm: M)

$\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))}$

(4)

Type: TYPE

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leftDiv(k:Union(divisible,"failed")):M == (k::divisible).lm

   Function declaration leftDiv : Union(Record(lm: FMONOID(OVAR([p,q,r]
      )),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r])
      ) has been added to workspace.

Type: VOID

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rightDiv(k:Union(divisible,"failed")):M == (k::divisible).rm

   Function declaration rightDiv : Union(Record(lm: FMONOID(OVAR([p,q,r
      ])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r]
      )) has been added to workspace.

Type: VOID

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K := FRAC POLY INT

$\label{eq5}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ }))$

(5)

Type: TYPE

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MK := FreeModule(K,M)

$\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))$

(6)

Type: TYPE

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coeff(x:MK):K == leadingCoefficient(x)

   Function declaration coeff : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FRAC(POLY(INT)) has been added to workspace.

Type: VOID

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monomial(x:MK):M == leadingMonomial(x)

   Function declaration monomial : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q
      ,r]))) -> FMONOID(OVAR([p,q,r])) has been added to workspace.

Type: VOID

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m(x:M):K == subscript('m,[retract(x)::Symbol])

   Function declaration m : FMONOID(OVAR([p,q,r])) -> FRAC(POLY(INT)) 
      has been added to workspace.

Type: VOID

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γ(x:M,y:M):K == subscript('γ,[concat(string retract x, string retract y)::Symbol])

   Function declaration γ : (FMONOID(OVAR([p,q,r])),FMONOID(OVAR([p,q,r
      ]))) -> FRAC(POLY(INT)) has been added to workspace.

Type: VOID

Basis

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

$\label{eq7}\left[ p , \: q , \: r , \:{p \ q}, \:{p \ r}, \:{q \ p}, \:{q \ r}, \:{r \ p}, \:{r \ q}\right]$

(7)

Type: LIST(FMONOID(OVAR([p,q,r])))

Idempotent

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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]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Type: VOID

Reduction

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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]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Type: VOID

Modulo fixed point of applied rules

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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]))
      ) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added to
      workspace.

Type: VOID

Matrix

Multiplication is monoidal concatenation modulo the fixed point

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MT := [[mod(i*j) for j in basis] for i in basis]

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Compiling function monomial with type FM(FRAC(POLY(INT)),FMONOID(
      OVAR([p,q,r]))) -> FMONOID(OVAR([p,q,r]))

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Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FRAC(POLY(INT))

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Compiling function m with type FMONOID(OVAR([p,q,r])) -> FRAC(POLY(
      INT))

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Compiling function γ with type (FMONOID(OVAR([p,q,r])),FMONOID(OVAR(
      [p,q,r]))) -> FRAC(POLY(INT))

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Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
      p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p
      ,q,r]))

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Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
      [p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([
      p,q,r]))

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Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

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Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

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Compiling function mod with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {��_{pp}}\ p}, \:{p \ q}, \:{p \ r}, \:{{m_{p}}\ {��_{pp}}\ p \ q}, \:{{m_{p}}\ {��_{pp}}\ p \ r}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\ p}, \: \right. \ \ \displaystyle \left.{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ p \ r}, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}\ p}, \:{{{{m_{r}}\ {��_{pr}}\ {��_{rq}}}\over{��_{pq}}}\ p \ q}\right]$

(8)

Type: LIST(LIST(FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))))

Structure Constants

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

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 0 1 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ }))$

(9)

Type: TYPE

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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])) -> LIST(LIST(FRAC
      (DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr
      ,*01γrp,*01γrq,*01γrr],INT)))) has been added to workspace.

Type: VOID

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ss:=map(mat3, basis)

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Compiling function mat3 with type FMONOID(OVAR([p,q,r])) -> LIST(
      LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT))))

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\left[{{m_{p}}\ {��_{pp}}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: 0, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}, \: 0, \: \right. \ \ \displaystyle \left.{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}, \: 0 \right]$

(10)

Type: LIST(LIST(LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))))

Algebra

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cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol

   Function declaration cats : FMONOID(OVAR([p,q,r])) -> SYMBOL has 
      been added to workspace.

Type: VOID

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A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI,map(cats,basis),ss::Vector(Matrix R))

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Compiling function cats with type FMONOID(OVAR([p,q,r])) -> SYMBOL

$\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 01 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ })) , 9, [ p , q , r , pq , pr , qp , qr , rp , rq ] , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ])$

(11)

Type: TYPE

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alternative?()$A

   algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

$\label{eq12} \mbox{\rm true}$

(12)

Type: BOOLEAN

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antiAssociative?()$A

   algebra is not anti-associative

$\label{eq13} \mbox{\rm false}$

(13)

Type: BOOLEAN

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antiCommutative?()$A

   algebra is not anti-commutative

$\label{eq14} \mbox{\rm false}$

(14)

Type: BOOLEAN

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associative?()$A

   algebra is associative

$\label{eq15} \mbox{\rm true}$

(15)

Type: BOOLEAN

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commutative?()$A

   algebra is not commutative

$\label{eq16} \mbox{\rm false}$

(16)

Type: BOOLEAN

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flexible?()$A

   algebra is flexible

$\label{eq17} \mbox{\rm true}$

(17)

Type: BOOLEAN

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jacobiIdentity?()$A

   Jacobi identity does not hold

$\label{eq18} \mbox{\rm false}$

(18)

Type: BOOLEAN

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jordanAdmissible?()$A

   algebra is not Jordan admissible

$\label{eq19} \mbox{\rm false}$

(19)

Type: BOOLEAN

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jordanAlgebra?()$A

   algebra is not commutative
   this is not a Jordan algebra

$\label{eq20} \mbox{\rm false}$

(20)

Type: BOOLEAN

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leftAlternative?()$A

   algebra is left alternative

$\label{eq21} \mbox{\rm true}$

(21)

Type: BOOLEAN

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lieAdmissible?()$A

   algebra is Lie admissible

$\label{eq22} \mbox{\rm true}$

(22)

Type: BOOLEAN

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lieAlgebra?()$A

   algebra is not anti-commutative
   this is not a Lie algebra

$\label{eq23} \mbox{\rm false}$

(23)

Type: BOOLEAN

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powerAssociative?()$A

   Internal Error
   The function powerAssociative? with signature hashcode is missing 
      from domain AlgebraGivenByStructuralConstants
      (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01m r) (*01γ pp) (*01γ pq) (*01γ pr) (*01γ qp) (*01γ qq) (*01γ qr) (*01γ rp) (*01γ rq) (*01γ rr)) (Integer)))
      9(p q r pq pr qp qr rp rq)UNPRINTABLE

Check Multiplication

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AB := entries basis()$A

$\label{eq24}\left[ p , \: q , \: r , \: pq , \: pr , \: qp , \: qr , \: rp , \: rq \right]$

(24)

Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[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,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9
      ,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR(
      [p,q,r]))) 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,
      *01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),
      FMONOID(OVAR([p,q,r])))

$\label{eq25} \mbox{\rm true}$

(25)

Type: BOOLEAN

Trace

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

$\label{eq26}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}\ {��_{pp}}}, \:{3 \ {m_{q}}\ {��_{qq}}}, \:{3 \ {m_{r}}\ {��_{rr}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}\right]$

(26)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

[leftTrace(i)$A for i in AB]

$\label{eq27}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}\ {��_{pp}}}, \:{3 \ {m_{q}}\ {��_{qq}}}, \:{3 \ {m_{r}}\ {��_{rr}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}\right]$

(27)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],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
      ,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FRAC(DMP([*01mp,*01mq,
      *01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,
      *01γrr],INT))

$\label{eq28}\begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {��_{pp}}}, \:{{m_{q}}\ {��_{qq}}}, \:{{m_{r}}\ {��_{rr}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \: \right. \ \ \displaystyle \left.{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}, \:{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}, \:{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}\right]$

(28)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

Scalar Product

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.1,AB.2,AB.3]]

$\label{eq29}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {{��_{pp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}} \ {{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{q}}^{2}}\ {{��_{qq}}^{2}}}&{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}} \ {{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}&{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{r}}^{2}}\ {{��_{rr}}^{2}}}$

(29)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.4,AB.5,AB.6]]

$\label{eq30}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}$

(30)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.7,AB.8,AB.9]]

$\label{eq31}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}$

(31)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.1,AB.2,AB.3]]

$\label{eq32}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}$

(32)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.4,AB.5,AB.6]]

$\label{eq33}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{rp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}} \ {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}$

(33)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.7,AB.8,AB.9]]

$\label{eq34}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}}$

(34)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.1,AB.2,AB.3]]

$\label{eq35}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}} \ {{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}} \ {{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}$

(35)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.4,AB.5,AB.6]]

$\label{eq36}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}}$

(36)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.7,AB.8,AB.9]]

$\label{eq37}\left[ \begin{array}{ccc} {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{rp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}} \ {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rq}}^{2}}}$

(37)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

Center

fricas

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

$\label{eq38}1$

(38)

Type: PI

fricas

c:=C(1)

$\label{eq39}\begin{array}{@{}l} \displaystyle rq +{{{-{{m_{q}}\ {��_{qp}}\ {{��_{rq}}^{2}}}+{{m_{q}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}}}\over{{{m_{p}}\ {��_{pp}}\ {��_{rp}}\ {��_{rq}}}-{{m_{p}}\ {��_{pq}}\ {{��_{rp}}^{2}}}}}\ rp}+ \ \ \displaystyle {{{{{��_{pp}}\ {��_{qr}}\ {��_{rq}}}-{{��_{pr}}\ {��_{qp}}\ {��_{rq}}}}\over{{{��_{pp}}\ {��_{qr}}\ {��_{rq}}}-{{��_{pq}}\ {��_{qr}}\ {��_{rp}}}}}\ qr}+{{{{{m_{r}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}}-{{m_{r}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}}\over{{{m_{p}}\ {��_{pp}}\ {��_{qp}}\ {��_{rq}}}-{{m_{p}}\ {��_{pq}}\ {��_{qp}}\ {��_{rp}}}}}\ qp}+ \ \ \displaystyle {{{-{{m_{q}}\ {��_{pq}}\ {��_{qr}}\ {��_{rq}}}+{{m_{q}}\ {��_{pr}}\ {��_{qq}}\ {��_{rq}}}}\over{{{m_{p}}\ {��_{pp}}\ {��_{pr}}\ {��_{rq}}}-{{m_{p}}\ {��_{pq}}\ {��_{pr}}\ {��_{rp}}}}}\ pr}+ \ \ \displaystyle {{{{{m_{r}}\ {��_{pq}}\ {��_{rq}}\ {��_{rr}}}-{{m_{r}}\ {��_{pr}}\ {{��_{rq}}^{2}}}}\over{{{m_{p}}\ {��_{pp}}\ {��_{pq}}\ {��_{rq}}}-{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{rp}}}}}\ pq}+ \ \ \displaystyle {{{-{{m_{q}}\ {��_{pp}}\ {��_{qq}}\ {��_{rq}}}+{{m_{q}}\ {��_{pq}}\ {��_{qp}}\ {��_{rq}}}}\over{{{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}}}\ r}+ \ \ \displaystyle {{{-{{m_{r}}\ {��_{pp}}\ {��_{rq}}\ {��_{rr}}}+{{m_{r}}\ {��_{pr}}\ {��_{rp}}\ {��_{rq}}}}\over{{{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}}}\ q}+ \ \ \displaystyle {{{-{{m_{q}}\ {m_{r}}\ {��_{qq}}\ {��_{rq}}\ {��_{rr}}}+{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {{��_{rq}}^{2}}}}\over{{{m_{p}}\ {��_{pp}}\ {��_{rq}}}-{{m_{p}}\ {��_{pq}}\ {��_{rp}}}}}\ p}$

(39)

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX])

fricas

[c*i-i*c for i in AB]

$\label{eq40}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(40)

fricas

test(c*c=c)

$\label{eq41} \mbox{\rm false}$

(41)

Type: BOOLEAN

Unit

fricas

rightTrace(c)

$\label{eq42}{\left( \begin{array}{@{}l} \displaystyle -{9 \ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qq}}\ {��_{rq}}\ {��_{rr}}}+{9 \ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qr}}\ {{��_{rq}}^{2}}}+ \ \ \displaystyle {9 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}}-{9 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}- \ \ \displaystyle {9 \ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {{��_{rq}}^{2}}}+{9 \ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}}$

(42)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

n := #basis / rightTrace(c) * c

$\label{eq43}\begin{array}{@{}l} \displaystyle {{{-{{��_{pp}}\ {��_{rq}}}+{{��_{pq}}\ {��_{rp}}}}\over{\left( \begin{array}{@{}l} \displaystyle {{m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qq}}\ {��_{rq}}\ {��_{rr}}}-{{m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{qr}}\ {{��_{rq}}^{2}}}- \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}}+{{m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {{��_{rq}}^{2}}}-{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qq}}\ {��_{rp}}\ {��_{rq}}}$

(43)

fricas

trace(n)

$\label{eq44}3$

(44)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

test(n*n=n)

$\label{eq45} \mbox{\rm true}$

(45)

Type: BOOLEAN

fricas

[n*i-i for i in AB]

$\label{eq46}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(46)

fricas

[i*n-i for i in AB]

$\label{eq47}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(47)

fricas

test(n=unit()$A)

$\label{eq48} \mbox{\rm true}$

(48)

Type: BOOLEAN

fricas

f:=gcd map(x+->denom x,coordinates(n))

$\label{eq49}{{��_{pp}}\ {��_{qq}}\ {��_{rr}}}-{{��_{pp}}\ {��_{qr}}\ {��_{rq}}}-{{��_{pq}}\ {��_{qp}}\ {��_{rr}}}+{{��_{pq}}\ {��_{qr}}\ {��_{rp}}}+{{��_{pr}}\ {��_{qp}}\ {��_{rq}}}-{{��_{pr}}\ {��_{qq}}\ {��_{rp}}}$

(49)

Type: DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)

fricas

ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]

$\label{eq50}\left[ \begin{array}{ccc} {��_{pp}}&{��_{pq}}&{��_{pr}} \ {��_{qp}}&{��_{qq}}&{��_{qr}} \ {��_{rp}}&{��_{rq}}&{��_{rr}}$

(50)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

test(f = determinant(ff))

$\label{eq51} \mbox{\rm true}$

(51)

Type: BOOLEAN

fricas

(f*n)::OutputForm / f::OutputForm

$\label{eq52}{\left( \begin{array}{@{}l} \displaystyle {{{-{{��_{pp}}\ {��_{rq}}}+{{��_{pq}}\ {��_{rp}}}}\over{{m_{q}}\ {m_{r}}\ {��_{rq}}}}\ rq}+{{{{{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}}\over{{m_{p}}\ {m_{r}}\ {��_{rp}}}}\ rp}+ \ \ \displaystyle {{{-{{��_{pp}}\ {��_{qr}}}+{{��_{pr}}\ {��_{qp}}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{{-{{��_{qp}}\ {��_{rr}}}+{{��_{qr}}\ {��_{rp}}}}\over{{m_{p}}\ {m_{q}}\ {��_{qp}}}}\ qp}+ \ \ \displaystyle {{{{{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+{{{-{{��_{pq}}\ {��_{rr}}}+{{��_{pr}}\ {��_{rq}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+ \ \ \displaystyle {{{{{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}}\over{m_{r}}}\ r}+{{{{{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}}\over{m_{q}}}\ q}+{{{{{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}}\over{m_{p}}}\ p}$

(52)

Type: OUTFORM

Symmetric γ

fricas

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

$\label{eq53}\begin{array}{@{}l} \displaystyle \left[{{��_{pp}}={��_{pp}}}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{pr}}={��_{pr}}}, \:{{��_{qp}}={��_{pq}}}, \:{{��_{qq}}={��_{qq}}}, \:{{��_{qr}}={��_{qr}}}, \:{{��_{rp}}={��_{pr}}}, \right. \ \ \displaystyle \left.\:{{��_{rq}}={��_{qr}}}, \:{{��_{rr}}={��_{rr}}}\right]$

(53)

Type: LIST(EQ(EXPR(INT)))

fricas

symm(x:R):R == subst(x::EXPR INT, eqSymm)::FRAC POLY INT

   Function declaration symm : FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γ
      pq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)) -> 
      FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,
      *01γqr,*01γrp,*01γrq,*01γrr],INT)) has been added to workspace.

Type: VOID

fricas

(symm(f)*map(symm ,coordinates(n))::A)::OutputForm / symm(f)::OutputForm

fricas

Compiling function symm with type FRAC(DMP([*01mp,*01mq,*01mr,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))
       -> FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

$\label{eq54}{\left( \begin{array}{@{}l} \displaystyle {{{-{{��_{pp}}\ {��_{qr}}}+{{��_{pq}}\ {��_{pr}}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{{{{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{{-{{��_{pp}}\ {��_{qr}}}+{{��_{pq}}\ {��_{pr}}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{{-{{��_{pq}}\ {��_{rr}}}+{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+ \ \ \displaystyle {{{{{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+{{{-{{��_{pq}}\ {��_{rr}}}+{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+ \ \ \displaystyle {{{{{��_{pp}}\ {��_{qq}}}-{{��_{pq}}^{2}}}\over{m_{r}}}\ r}+{{{{{��_{pp}}\ {��_{rr}}}-{{��_{pr}}^{2}}}\over{m_{q}}}\ q}+{{{{{��_{qq}}\ {��_{rr}}}-{{��_{qr}}^{2}}}\over{m_{p}}}\ p}$

(54)

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]

$\label{eq55}\begin{array}{@{}l} \displaystyle \left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{pr}}={��_{pr}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}, \:{{��_{qr}}={��_{qr}}}, \: \right. \ \ \displaystyle \left.{{��_{rp}}= -{��_{pr}}}, \:{{��_{rq}}= -{��_{qr}}}, \:{{��_{rr}}= 0}\right]$

(55)

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,*01γpp,*01γ
      pq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)) -> 
      FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,
      *01γqr,*01γrp,*01γrq,*01γrr],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,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))
       -> FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT)) 

   >> Error detected within library code:
   catdef: division by zero

Momentum

fricas

P:=reduce(+,concat [[(1/γ(basis(i),basis(j)))::R*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V])

$\label{eq56}\begin{array}{@{}l} \displaystyle {{1 \over{��_{rq}}}\ rq}+{{1 \over{��_{rp}}}\ rp}+{{1 \over{��_{qr}}}\ qr}+{{1 \over{��_{qp}}}\ qp}+{{1 \over{��_{pr}}}\ pr}+{{1 \over{��_{pq}}}\ pq}+{{m_{r}}\ r}+ \ \ \displaystyle {{m_{q}}\ q}+{{m_{p}}\ p}$

(56)

fricas

trace(P)

$\label{eq57}\begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}}+{{m_{p}}\ {m_{r}}\ {��_{rp}}}+ \ \ \displaystyle {{{m_{q}}^{2}}\ {��_{qq}}}+{{m_{q}}\ {m_{r}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {��_{rq}}}+{{{m_{r}}^{2}}\ {��_{rr}}}$

(57)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

c:=1/trace(P)*P

$\label{eq58}\begin{array}{@{}l} \displaystyle {{1 \over{\left( \begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {��_{pp}}\ {��_{rq}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{rq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}\ {��_{rq}}}+ \ \ \displaystyle {{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rq}}}+{{m_{p}}\ {m_{r}}\ {��_{rp}}\ {��_{rq}}}+{{{m_{q}}^{2}}\ {��_{qq}}\ {��_{rq}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}+{{m_{q}}\ {m_{r}}\ {{��_{rq}}^{2}}}+{{{m_{r}}^{2}}\ {��_{rq}}\ {��_{rr}}}$

(58)

fricas

c*c-c

$\label{eq59}0$

(59)

fricas

trace(c)

$\label{eq60}1$

(60)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

Scalar Product

fricas

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

$\label{eq61}\left[ \begin{array}{ccccccccc} {{{m_{p}}^{2}}\ {{��_{pp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}} \ {{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{q}}^{2}}\ {{��_{qq}}^{2}}}&{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}} \ {{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{rp}}}&{{m_{q}}\ {m_{r}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{r}}^{2}}\ {{��_{rr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}} \ {{{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}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}} \ {{{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}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{rp}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}\ {��_{rr}}} \ {{{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}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pr}}\ {��_{qp}}\ {��_{rq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{{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}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{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}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{qp}}\ {��_{qr}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}} \ {{{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}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pp}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {��_{pp}}\ {��_{pr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qp}}\ {��_{rp}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{qr}}\ {��_{rp}}\ {��_{rr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{rp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}} \ {{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_{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}}^{2}}\ {m_{r}}\ {��_{pr}}\ {��_{qp}}\ {��_{qq}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {��_{qq}}\ {��_{qr}}\ {��_{rq}}\ {��_{rr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pr}}\ {��_{qr}}\ {��_{rp}}\ {��_{rq}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rq}}^{2}}}$

(61)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

determinant(S)/f^(2*#gens)

$\label{eq62}-{{{m_{p}}^{10}}\ {{m_{q}}^{10}}\ {{m_{r}}^{10}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qp}}^{2}}\ {{��_{qr}}^{2}}\ {{��_{rp}}^{2}}\ {{��_{rq}}^{2}}}$

(62)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

C:=map(x+->factor(numer(x))/factor(denom(x)),inverse(S/f^2))

\label{eq63}\left[
\begin{array}{ccccccccc}
{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}^{2}}\over{{m_{p}}^{2}}}&{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}}\over{{m_{q}}\ {m_{p}}}}&{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{m_{r}}\ {m_{p}}}}& -{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}}\over{{��_{pq}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}}\over{{��_{qp}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {m_{r}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}}\over{{��_{rq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}
\
{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}}\over{{m_{q}}\ {m_{p}}}}&{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}^{2}}\over{{m_{q}}^{2}}}&{{{\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{m_{r}}\ {m_{q}}}}& -{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{pq}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{qp}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {{m_{q}}^{2}}}}&{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {m_{r}}\ {{m_{q}}^{2}}}}
\
{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{m_{r}}\ {m_{p}}}}&{{{\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{m_{r}}\ {m_{q}}}}&{{{\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}^{2}}\over{{m_{r}}^{2}}}& -{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{pq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{pr}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qp}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}}}&{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rp}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rq}}\ {{m_{r}}^{2}}\ {m_{q}}}}
\
-{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}}\over{{��_{pq}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{pq}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{pq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}^{2}}\over{{{��_{pq}}^{2}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{qp}}\ {��_{pq}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}
\
{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{pr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{pr}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{��_{pr}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}^{2}}\over{{{��_{pr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qp}}\ {��_{pr}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rp}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rq}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}
\
-{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}}\over{{��_{qp}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{qp}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qp}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{qp}}\ {��_{pq}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qp}}\ {��_{pr}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}^{2}}\over{{{��_{qp}}^{2}}\ {{m_{q}}^{2}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {��_{qp}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {��_{qp}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {��_{qp}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}
\
-{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {m_{r}}\ {{m_{q}}^{2}}}}& -{{{\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}}}&{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{qr}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{pq}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qq}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}}\over{{��_{qr}}\ {��_{qp}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pp}}\ {��_{qr}}}-{{��_{pr}}\ {��_{qp}}}\right)}^{2}}\over{{{��_{qr}}^{2}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rp}}\ {��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rq}}\ {��_{qr}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}
\
{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {m_{r}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rp}}\ {{m_{r}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {��_{pq}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}&{{{\left({{��_{qq}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rp}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}}\over{{��_{rp}}\ {��_{qp}}\ {m_{r}}\ {m_{q}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rr}}}-{{��_{qr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rp}}\ {��_{qr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}^{2}}\over{{{��_{rp}}^{2}}\ {{m_{r}}^{2}}\ {{m_{p}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {��_{rp}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}
\
-{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}}\over{{��_{rq}}\ {m_{r}}\ {m_{q}}\ {m_{p}}}}& -{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {m_{r}}\ {{m_{q}}^{2}}}}& -{{{\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rq}}\ {{m_{r}}^{2}}\ {m_{q}}}}&{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {��_{pq}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}& -{{{\left({{��_{pq}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rq}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rq}}\ {��_{pr}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {��_{qp}}\ {m_{r}}\ {{m_{q}}^{2}}\ {m_{p}}}}&{{{\left({{��_{pp}}\ {��_{rr}}}-{{��_{pr}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}\right)}}\over{{��_{rq}}\ {��_{qr}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}& -{{{\left({{��_{qp}}\ {��_{rq}}}-{{��_{qq}}\ {��_{rp}}}\right)}\ {\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}}\over{{��_{rq}}\ {��_{rp}}\ {{m_{r}}^{2}}\ {m_{q}}\ {m_{p}}}}&{{{\left({{��_{pp}}\ {��_{rq}}}-{{��_{pq}}\ {��_{rp}}}\right)}^{2}}\over{{{��_{rq}}^{2}}\ {{m_{r}}^{2}}\ {{m_{q}}^{2}}}}

(63)

Type: MATRIX(FRAC(FR(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))))