# MAPLE PROGRAM FOR MAKING THE TWO FREE GLUEBALL STATES ############################################################### # # Specification of the O_h symmetry group # # symmetry operations of the octahedral group with inversion (O_h) # (for the single-valued representations) # # Notation: Id = identity group element # I = spatial inversion # Cnj = proper rotation through angle 2*Pi/n in a # right-hand screw sense about the axis Oj, where # O is the origin. # inv = refers to the inverse operation # # Various axes: Ox = (1,0,0) Oy = (0,1,0) Oz = (0,0,1) # Oa = (1,1,0) Ob = (1,-1,0) Oc = (1,0,1) # Od = (-1,0,1) Oe = (0,1,1) Of = (0,1,-1) # Oalpha = (-1,-1,1) Obeta = (-1,1,-1) # Ogamma = (1,-1,-1) Odelta = (1,1,1) # # Below are all of the group elements of the group O_h, # the class structure, the irreps, and the irrep characters. # The group multiplication table is given, and the inverses # of each group element are specified. # ################################################################ Oh:={Id, C4x, C4y, C4z, C2x, C2y, C2z, C4xinv, C4yinv, C4zinv, C3alpha, C3beta, C3gamma, C3delta, C3alphainv, C3betainv, C3gammainv, C3deltainv, C2a, C2b, C2c, C2d, C2e, C2f, Is, IC4x, IC4y, IC4z, IC2x, IC2y, IC2z, IC4xinv, IC4yinv, IC4zinv, IC3alpha, IC3beta, IC3gamma, IC3delta, IC3alphainv, IC3betainv, IC3gammainv, IC3deltainv, IC2a, IC2b, IC2c, IC2d, IC2e, IC2f}: ########################### # # The classes of O_h (ordering must be maintained) # ########################### Oh_classes:=[{Id}, {C3alphainv, C3betainv, C3gammainv, C3deltainv, C3alpha, C3beta, C3gamma, C3delta}, {C2y, C2z, C2x}, {C4zinv, C4xinv, C4yinv, C4z, C4y, C4x}, {C2f, C2b, C2c, C2d, C2e, C2a}, {Is}, {IC3alpha, IC3gamma, IC3delta, IC3alphainv, IC3betainv, IC3gammainv, IC3deltainv, IC3beta}, {IC2x, IC2y, IC2z}, {IC4z, IC4xinv, IC4yinv, IC4zinv, IC4x, IC4y}, {IC2e, IC2f, IC2a, IC2c, IC2d, IC2b} ]: Oh_class_dims:=map(nops,Oh_classes): # if A is an element of O_h, then the class to which A belongs # is class(A). The index of class(A) in the list "Oh_classes" # is then given in Oh_class_index[A]: Oh_class_index := table([(C2y)=3,(C2z)=3,(C4xinv)=4,(IC3alpha)=7,(C4yinv)=4,( C4zinv)=4,(C3alpha)=2,(IC2c)=10,(Is)=6,(IC2e)=10,(IC3betainv)=7,(IC2d)=10,( IC4yinv)=9,(C3alphainv)=2,(IC3beta)=7,(IC2b)=10,(IC4zinv)=9,(IC4y)=9,(IC4z)=9,( IC3gamma)=7,(IC3gammainv)=7,(C2d)=5,(C3beta)=2,(C2e)=5,(C3gamma)=2,(Id)=1,(C2f) =5,(IC4x)=9,(IC2f)=10,(IC3delta)=7,(C4z)=4,(C3deltainv)=2,(C4y)=4,(C2x)=3,( C3gammainv)=2,(IC3alphainv)=7,(IC3deltainv)=7,(IC2x)=8,(IC2a)=10,(IC2y)=8,( C3betainv)=2,(IC2z)=8,(IC4xinv)=9,(C4x)=4,(C3delta)=2,(C2a)=5,(C2b)=5,(C2c)=5]): ############################################ # # The characters of the irreps of O_h # ############################################ Oh_irreps:=[A1g,A2g,Eg,T1g,T2g,A1u,A2u,Eu,T1u,T2u]: Oh_irrep_characters[A1g]:=[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]: Oh_irrep_characters[A2g]:=[ 1, 1, 1,-1,-1, 1, 1, 1,-1,-1 ]: Oh_irrep_characters[Eg ]:=[ 2,-1, 2, 0, 0, 2,-1, 2, 0, 0 ]: Oh_irrep_characters[T1g]:=[ 3, 0,-1, 1,-1, 3, 0,-1, 1,-1 ]: Oh_irrep_characters[T2g]:=[ 3, 0,-1,-1, 1, 3, 0,-1,-1, 1 ]: Oh_irrep_characters[A1u]:=[ 1, 1, 1, 1, 1, -1,-1,-1,-1,-1 ]: Oh_irrep_characters[A2u]:=[ 1, 1, 1,-1,-1, -1,-1,-1, 1, 1 ]: Oh_irrep_characters[Eu ]:=[ 2,-1, 2, 0, 0, -2, 1,-2, 0, 0 ]: Oh_irrep_characters[T1u]:=[ 3, 0,-1, 1,-1, -3, 0, 1,-1, 1 ]: Oh_irrep_characters[T2u]:=[ 3, 0,-1,-1, 1, -3, 0, 1, 1,-1 ]: # The group multiplication table is given below. If A and B # are elements of Oh, the gprod[A,B] contains the product A*B # which must also be an element of the group. gprod := table([(C4x, IC2y)=IC2e,(C2y, C2z)=C2x,(C3gammainv, C4z)=C2e,( C3deltainv, C3alpha)=C2x,(C2e, IC3alphainv)=IC2b,(IC4x, C3deltainv)=IC4zinv,( IC4yinv, C2a)=IC3delta,(IC3beta, IC2b)=C4yinv,(IC3gamma, IC2d)=C4xinv,( IC3deltainv, IC3delta)=Id,(C4zinv, C3deltainv)=C2c,(C4zinv, C2c)=C3betainv,( C3gamma, C2b)=C4y,(C3gamma, IC3beta)=IC3alphainv,(C3alphainv, IC2d)=IC4z,( C3deltainv, C3betainv)=C3alpha,(C2e, Is)=IC2e,(IC4x, C4xinv)=Is,(IC4zinv, C3beta)=IC2f,(IC3beta, IC3beta)=C3betainv,(IC2d, IC2y)=C2c,(C4z, IC2f)=IC3beta, (C2x, IC2x)=Is,(C2a, IC4z)=IC2x,(C2b, IC4x)=IC3beta,(C2c, C2y)=C2d,(Is, Is)=Id, (IC4x, IC2c)=C3beta,(IC4xinv, C2z)=IC2e,(IC3gammainv, IC4yinv)=C4z,(IC3deltainv , IC4yinv)=C2a,(C2y, C4xinv)=C2e,(C2y, C3delta)=C3alpha,(C3betainv, C4xinv)= C4yinv,(C2a, IC2z)=IC2b,(IC4y, IC4z)=C3delta,(IC4y, IC2b)=C3gamma,(IC2y, C4xinv )=IC2e,(IC3alpha, Id)=IC3alpha,(IC3beta, C3beta)=IC3betainv,(IC3delta, IC4x)= C2a,(IC2a, C3alphainv)=IC4yinv,(C4y, IC3alphainv)=IC2f,(C4xinv, IC2a)= IC3alphainv,(C4zinv, C2y)=C2a,(C3beta, IC3alpha)=IC3gammainv,(C2f, IC2z)=IC4x,( IC2x, IC4y)=C2c,(IC2z, IC2a)=C2b,(IC4yinv, C3betainv)=IC2f,(IC3delta, C3deltainv)=Is,(IC3alphainv, C3betainv)=IC3gamma,(C4z, IC3alpha)=IC2f,(C3beta, IC2d)=IC2e,(C3gamma, IC4zinv)=IC2d,(C3deltainv, IC3deltainv)=IC3delta,(C2c, C3deltainv)=C4x,(Is, C2e)=IC2e,(IC4x, IC4x)=C2x,(IC2x, C2f)=IC2e,(IC4yinv, IC2b )=C3beta,(IC4zinv, IC4z)=Id,(IC2e, IC4z)=C3deltainv,(C4y, C4xinv)=C3gammainv,( C2e, IC4xinv)=IC2z,(IC4y, IC4x)=C3alphainv,(IC4y, IC2y)=C4yinv,(IC2z, IC4xinv)= C2f,(IC3delta, C2z)=IC3beta,(IC3deltainv, C2x)=IC3betainv,(IC2e, IC2a)= C3betainv,(C4z, IC4z)=IC2z,(C2y, C2a)=C4z,(C2z, IC2y)=IC2x,(C4xinv, IC3alphainv )=IC4zinv,(C2c, IC2e)=IC3alphainv,(C2d, IC3beta)=IC2a,(IC2y, IC2a)=C4z,(IC2a, C2x)=IC4z,(IC2d, C2d)=Is,(IC2d, IC3delta)=C2b,(C4x, C4x)=C2x,(C4y, IC2z)=IC2c,( C2y, IC2e)=IC4xinv,(C4zinv, IC4y)=IC3alphainv,(C3gamma, C2c)=C2f,(C3delta, IC4z )=IC2c,(C3alphainv, IC2a)=IC4xinv,(C2c, Is)=IC2c,(Is, C3gammainv)=IC3gammainv,( IC4y, IC4y)=C2y,(IC4zinv, C3gamma)=IC2e,(IC3delta, IC2e)=C4zinv,(IC2d, IC3gamma )=C4z,(C2y, C2f)=C4x,(IC4y, C4yinv)=Is,(IC4z, C4xinv)=IC3alpha,(IC3beta, C3gamma)=IC3deltainv,(IC3gamma, C2d)=IC4xinv,(C4x, C4xinv)=Id,(C4x, C3alpha)= C4yinv,(C2y, IC2x)=IC2z,(C4zinv, C3alphainv)=C2d,(C3gamma, IC3alphainv)=IC2x,( IC4xinv, IC4xinv)=C2x,(IC2f, C2x)=IC2e,(Id, IC3deltainv)=IC3deltainv,(C4xinv, IC4xinv)=IC2x,(C2c, IC2z)=IC4y,(IC4y, IC3alpha)=C4z,(IC2x, C3gammainv)= IC3betainv,(IC4yinv, IC3deltainv)=C2e,(IC3betainv, C4z)=IC2f,(IC3deltainv, C4z) =IC4xinv,(IC2b, IC3alphainv)=C2c,(C4y, IC3gammainv)=IC2e,(C2y, IC3betainv)= IC3alphainv,(C4xinv, IC2x)=IC4x,(C4yinv, Id)=C4yinv,(C4zinv, C2d)=C3gammainv,( C3delta, C3betainv)=C2x,(IC4xinv, IC3beta)=C2c,(IC4yinv, C4zinv)=IC3gamma,(IC2a , C2a)=Is,(IC2a, IC4zinv)=C2y,(IC2c, IC4y)=C2z,(C4xinv, IC2y)=IC2f,(C3betainv, IC2y)=IC3gammainv,(C3gammainv, IC4xinv)=IC2d,(IC2x, C4xinv)=IC4x,(IC2x, IC3deltainv)=C3alphainv,(IC3beta, IC2z)=C3delta,(IC3delta, IC3beta)=C3gammainv, (IC3betainv, IC2d)=C2a,(C2y, IC4z)=IC2a,(C3gamma, IC3delta)=IC3betainv,( C3betainv, C3betainv)=C3beta,(C2d, C3gammainv)=C4x,(C2f, IC3alphainv)=IC4z,(Is , C3gamma)=IC3gamma,(IC4xinv, IC3deltainv)=C2a,(IC3alpha, IC2a)=C4y,(IC3delta, C2a)=IC4yinv,(IC3gammainv, C3alpha)=IC2z,(C4y, IC3betainv)=IC4x,(C4z, C3gamma)= C4x,(IC3alpha, C3gammainv)=IC2x,(Id, IC2e)=IC2e,(C2z, C4xinv)=C2f,(C3gammainv, IC2y)=IC3betainv,(C2c, C3delta)=C4zinv,(IC3alpha, IC3beta)=C3deltainv,(IC3gamma , C2b)=IC4y,(IC2a, IC3beta)=C2e,(Id, C2z)=C2z,(C4y, C3alphainv)=C2f,(C2x, C4x)= C4xinv,(IC4y, IC2d)=C2z,(IC4zinv, IC2d)=C3gammainv,(IC3betainv, IC2e)=C2d,(C4x , C4z)=C3betainv,(C4z, Is)=IC4z,(C2x, C2c)=C4y,(C2x, IC4x)=IC4xinv,(C3beta, C4x )=C4zinv,(C3gamma, C4yinv)=C2e,(C3alphainv, IC2y)=IC3deltainv,(C2b, C2e)= C3alpha,(C2d, IC2x)=IC4y,(IC4y, C3delta)=IC2a,(IC4zinv, C2a)=IC2x,(IC3gamma, C2e)=IC4z,(IC3gammainv, IC4y)=C2b,(IC2b, IC4y)=C3betainv,(IC2d, IC2x)=C4y,( C4xinv, C4z)=C3gammainv,(C4zinv, C4y)=C3alphainv,(C3delta, C3alphainv)=C2y,( C3alphainv, IC2b)=IC2e,(C3deltainv, IC2a)=IC4x,(C2b, IC2e)=IC3alpha,(C2d, IC2d) =Is,(Is, C2z)=IC2z,(IC4z, C4yinv)=IC3betainv,(IC2x, IC4x)=C4xinv,(IC4zinv, IC2y )=C2a,(IC3alpha, IC2b)=C2c,(IC3delta, C3beta)=IC3gammainv,(IC3delta, IC3betainv )=C2x,(IC3gammainv, C3betainv)=IC3delta,(IC3gammainv, C2d)=IC4zinv,(C4x, IC3gammainv)=IC4z,(C3gamma, C2x)=C3beta,(C3gamma, IC4yinv)=IC2e,(C3delta, IC4zinv)=IC4y,(C2a, IC3gamma)=IC2f,(C2c, C3gammainv)=C2f,(C2c, IC2d)=IC2y,(IC2f , C2f)=Is,(Id, C3delta)=C3delta,(C4y, Is)=IC4y,(C3gamma, C3deltainv)=C2z,(C2c, IC4x)=IC3deltainv,(IC2x, C4z)=IC2b,(IC3alpha, C4yinv)=IC2f,(IC3beta, IC4xinv)= C2b,(IC2b, IC2c)=C3alphainv,(IC2c, IC2y)=C2d,(C4x, IC2c)=IC3beta,(C4z, IC3deltainv)=IC4yinv,(C3beta, IC3alphainv)=IC2z,(IC4x, C3delta)=IC2c,(IC3beta, IC3alphainv)=C2z,(IC3betainv, IC4xinv)=C4yinv,(IC3betainv, IC3delta)=C2z,(IC2e , IC4xinv)=C2z,(IC2f, C2e)=IC2x,(IC2f, IC3gamma)=C2c,(C3beta, C2b)=C4yinv,( C3gamma, IC2a)=IC2c,(C3deltainv, IC2z)=IC3gammainv,(C2e, Id)=C2e,(C2e, C2x)=C2f ,(IC2x, IC3alpha)=C3gamma,(IC4yinv, IC4yinv)=C2y,(IC4zinv, IC2z)=C4z,(IC2b, IC4yinv)=C3gammainv,(IC2e, C4x)=IC2y,(C4z, IC3beta)=IC4xinv,(C2y, IC3gamma)= IC3beta,(C3gamma, IC2d)=IC4xinv,(C3alphainv, C2f)=C4yinv,(C3alphainv, IC3betainv)=IC3gamma,(C3betainv, C2f)=C4y,(C3gammainv, C3gamma)=Id,(C2f, IC3alpha)=IC4y,(IC2z, C2a)=IC2b,(IC3alpha, C2y)=IC3beta,(IC3gamma, IC3alphainv) =C2x,(IC3delta, Is)=C3delta,(IC2b, C3deltainv)=IC2d,(IC2d, C2f)=IC3deltainv,( IC2f, C4yinv)=IC3beta,(C2y, C4zinv)=C2b,(C2z, IC2b)=IC2a,(C3gammainv, IC2z)= IC3deltainv,(C2b, IC3deltainv)=IC2d,(C2e, IC2d)=IC3beta,(IC4y, C2b)=IC3gamma,( IC4xinv, C3delta)=IC4y,(IC3beta, IC4zinv)=C2c,(IC3gamma, IC3deltainv)=C2z,( IC3betainv, C4xinv)=IC4yinv,(IC3deltainv, C2a)=IC4x,(IC2a, IC3deltainv)=C4y,( IC2c, C3beta)=IC4z,(C4z, C2b)=C2x,(C3alpha, IC2f)=IC4zinv,(C3delta, IC4x)=IC2a, (C3alphainv, C4z)=C4x,(C3deltainv, C2f)=C2d,(C2e, IC3beta)=IC2d,(IC4xinv, C3alpha)=IC2d,(IC3beta, IC3delta)=C3alphainv,(IC3delta, IC2z)=C3beta,(IC2d, IC3alphainv)=C4xinv,(C3alpha, C2e)=C2b,(C2c, IC3betainv)=IC4xinv,(IC2x, IC4xinv )=C4x,(IC3alphainv, IC3beta)=C2y,(IC3deltainv, IC2z)=C3gammainv,(C2x, IC4zinv)= IC2a,(C4yinv, C2d)=C2x,(C3betainv, Is)=IC3betainv,(C3deltainv, C3beta)=C2z,(C2d , C4z)=C3gamma,(IC3deltainv, IC2d)=C2b,(IC2b, IC2f)=C3delta,(IC2e, IC2x)=C2f,( C4z, IC4zinv)=Is,(C2z, C2y)=C2x,(C4yinv, C2f)=C3alphainv,(C3gammainv, C4y)=C2b, (C3deltainv, C2y)=C3alphainv,(C2e, C3gamma)=C4y,(Is, IC4x)=C4x,(IC2y, IC3alphainv)=C3betainv,(IC3gamma, IC2b)=C4y,(IC3betainv, IC3alphainv)=C3delta,( IC3deltainv, C4xinv)=IC2c,(IC2b, Is)=C2b,(IC2d, IC4xinv)=C3alphainv,(IC2f, IC3betainv)=C4zinv,(Id, C2d)=C2d,(C2x, C3beta)=C3delta,(C2x, IC3alpha)=IC3gamma ,(C2x, IC2b)=IC4z,(C2y, C2c)=C2d,(C3alphainv, IC4yinv)=IC4zinv,(C3betainv, IC3betainv)=IC3beta,(C2a, IC4x)=IC3alpha,(C2d, C3alpha)=C4zinv,(C2d, IC2y)=IC2c ,(C2e, C3betainv)=C2a,(C2f, Id)=C2f,(C2f, C3betainv)=C4zinv,(IC4x, C2y)=IC2e,( IC4yinv, IC2c)=C2z,(IC3alpha, C2x)=IC3delta,(IC3delta, Id)=IC3delta,(IC2e, IC2e )=Id,(IC2f, C4z)=IC3alphainv,(Id, IC3gammainv)=IC3gammainv,(C4yinv, C3gammainv) =C4xinv,(C4zinv, IC3betainv)=IC4yinv,(IC3gamma, C4y)=IC4x,(IC3gammainv, IC3beta )=C2x,(IC2d, IC2f)=C3deltainv,(C4z, C4x)=C3delta,(C4yinv, IC3delta)=IC4z,( C4zinv, IC3beta)=IC2f,(C3deltainv, IC3betainv)=IC3alpha,(C2a, C2f)=C3gamma,(C2c , C4z)=C3beta,(IC4zinv, IC3beta)=C2f,(IC3gamma, C2y)=IC3delta,(C4x, IC2x)= IC4xinv,(C2z, C3beta)=C3alpha,(C4yinv, C3alpha)=C2a,(C4yinv, IC4zinv)=IC3gamma, (C3delta, C2a)=C4yinv,(C3deltainv, C4yinv)=C2a,(C2c, IC3alphainv)=IC2e,(C2f, IC2e)=IC2x,(IC4x, C3gammainv)=IC4z,(IC4xinv, C3betainv)=IC4z,(IC3beta, C2d)= IC2e,(IC3betainv, IC2c)=C4zinv,(C4yinv, IC3deltainv)=IC2e,(C3delta, C2z)=C3beta ,(C3betainv, C2c)=C4zinv,(C3deltainv, C3delta)=Id,(C2a, C4y)=C3deltainv,(IC4x, C2e)=IC2z,(IC4yinv, IC3alphainv)=C4x,(IC3beta, IC2f)=C4z,(IC2e, C2c)=IC3alpha,( C3alphainv, C2x)=C3gammainv,(C2a, IC2c)=IC3gammainv,(C2b, C4zinv)=C2x,(C2c, C2x )=C4yinv,(IC4zinv, C3delta)=IC4x,(IC3gamma, C3beta)=IC3alphainv,(IC3deltainv, IC2a)=C4x,(IC2a, C3gamma)=IC2f,(IC2e, IC2z)=C4xinv,(C4x, C3delta)=C2c,(C4xinv, IC2c)=IC3delta,(C4yinv, IC2e)=IC3gammainv,(C3gammainv, C3betainv)=C3delta,(C2b , C3gamma)=C4xinv,(C2e, C4z)=C3deltainv,(IC2x, C2c)=IC4y,(IC2z, C3betainv)= IC3deltainv,(IC4xinv, IC3betainv)=C4z,(IC2e, C2x)=IC2f,(C2y, IC4x)=IC2f,(C4zinv , C3delta)=C4x,(C3alpha, IC3alphainv)=Is,(C2c, C2c)=Id,(Is, IC2b)=C2b,(IC3alpha , C3gamma)=IC3betainv,(IC3betainv, C2z)=IC3alphainv,(IC2e, IC3delta)=C4yinv,( IC2f, IC2f)=Id,(C3alphainv, C2e)=C2c,(C3betainv, C4zinv)=C4x,(C2a, IC3delta)= IC4xinv,(C2f, C2b)=C3deltainv,(C2f, IC2x)=IC2e,(IC4yinv, C3gammainv)=IC4xinv,( C4x, Is)=IC4x,(C2x, IC3deltainv)=IC3alphainv,(C3delta, C4xinv)=C4z,(C3gammainv , C2f)=C2c,(IC4x, C3alphainv)=IC2a,(IC4y, IC3gamma)=C4zinv,(IC3betainv, IC2z)= C3alphainv,(IC2f, C4xinv)=IC2y,(C3alphainv, C2z)=C3betainv,(C3deltainv, IC3gammainv)=IC3beta,(C2e, IC2b)=IC3alphainv,(IC4x, C3betainv)=IC2b,(IC4yinv, IC3gammainv)=C4xinv,(IC4zinv, C2x)=IC2b,(IC4zinv, IC4yinv)=C3deltainv,(IC3alpha , C3alphainv)=Is,(IC3delta, C3alphainv)=IC2y,(IC3deltainv, C2b)=IC2f,(IC2b, C3alpha)=IC2e,(C2x, IC4yinv)=IC2d,(C4xinv, IC3beta)=IC2c,(C3alpha, IC2d)=IC4x,( C3alphainv, IC3deltainv)=IC3beta,(C2b, C4z)=C2y,(C2b, IC3betainv)=IC4y,(C2d, IC2c)=IC2y,(IC4x, C3alpha)=IC4yinv,(IC3gamma, IC4yinv)=C2e,(IC2b, IC2e)=C3alpha ,(C2x, C2z)=C2y,(C4yinv, IC2z)=IC2d,(C2c, IC4yinv)=IC2x,(IC4z, C3alphainv)=IC4y ,(IC4xinv, IC2b)=C3betainv,(IC2c, IC4xinv)=C3betainv,(C4xinv, C2c)=C3delta,( C3delta, IC3betainv)=IC2x,(Is, IC4xinv)=C4xinv,(IC4z, C3alpha)=IC2f,(IC4zinv, IC2a)=C2x,(IC3delta, C3gammainv)=IC2z,(IC2a, C3alpha)=IC4x,(IC2c, IC3deltainv)= C4x,(C4xinv, C3deltainv)=C2a,(C4zinv, C2x)=C2b,(C3alphainv, C2a)=C4xinv,( C3alphainv, IC3delta)=IC2x,(C2e, C4x)=C2y,(C2f, C3gammainv)=C2a,(IC3gamma, IC2a )=C2c,(IC2b, IC3alpha)=C2e,(IC2d, C3alphainv)=IC4xinv,(C2x, C4zinv)=C2a,(C2z, C3alphainv)=C3gammainv,(C3gammainv, IC3delta)=IC2y,(Is, C4yinv)=IC4yinv,(IC2x, Is)=C2x,(IC4xinv, C2c)=IC3delta,(IC3delta, C2y)=IC3gamma,(IC3delta, C2c)= IC4xinv,(IC2a, IC2d)=C3betainv,(IC2b, IC2a)=C2z,(IC2c, IC3gammainv)=C2f,(C2z, C2z)=Id,(C3beta, C4z)=C4y,(IC2c, C4xinv)=IC3betainv,(IC2c, C2d)=IC2y,(Id, C2f)= C2f,(C2y, Id)=C2y,(C2z, IC4y)=IC2d,(C3beta, C3betainv)=Id,(C3gammainv, C4yinv)= C4z,(C3gammainv, IC4zinv)=IC4xinv,(C2b, C2x)=C4zinv,(C2b, C3alpha)=C2e,(IC4z, IC4x)=C3delta,(IC3deltainv, Id)=IC3deltainv,(IC2e, C2f)=IC2x,(IC2f, IC2d)= C3delta,(C3gamma, IC2e)=IC4z,(C3gammainv, IC3alpha)=IC2z,(C2e, IC2a)=IC3betainv ,(IC2x, C2z)=IC2y,(IC2y, IC3betainv)=C3alphainv,(IC4yinv, Is)=C4yinv,(IC2e, IC4y)=C3gamma,(C4z, C3betainv)=C2c,(C2d, IC2a)=IC3beta,(IC3alpha, C2z)=IC3gamma ,(IC3alpha, C4zinv)=IC4yinv,(IC3alpha, C3alpha)=IC3alphainv,(C4x, C2b)= C3gammainv,(C3delta, IC3delta)=IC3deltainv,(C3alphainv, C2d)=C4z,(Is, IC2c)=C2c ,(IC3betainv, C4yinv)=IC2b,(Id, IC4x)=IC4x,(C4y, IC4yinv)=Is,(C2x, IC2e)=IC2f,( C3beta, IC3deltainv)=IC2x,(C3deltainv, IC4x)=IC4yinv,(C2f, IC2b)=IC3deltainv,( IC2y, IC3alpha)=C3delta,(IC2z, IC3deltainv)=C3betainv,(IC2b, C4xinv)=IC3gamma,( IC2b, C3alphainv)=IC2c,(IC2d, C4x)=IC3gammainv,(C3gammainv, IC3alphainv)= IC3beta,(IC4x, C4z)=IC3betainv,(IC4z, C4zinv)=Is,(IC2z, C4xinv)=IC2f,(IC4yinv, IC4xinv)=C3deltainv,(IC4zinv, C3deltainv)=IC2c,(C4y, C3betainv)=C4x,(C3beta, IC4x)=IC4zinv,(C3gamma, C2y)=C3delta,(C3deltainv, Id)=C3deltainv,(C2d, IC3alpha )=IC4zinv,(C2f, C3alphainv)=C4z,(IC2x, IC2e)=C2f,(IC3gammainv, C2y)=IC3betainv, (IC2d, C4y)=IC2x,(IC2f, IC2c)=C3gamma,(Id, IC3alpha)=IC3alpha,(C4x, IC4y)= IC3delta,(C2x, C4z)=C2b,(C2x, IC2f)=IC2e,(C2a, C4xinv)=C3delta,(IC4xinv, C4x)= Is,(IC3beta, Is)=C3beta,(C4y, C4yinv)=Id,(C2y, C3betainv)=C3alphainv,(C2z, C3delta)=C3gamma,(C4xinv, IC2b)=IC3betainv,(C3beta, C2a)=C2d,(C3beta, IC2c)= IC4x,(C3alphainv, C3gamma)=C2z,(C2e, C3alphainv)=C2b,(C2e, IC2c)=IC3alpha,(Is, IC3gamma)=C3gamma,(IC4x, IC3alphainv)=C2a,(IC2y, IC4yinv)=C4y,(IC4yinv, C3delta )=IC4z,(IC3gammainv, C3beta)=IC2x,(C2z, C4x)=C2e,(C4zinv, C2f)=C3alpha,(C3alpha , IC4y)=IC4xinv,(C3gamma, C3alphainv)=C2x,(C2d, C3delta)=C2b,(Is, C3betainv)= IC3betainv,(IC4x, C2b)=IC3gammainv,(IC2x, IC3betainv)=C3gammainv,(IC2x, IC2b)= C4z,(IC2z, IC4y)=C2d,(IC2z, IC3alphainv)=C3gammainv,(IC4xinv, C3beta)=IC2c,( IC4yinv, C3deltainv)=IC2e,(IC3betainv, C2e)=IC2d,(IC3deltainv, C3beta)=IC2z,( IC2b, C2c)=IC3alphainv,(C2y, C2y)=Id,(C3delta, IC3gamma)=IC3alphainv,(C3betainv , IC2d)=IC2a,(C2b, IC3beta)=IC4x,(IC2x, IC2x)=Id,(IC2y, C3alphainv)=IC3betainv, (IC3beta, C4xinv)=IC2b,(IC2a, Id)=IC2a,(IC2a, C2y)=IC4zinv,(IC2d, IC2e)= C3betainv,(Id, IC2z)=IC2z,(IC4y, IC2c)=C2x,(IC3beta, IC3gamma)=C3deltainv,(C4z , C2c)=C3deltainv,(C4yinv, C2b)=C3beta,(C3gammainv, IC3gamma)=Is,(IC4y, C3betainv)=IC4x,(IC3alpha, C2d)=IC4x,(IC2b, C3gamma)=IC4xinv,(C2x, IC2a)= IC4zinv,(C4zinv, C3beta)=C2f,(C4zinv, C3gamma)=C2e,(C3gammainv, IC4y)=IC2b,( C3gammainv, IC2c)=IC2a,(C2e, C3deltainv)=C4z,(C2e, IC2y)=IC4x,(C2f, C4yinv)= C3beta,(IC4yinv, IC4z)=C3alpha,(IC2c, IC4z)=C3beta,(Id, C3betainv)=C3betainv,( C2y, IC4xinv)=IC2e,(C3delta, C2y)=C3gamma,(IC4y, IC3beta)=C2b,(IC2x, C3betainv) =IC3gammainv,(IC2x, C2e)=IC2f,(IC3delta, IC2a)=C4yinv,(IC3alphainv, IC3gammainv )=C3delta,(IC2e, IC3deltainv)=C4z,(C2x, C2y)=C2z,(C3alpha, C2x)=C3delta,( C3betainv, IC3delta)=IC2z,(C2a, C2b)=C2z,(C2c, C4y)=C2z,(C2d, C3betainv)=C2e,( IC2y, IC2z)=C2x,(IC2z, IC2x)=C2y,(IC4zinv, IC4x)=C3gamma,(C3alpha, IC4zinv)= IC4yinv,(C3beta, IC2e)=IC2a,(C3betainv, IC4xinv)=IC4yinv,(C3deltainv, C2a)=C4x, (C2b, IC4zinv)=IC2x,(C2d, C2z)=C4yinv,(IC4x, C2c)=IC3beta,(IC4yinv, C2e)= IC3gammainv,(IC3betainv, Is)=C3betainv,(IC3gammainv, Id)=IC3gammainv,(IC2a, IC2x)=C4z,(IC2e, IC3gammainv)=C4zinv,(Id, C3gammainv)=C3gammainv,(C4y, C4x)= C3alphainv,(C4y, IC2f)=IC3betainv,(C4yinv, C3deltainv)=C2e,(C3beta, IC2x)= IC3gamma,(C3betainv, IC4zinv)=IC4x,(C3deltainv, IC4z)=IC4xinv,(C2a, C3gammainv) =C2c,(C2c, IC3deltainv)=IC4x,(C2f, C4zinv)=C3betainv,(C2f, IC4y)=IC3alpha,(IC4x , IC2d)=C3alpha,(IC2y, C2b)=IC4zinv,(IC4zinv, IC3gammainv)=C4y,(IC3alphainv, C4yinv)=IC4zinv,(IC3alphainv, IC2e)=C2c,(Id, IC4y)=IC4y,(C2x, IC4y)=IC2c,( C4yinv, C2x)=C2c,(C3deltainv, C4x)=C4yinv,(C2a, C3betainv)=C2d,(IC2z, IC3delta) =C3gamma,(IC4yinv, C2x)=IC2c,(IC2a, C4yinv)=IC3alphainv,(IC2b, C4z)=IC2y,(C4y, IC4zinv)=IC3beta,(C3beta, C4xinv)=C2b,(C2d, C2f)=C3deltainv,(IC4y, IC4yinv)=Id, (IC2y, C3gammainv)=IC3deltainv,(IC2y, IC2b)=C4zinv,(IC2e, IC4yinv)=C3delta,(Id , C4z)=C4z,(C4x, C3betainv)=C2b,(C2z, Is)=IC2z,(Is, IC3alpha)=C3alpha,(IC2x, C3delta)=IC3beta,(IC2c, C4z)=IC3beta,(Id, C2y)=C2y,(C4x, IC3delta)=IC2c,(C4z, C2a)=C2y,(C4xinv, C3delta)=C4y,(C3alphainv, C2y)=C3deltainv,(C3gammainv, C2z)= C3deltainv,(C2a, IC4zinv)=IC2y,(C2d, IC3alphainv)=IC4xinv,(C2d, IC3gammainv)= IC4x,(IC2x, IC4yinv)=C2d,(IC4xinv, IC4yinv)=C3alpha,(IC4xinv, IC2e)=C2y,( IC3gamma, IC4xinv)=C4zinv,(IC3gamma, IC3betainv)=C2y,(Id, C3gamma)=C3gamma,(C2x , Id)=C2x,(C3alpha, C2d)=C4x,(C3gammainv, IC3deltainv)=IC3alpha,(IC2x, C4x)= IC4xinv,(IC4yinv, C4yinv)=IC2y,(IC4yinv, C3alphainv)=IC4x,(IC4yinv, IC3gamma)= C2b,(IC4zinv, IC3gamma)=C2e,(IC3beta, IC3betainv)=Id,(IC2a, C4z)=IC2x,(IC2a, IC4z)=C2x,(IC2b, C3betainv)=IC4y,(IC2c, IC2z)=C4y,(IC2d, IC2d)=Id,(Id, C4xinv)= C4xinv,(C2y, C3deltainv)=C3gammainv,(C3gammainv, C3deltainv)=C3alpha,(IC4y, C3alpha)=IC4z,(IC3delta, IC2y)=C3gamma,(IC3deltainv, C3delta)=Is,(IC2b, C3beta) =IC4x,(IC2f, C4y)=IC3alpha,(C4z, IC3gammainv)=IC2d,(C2y, IC3beta)=IC3gamma,( C4zinv, IC2x)=IC2b,(C3alpha, C3gammainv)=C2x,(C3alphainv, Is)=IC3alphainv,( C3alphainv, IC3alpha)=Is,(C3gammainv, C3gammainv)=C3gamma,(C2d, IC3gamma)=IC4z, (C2e, IC4z)=IC3deltainv,(IC3alphainv, C3gammainv)=IC3delta,(IC2a, C4y)= IC3deltainv,(Id, C2e)=C2e,(C2x, C3delta)=C3beta,(C2x, C2f)=C2e,(C2y, IC4y)= IC4yinv,(C4yinv, IC4y)=Is,(C2d, C4x)=C3gammainv,(IC4z, C2z)=IC4zinv,(IC2y, IC4xinv)=C2e,(IC4zinv, IC3deltainv)=C2c,(IC3gamma, IC4y)=C4x,(IC3betainv, C2f)= IC4y,(IC2a, IC3gamma)=C2f,(IC2f, C2a)=IC3gammainv,(C4z, C2z)=C4zinv,(C4xinv, C2b)=C3betainv,(C3gammainv, IC3gammainv)=IC3gamma,(C3deltainv, IC4yinv)=IC2a,( Is, C2a)=IC2a,(IC2z, C2b)=IC2a,(IC4xinv, IC2a)=C3alphainv,(IC2c, IC3alphainv)= C2e,(Id, C3beta)=C3beta,(C2c, C3alpha)=C2b,(IC4y, IC2z)=C2c,(IC4z, C4x)= IC3delta,(IC3gamma, C2z)=IC3alpha,(IC3gammainv, C2z)=IC3deltainv,(IC2e, C2y)= IC4x,(C4z, C3alpha)=C2f,(C4xinv, IC3delta)=IC4y,(C2c, C4zinv)=C3delta,(IC2y, C2a)=IC4z,(IC2z, Is)=C2z,(IC3alpha, IC3alpha)=C3alphainv,(IC3gamma, IC2e)=C4z,( IC3delta, C3alpha)=IC3betainv,(IC3betainv, C3alphainv)=IC3delta,(IC3betainv, IC2x)=C3deltainv,(IC3gammainv, IC2y)=C3betainv,(IC2d, C2b)=IC3delta,(IC2f, C3gamma)=IC2c,(C4z, C2y)=C2b,(C4xinv, C3alpha)=C2d,(C4xinv, C2f)=C2z,(C3beta, C4zinv)=C2c,(C3alphainv, C2b)=C2e,(C2a, IC2e)=IC3beta,(IC2z, IC2c)=C4yinv,( IC4zinv, C2d)=IC3gammainv,(IC3alphainv, C2e)=IC2c,(IC3betainv, IC4zinv)=C4x,( IC2e, C4y)=IC3gamma,(C2d, IC4yinv)=IC2z,(IC2a, IC2a)=Id,(IC2f, IC4y)=C3alpha,( C2x, C4y)=C2c,(C4zinv, C2a)=C2x,(C3gamma, C3gamma)=C3gammainv,(C3gamma, IC4xinv )=IC4zinv,(C3betainv, IC2a)=IC2e,(IC2z, IC4z)=C4zinv,(IC2a, C2b)=IC2z,(C4zinv, C4z)=Id,(C2c, C3gamma)=C2a,(C2e, IC3alpha)=IC2c,(IC4z, C2a)=IC2y,(C2y, IC3deltainv)=IC3gammainv,(C4zinv, IC4z)=Is,(C3gamma, IC4z)=IC4yinv,(IC4y, C2f)= IC3betainv,(IC2y, C2c)=IC2d,(IC3beta, IC2y)=C3alpha,(IC3alphainv, IC2d)=C4z,( IC2b, C4zinv)=IC2x,(C3beta, C3beta)=C3betainv,(C3deltainv, C2x)=C3betainv,(IC2x , IC3alphainv)=C3deltainv,(IC2y, Id)=IC2y,(IC4xinv, C2b)=IC3betainv,(IC4yinv, C2z)=IC2d,(IC3deltainv, C3alphainv)=IC3gamma,(IC2a, C4x)=IC3alpha,(C2z, IC2c)= IC4yinv,(C4xinv, IC4y)=IC3beta,(C4zinv, C4xinv)=C3beta,(C3alpha, IC3gammainv)= IC2x,(IC4z, IC4yinv)=C3betainv,(IC4z, IC3deltainv)=C4yinv,(IC3gamma, C3delta)= IC3betainv,(IC3deltainv, C2f)=IC2d,(IC2e, Is)=C2e,(C2x, IC2z)=IC2y,(C2z, IC2e)= IC4x,(C3beta, C4y)=C2f,(C3gamma, C3betainv)=C2y,(C2a, IC3alpha)=IC4x,(C2b, Is)= IC2b,(C2b, IC3delta)=IC2f,(C2d, C4yinv)=C2z,(C2f, C3alpha)=C4y,(IC4z, IC2c)= C3deltainv,(IC4xinv, IC4zinv)=C3deltainv,(IC3delta, C3gamma)=IC3alphainv,( IC3alphainv, C2x)=IC3gammainv,(IC3alphainv, IC2c)=C2b,(IC2b, Id)=IC2b,(IC2c, IC3delta)=C4zinv,(C4y, C3deltainv)=C4xinv,(C4yinv, C4xinv)=C3deltainv,(C3beta, C4yinv)=C4xinv,(C2a, IC3alphainv)=IC4yinv,(Is, C4xinv)=IC4xinv,(IC4zinv, IC3alphainv)=C2d,(IC3beta, C4y)=IC2f,(C4zinv, C2z)=C4z,(C4zinv, IC2y)=IC2a,( C3alpha, C3alpha)=C3alphainv,(C3beta, IC4zinv)=IC2c,(IC2y, IC2f)=C4x,(IC2z, IC4zinv)=C4z,(IC3deltainv, IC2e)=C4y,(IC2b, IC4xinv)=C3gamma,(C4yinv, C4y)=Id,( C3alpha, C3gamma)=C3betainv,(C3delta, C2c)=C4xinv,(C3deltainv, IC3alpha)=IC2x,( C2c, C4xinv)=C3betainv,(C2d, Is)=IC2d,(IC4xinv, IC2y)=C2f,(IC3alpha, IC2f)= C4zinv,(IC3beta, C2x)=IC3gamma,(IC2a, IC3gammainv)=C2c,(C4z, IC3delta)=IC2e,( C3beta, C2d)=C2e,(C3beta, IC3beta)=IC3betainv,(C3gamma, IC4x)=IC2b,(IC4y, C4z)= IC3delta,(IC4xinv, C4zinv)=IC3deltainv,(IC3betainv, IC4y)=C4z,(IC2b, C2b)=Is,( IC2e, Id)=IC2e,(C4xinv, C2x)=C4x,(C3alpha, C4yinv)=C2f,(C2a, IC4xinv)=IC3delta, (C2e, C3alpha)=C2c,(IC4y, IC3delta)=C2a,(IC4y, IC3gammainv)=C2e,(IC4y, IC2e)= C3deltainv,(IC3gamma, IC3alpha)=C3deltainv,(IC3delta, C4zinv)=IC4y,(IC2b, C4x)= IC3beta,(C4xinv, IC3betainv)=IC4z,(C3alpha, C4z)=C2d,(C3gamma, IC3gamma)= IC3gammainv,(C3gammainv, C3delta)=C2y,(C3deltainv, IC4y)=IC4zinv,(IC2z, C4x)= IC2e,(IC2d, C2x)=IC4y,(IC2d, C4zinv)=IC3alpha,(IC2f, C3alpha)=IC4y,(C2x, C4xinv )=C4x,(C4xinv, IC2f)=IC2z,(C4zinv, IC4zinv)=IC2z,(C3alpha, IC2c)=IC2e,( C3deltainv, C2c)=C4z,(C2a, C2y)=C4zinv,(C2b, IC3gamma)=IC4xinv,(C2f, IC3deltainv)=IC2b,(IC2z, C3beta)=IC3alpha,(IC3gamma, IC2x)=C3beta,(IC3delta, C2d)=IC2f,(IC3gammainv, C4xinv)=IC2d,(IC3deltainv, C3deltainv)=IC3delta,(IC2e, C3alphainv)=IC2b,(C3alpha, C3delta)=C3gammainv,(C3alpha, IC3beta)=IC3deltainv,( C3alpha, IC3delta)=IC3gammainv,(C2b, C4y)=C3betainv,(IC2x, C3beta)=IC3delta,( IC3beta, C4yinv)=IC4xinv,(IC2f, C2b)=IC3deltainv,(C4z, IC2y)=IC2b,(C4xinv, C3beta)=C2c,(C3deltainv, C4z)=C4xinv,(C2e, C4zinv)=C3gammainv,(C2e, IC2e)=Is,( Is, C2c)=IC2c,(IC4z, Is)=C4z,(IC2x, C4zinv)=IC2a,(IC2y, IC2d)=C2c,(IC4zinv, IC2f)=C3alpha,(IC3alpha, C2b)=IC2c,(IC3alpha, IC4yinv)=C2f,(C4y, IC2x)=IC2d,( C4yinv, IC2b)=IC3beta,(Is, IC2a)=C2a,(IC4x, IC4yinv)=C3gamma,(IC3gammainv, IC3delta)=C2y,(IC2a, IC4x)=C3alpha,(IC2b, IC3beta)=C4x,(Id, C4yinv)=C4yinv,( C4zinv, C3gammainv)=C4y,(Is, C2f)=IC2f,(IC4xinv, C2y)=IC2f,(IC2f, C2c)=IC3gamma ,(Id, IC4zinv)=IC4zinv,(C4y, IC2b)=IC3gamma,(C2z, IC2z)=Is,(C3gammainv, C2c)= C2a,(C3deltainv, C4zinv)=C2e,(IC3betainv, C3delta)=IC2z,(IC2f, C2y)=IC4xinv,( C4y, IC4z)=IC3delta,(C4xinv, Id)=C4xinv,(C4xinv, IC3gammainv)=IC2b,(C3alpha, C3deltainv)=C2y,(C3gamma, IC3alpha)=IC3deltainv,(C3delta, C3deltainv)=Id,(C2c, Id)=C2c,(IC4y, Id)=IC4y,(IC2z, IC3gammainv)=C3alphainv,(IC4zinv, C4y)= IC3alphainv,(IC4zinv, C2z)=IC4z,(IC4zinv, IC4y)=C3alphainv,(IC3alphainv, C4x)= IC2d,(IC2a, IC2y)=C4zinv,(C4xinv, IC2d)=IC3gamma,(C2a, IC2y)=IC4zinv,(IC4x, IC4zinv)=C3alphainv,(IC3gamma, C3betainv)=IC2y,(IC2b, C2y)=IC4z,(IC2e, C3betainv)=IC2a,(C4x, C2d)=C3alpha,(C4y, C2x)=C2d,(C4y, C3gammainv)=C2e,(C4xinv , C2e)=C2y,(C3alphainv, C3alphainv)=C3alpha,(Is, C3alpha)=IC3alpha,(IC4x, Is)= C4x,(IC3delta, C2x)=IC3alpha,(IC3deltainv, C3gamma)=IC2y,(IC2c, C2y)=IC2d,(C4z , Id)=C4z,(C4z, IC2b)=IC2x,(C3alpha, C2a)=C4y,(C2b, IC2a)=IC2z,(C2c, IC3alpha)= IC2b,(C2f, IC2c)=IC3gamma,(IC4y, C3deltainv)=IC4xinv,(IC2x, IC3gamma)=C3alpha,( IC2d, C3gammainv)=IC4x,(IC2f, IC3delta)=C2d,(C4x, C3gamma)=C2d,(C4yinv, IC4yinv )=IC2y,(C2c, IC2f)=IC3gammainv,(C2f, C2f)=Id,(IC2y, C2y)=Is,(IC3gamma, C3gamma) =IC3gammainv,(IC3alphainv, IC3alphainv)=C3alpha,(C4x, C2y)=C2e,(C4x, C4zinv)= C3alphainv,(C4y, IC2y)=IC4yinv,(C4yinv, IC2y)=IC4y,(C3beta, IC2b)=IC4yinv,( C3gammainv, IC2e)=IC4yinv,(C2c, IC4z)=IC3beta,(IC4x, IC4z)=C3betainv,(IC2d, IC4x)=C3gammainv,(Id, C3alphainv)=C3alphainv,(C4x, IC2f)=IC2y,(C4y, IC2e)= IC3deltainv,(C3alpha, IC3betainv)=IC2z,(C3alphainv, C4zinv)=C2f,(C3betainv, C2d )=C2a,(C2d, C2y)=C2c,(C2f, C4xinv)=C2y,(C2f, IC2a)=IC3gammainv,(IC4z, C3betainv )=IC2c,(IC3alpha, IC2z)=C3gamma,(IC3delta, C4yinv)=IC4x,(IC3betainv, IC2b)= C4xinv,(IC2a, Is)=C2a,(IC2c, C2z)=IC4y,(C3gamma, C4z)=C4yinv,(IC2x, IC3gammainv )=C3betainv,(IC3alphainv, C2c)=IC2b,(IC3deltainv, C2e)=IC4y,(IC2e, C3beta)=IC2d ,(IC2e, C2e)=Is,(IC2f, C3delta)=IC2d,(IC2f, IC3deltainv)=C2b,(C4y, IC4y)=IC2y,( C3gammainv, C2a)=C2f,(C3gammainv, IC3betainv)=IC3delta,(IC4z, IC2a)=C2y,( IC3beta, IC4z)=C4y,(IC3gammainv, IC3alpha)=C2z,(IC3deltainv, IC4zinv)=C2e,(IC2b , IC4x)=C3beta,(C2x, C2b)=C4z,(C2z, Id)=C2z,(C2z, C3deltainv)=C3betainv,(C3beta , Is)=IC3beta,(C3gamma, Id)=C3gamma,(C2c, IC2x)=IC4yinv,(C2e, C4y)=C3gamma,( IC2x, C4y)=IC2c,(IC2y, IC3beta)=C3gamma,(IC3alpha, IC3alphainv)=Id,(IC3beta, C2z)=IC3delta,(IC3alphainv, C2z)=IC3betainv,(IC3alphainv, C4xinv)=IC4y,(IC2a, C2f)=IC3gamma,(IC2a, IC4y)=C3deltainv,(IC2a, IC2f)=C3gamma,(C3alpha, IC4yinv)= IC2f,(C3beta, C2z)=C3delta,(IC4x, C2z)=IC2f,(IC4z, C2e)=IC3gamma,(IC2x, C3alphainv)=IC3deltainv,(IC3alpha, C3beta)=IC3deltainv,(IC3beta, IC2x)=C3gamma, (IC2c, C2e)=IC3alphainv,(C4z, IC2e)=IC3gamma,(C2z, C3betainv)=C3deltainv,( C4yinv, IC4xinv)=IC3deltainv,(C3betainv, C4y)=C4z,(C2a, C3alpha)=C4x,( IC3gammainv, IC4z)=C2e,(IC3deltainv, IC2b)=C2f,(IC2b, C2f)=IC3delta,(IC2f, IC4zinv)=C3betainv,(C2y, IC2d)=IC2c,(C4zinv, C2e)=C3delta,(C2d, IC4y)=IC2x,( IC4z, C3delta)=IC2e,(IC4xinv, Is)=C4xinv,(IC3delta, C2b)=IC2d,(IC3betainv, IC2a )=C2e,(IC2a, IC2c)=C3gammainv,(C2b, IC2c)=IC3alphainv,(C2f, IC2f)=Is,(Is, IC3alphainv)=C3alphainv,(IC3gamma, IC3gammainv)=Id,(IC3betainv, C2c)=IC4zinv,( IC3gammainv, C3gammainv)=IC3gamma,(C2y, C2b)=C4zinv,(C2z, C4yinv)=C2c,(C4zinv, IC2d)=IC3gammainv,(C2e, IC4x)=IC2y,(C2f, IC4zinv)=IC3betainv,(IC3gamma, C2f)= IC2a,(IC2e, IC3beta)=C2d,(Id, C2b)=C2b,(C3gamma, IC2c)=IC2f,(C3betainv, C2e)= C2d,(C2f, C2x)=C2e,(IC2y, C3beta)=IC3gamma,(IC4xinv, C4z)=IC3gammainv,(IC2c, C4x)=IC3deltainv,(IC2e, C3deltainv)=IC4z,(Is, IC3betainv)=C3betainv,(IC3delta, IC2c)=C4xinv,(IC2c, IC3gamma)=C2a,(C4y, IC3deltainv)=IC4xinv,(C4z, C2d)= C3alphainv,(C2x, IC3delta)=IC3beta,(C4zinv, IC2f)=IC3alpha,(C3betainv, IC2c)= IC4zinv,(C3deltainv, C3gammainv)=C3beta,(IC2y, IC2y)=Id,(IC2z, C3alphainv)= IC3gammainv,(Id, Id)=Id,(C2y, Is)=IC2y,(C3alpha, IC4xinv)=IC2a,(C3delta, C2d)= C2f,(C3alphainv, IC3alphainv)=IC3alpha,(C2d, IC4z)=IC3gamma,(C2e, C3beta)=C2d,( C2f, IC4yinv)=IC3beta,(C2f, IC3beta)=IC4yinv,(IC4yinv, IC3betainv)=C2f,( IC3gamma, Is)=C3gamma,(IC3betainv, C2b)=IC4xinv,(C4z, IC4xinv)=IC3alpha,(C2x, IC3beta)=IC3delta,(C4yinv, C3alphainv)=C4x,(C4zinv, C3alpha)=C4xinv,(C2b, C3gammainv)=C4yinv,(IC2x, IC2c)=C4y,(IC2a, IC4xinv)=C3delta,(IC2e, IC3betainv)= C2a,(C4y, C2d)=C2z,(C3deltainv, IC2x)=IC3betainv,(C3deltainv, IC3delta)=Is,( IC4y, IC4zinv)=C3beta,(IC2y, C2z)=IC2x,(IC2y, C3alpha)=IC3delta,(IC3alpha, Is)= C3alpha,(IC3beta, C3delta)=IC3alphainv,(C2z, C3gamma)=C3delta,(C3delta, IC2d)= IC2f,(C3deltainv, IC2b)=IC2f,(C2d, C3alphainv)=C4xinv,(IC2x, IC3beta)=C3delta,( C4z, IC4x)=IC3delta,(C2z, C2f)=C4xinv,(C2z, IC3gammainv)=IC3alphainv,(C3gamma, C3gammainv)=Id,(C2f, C3deltainv)=C2b,(C2f, C2a)=C3gammainv,(IC2x, C2a)=IC4zinv, (IC4zinv, C4x)=IC3gamma,(IC4zinv, C2y)=IC2a,(IC3gamma, C3alphainv)=IC2x,( IC3gammainv, C3gamma)=Is,(IC2a, C2e)=IC3beta,(Id, IC3beta)=IC3beta,(C4x, C2f)= C2y,(C4yinv, C3gamma)=C2b,(C4zinv, IC3deltainv)=IC2c,(C3gamma, C2f)=C2a,( C3alphainv, C4xinv)=C4y,(C3betainv, IC3alphainv)=IC3delta,(C2b, C4yinv)= C3gammainv,(C2f, IC3gamma)=IC2c,(IC3alpha, C3betainv)=IC2z,(IC3betainv, IC2y)= C3gammainv,(IC3betainv, IC3deltainv)=C3gamma,(IC2d, C2z)=IC4yinv,(C3alpha, IC2b )=IC2c,(C3betainv, IC4y)=IC4z,(C2a, IC3betainv)=IC2d,(IC2z, C2z)=Is,(IC3gamma, Id)=IC3gamma,(IC3delta, IC4y)=C2e,(IC2a, C2z)=IC2b,(C2x, C4yinv)=C2d,(C2y, IC3gammainv)=IC3deltainv,(IC4x, IC3beta)=C4y,(IC4y, C3beta)=IC2b,(IC4z, C2c)= IC3deltainv,(IC4z, IC3beta)=C4xinv,(IC4yinv, IC2x)=C2c,(IC2d, C4yinv)=IC2z,( C3delta, IC2f)=IC2b,(C2d, IC3delta)=IC2b,(IC3beta, IC3deltainv)=C2x,(IC3gamma, C4z)=IC4yinv,(IC3betainv, IC3alpha)=C2y,(IC2e, C2b)=IC3alphainv,(C2x, C3deltainv)=C3alphainv,(C2y, C3gamma)=C3beta,(C4xinv, IC4x)=Is,(C3deltainv, C4y )=C4zinv,(IC4x, IC2b)=C3gammainv,(C4x, C2c)=C3beta,(C4z, C4zinv)=Id,(C2x, IC4xinv)=IC4x,(C2z, IC2x)=IC2y,(C3gamma, C2z)=C3alpha,(C3gamma, C3beta)= C3alphainv,(C3gammainv, C2d)=C4zinv,(Is, C4y)=IC4y,(IC2b, C3gammainv)=IC4yinv,( IC2e, C4zinv)=IC3gammainv,(C4y, C3gamma)=C4zinv,(C2y, IC3delta)=IC3alpha,(C2z, IC4xinv)=IC2f,(C2z, IC3alpha)=IC3beta,(C4yinv, IC3beta)=IC4zinv,(C3betainv, C4yinv)=C2b,(C2f, IC3delta)=IC2d,(IC4x, IC2x)=C4xinv,(IC4z, IC4y)=C3gammainv,( IC3alpha, IC2y)=C3beta,(IC3betainv, IC4x)=C2c,(IC2c, IC2x)=C4yinv,(C3delta, C4z )=C2c,(Is, IC2z)=C2z,(IC4xinv, C4yinv)=IC3alpha,(IC3beta, IC2a)=C2d,(IC3beta, IC2c)=C4x,(IC3deltainv, C2d)=IC2b,(IC2d, IC2a)=C3beta,(C3gamma, IC4y)=IC4x,( C3delta, IC3alphainv)=IC2y,(C2e, C4yinv)=C3delta,(Is, C2d)=IC2d,(IC4zinv, C3gammainv)=IC4y,(IC3gammainv, IC2c)=C2a,(IC2f, IC2x)=C2e,(C3beta, IC3gammainv) =IC2y,(Is, C3beta)=IC3beta,(IC4x, C4y)=IC3delta,(IC4xinv, IC2z)=C2e,(IC4zinv, C2c)=IC3betainv,(IC4zinv, IC2x)=C2b,(IC3alphainv, C3gamma)=IC2z,(IC3gammainv, C3alphainv)=IC3beta,(IC2d, IC4yinv)=C2z,(C4yinv, IC4z)=IC3alpha,(C3alphainv, IC4y)=IC2a,(C3betainv, C3beta)=Id,(IC4xinv, C3gamma)=IC4yinv,(IC2b, C2z)=IC2a,( C4z, IC4yinv)=IC3betainv,(C3beta, IC4z)=IC4y,(C2e, IC4zinv)=IC3gammainv,(IC2b, IC3gammainv)=C4yinv,(IC2c, IC2b)=C3alpha,(C2b, C2a)=C2z,(C2e, C2c)=C3alpha,( IC4xinv, IC3gammainv)=C2b,(IC4yinv, C2c)=IC2z,(IC3delta, C4y)=IC2e,(IC2c, Is)= C2c,(C4x, IC3betainv)=IC2b,(C4xinv, C3gamma)=C4yinv,(C3gamma, C4y)=C4x,(C3gamma , Is)=IC3gamma,(C2a, C2e)=C3beta,(C2b, C2y)=C4z,(C2b, C2c)=C3alphainv,(C2c, C3beta)=C4z,(IC4z, IC3gamma)=C4x,(IC3alpha, IC3gammainv)=C2x,(IC3gammainv, C3delta)=IC2y,(IC2a, C3betainv)=IC2d,(IC2c, C4y)=IC2z,(IC2c, C4yinv)=IC2x,(C4z , C3delta)=C2e,(C2z, C4z)=C4zinv,(C3betainv, IC2e)=IC2d,(IC4z, C4z)=IC2z,(IC2y , C2e)=IC4xinv,(IC4zinv, C2e)=IC3delta,(IC3betainv, IC4z)=C2f,(IC3gammainv, Is) =C3gammainv,(IC3deltainv, C4x)=IC4yinv,(IC2f, IC2e)=C2x,(C4x, C3gammainv)=C4z,( C2x, C2e)=C2f,(C3deltainv, IC2y)=IC3alphainv,(C2a, Is)=IC2a,(C2e, C2f)=C2x,( IC4xinv, C3gammainv)=IC2b,(IC3betainv, C3gammainv)=IC3alpha,(C4zinv, C3betainv) =C4yinv,(C3gamma, C3alpha)=C3deltainv,(C3delta, Is)=IC3delta,(C3betainv, IC2b)= IC4xinv,(C2d, IC2f)=IC3deltainv,(Is, IC2d)=C2d,(IC4z, IC4z)=C2z,(IC4xinv, C2x)= IC4x,(IC3beta, C2a)=IC2d,(IC3alphainv, IC3gamma)=C2z,(IC2b, C4y)=IC3betainv,( IC2f, IC4xinv)=C2y,(C4x, IC4x)=IC2x,(C3delta, IC2e)=IC4zinv,(C2b, IC2d)= IC3deltainv,(IC4z, IC2d)=C3alphainv,(IC2y, C4y)=IC4yinv,(IC4yinv, IC2z)=C2d,( C2x, IC2c)=IC4y,(C3alphainv, C3deltainv)=C3beta,(C2a, C2x)=C4z,(IC4x, IC2y)=C2e ,(IC2y, IC3gamma)=C3beta,(IC3alpha, C4xinv)=IC2a,(IC2d, IC3deltainv)=C2f,(IC2e , IC2y)=C4x,(C2x, IC2y)=IC2z,(C2z, IC2d)=IC4y,(C4xinv, C2d)=C3gamma,(C2b, IC4xinv)=IC3gamma,(C2f, IC4z)=IC3alphainv,(Is, C2b)=IC2b,(IC4z, C2d)= IC3alphainv,(IC2y, C2f)=IC4x,(IC4zinv, IC3alpha)=C4xinv,(C2x, C2d)=C4yinv,( C3betainv, C3deltainv)=C3gamma,(C2e, C2z)=C4xinv,(IC2z, IC2z)=Id,(IC4zinv, IC2e )=C3delta,(IC3beta, C3deltainv)=IC2x,(IC3delta, IC2x)=C3alpha,(IC3alphainv, C4y )=IC2a,(IC3deltainv, IC3beta)=C2z,(C2y, IC2a)=IC4z,(C4xinv, IC4zinv)= IC3deltainv,(C4zinv, IC3delta)=IC4x,(IC4y, C4xinv)=IC3gammainv,(IC3beta, C2e)= IC2a,(IC3delta, IC3alphainv)=C2y,(IC3alphainv, C2d)=IC4z,(IC3betainv, C2x)= IC3deltainv,(IC3gammainv, C2a)=IC2f,(IC2c, C2c)=Is,(C2y, C3gammainv)=C3deltainv ,(C4zinv, IC4x)=IC3gamma,(C3gamma, IC3betainv)=IC2y,(C2a, IC3deltainv)=IC4y,( IC3alpha, C4x)=IC4z,(IC3gamma, C2a)=IC2c,(IC2b, C2a)=IC2z,(C4x, IC4z)= IC3betainv,(C3alpha, Is)=IC3alpha,(C3beta, C2e)=C2a,(C3gamma, C3delta)= C3betainv,(C3delta, IC2x)=IC3alpha,(C3delta, IC3alpha)=IC3betainv,(C3deltainv, IC4zinv)=IC2e,(C2a, IC2d)=IC3betainv,(Is, C4x)=IC4x,(Is, C3alphainv)= IC3alphainv,(IC4zinv, C3alphainv)=IC2d,(IC3deltainv, IC4xinv)=C2c,(C4x, IC3gamma)=IC2d,(C3delta, C2x)=C3alpha,(IC4x, IC3alpha)=C4yinv,(IC4y, C2c)=IC2x, (IC2z, C4zinv)=IC4z,(IC3alphainv, C3deltainv)=IC3beta,(IC3deltainv, IC4z)= C4xinv,(IC2b, IC4z)=C2y,(IC2b, IC3delta)=C2f,(C2z, IC2a)=IC2b,(C4xinv, C4x)=Id, (C4yinv, C2y)=C4y,(C4yinv, IC3alpha)=IC2a,(C3alpha, IC4z)=IC2d,(C3betainv, IC3alpha)=IC2y,(C3deltainv, C3gamma)=C2y,(C3deltainv, IC3gamma)=IC2y,(C2c, C2f) =C3gammainv,(C2d, C4y)=C2x,(C2e, IC4y)=IC3gamma,(C2f, IC4x)=IC2z,(C3gamma, C4zinv)=C2d,(IC2x, C3alpha)=IC3gamma,(IC4xinv, C2f)=IC2z,(IC3alpha, IC2c)=C2e,( IC3gammainv, C2b)=IC4x,(IC3deltainv, C2y)=IC3alphainv,(C4y, C4z)=C3delta,(C4y, C2c)=C2x,(C2y, IC2y)=Is,(C2b, C3beta)=C4x,(C2f, IC3gammainv)=IC2a,(IC4x, C4x)= IC2x,(IC4y, C2z)=IC2c,(IC4yinv, IC4y)=Id,(IC4zinv, IC3delta)=C4x,(IC3alpha, C2e )=IC2b,(IC2e, C4z)=IC3deltainv,(C4xinv, C3gammainv)=C2b,(C3gammainv, IC2b)=IC4x ,(C3deltainv, IC3alphainv)=IC3gamma,(C2a, C3delta)=C4xinv,(IC4y, C3gammainv)= IC2e,(IC4zinv, C4z)=Is,(IC3alpha, IC3betainv)=C2z,(IC3gamma, IC4z)=C4yinv,( IC3betainv, C3beta)=Is,(IC3deltainv, IC3betainv)=C3alpha,(IC2d, C2c)=IC2y,(IC2e , C3gammainv)=IC4zinv,(C4x, IC2z)=IC2f,(C2x, IC3alphainv)=IC3deltainv,(C3delta , Id)=C3delta,(C2d, IC2z)=IC4yinv,(C2f, C2e)=C2x,(Is, C3delta)=IC3delta,(IC2x, IC2z)=C2y,(IC3alphainv, IC4xinv)=C4y,(IC3gammainv, C4y)=IC2b,(IC3gammainv, IC4x )=C4y,(IC2d, C3betainv)=IC2e,(IC2f, C4zinv)=IC3betainv,(C3alpha, IC3deltainv)= IC2y,(C2c, C2d)=C2y,(IC4x, IC3deltainv)=C4zinv,(IC4y, IC2f)=C3betainv,(IC2x, IC2f)=C2e,(IC3alpha, C2c)=IC2e,(IC3deltainv, IC2x)=C3betainv,(IC2c, C3deltainv) =IC4x,(C4y, C2a)=C3alpha,(C4y, IC3alpha)=IC4z,(C4xinv, C2y)=C2f,(C4xinv, C4zinv )=C3deltainv,(C4yinv, IC3gamma)=IC2b,(C4yinv, IC2d)=IC2x,(C3delta, IC4y)=IC2e,( C2a, IC3beta)=IC2e,(C2d, IC3betainv)=IC2e,(Is, Id)=Is,(IC4z, IC2f)=C3beta,( IC4zinv, Id)=IC4zinv,(IC3gamma, IC2z)=C3alpha,(IC2e, C2d)=IC3beta,(C4z, C3beta) =C4xinv,(C4yinv, C4yinv)=C2y,(C3alphainv, C4yinv)=C4zinv,(C3alphainv, IC4z)= IC4x,(C2c, IC3beta)=IC4z,(IC3alpha, IC4x)=C4z,(IC3alphainv, C2f)=IC4yinv,( C4yinv, IC2x)=IC2c,(C3alpha, C2z)=C3gamma,(C3gamma, IC2x)=IC3beta,(IC4z, IC3betainv)=C2c,(IC2z, C2x)=IC2y,(IC3gammainv, C3deltainv)=IC3alpha,(IC2d, IC4y )=C2x,(C4x, C2e)=C2z,(C3gammainv, IC4yinv)=IC4z,(C2d, IC4x)=IC3gammainv,(C2e, IC3gammainv)=IC4zinv,(IC3delta, C2f)=IC2b,(Id, C2a)=C2a,(C3alpha, C4y)=C4xinv,( C3deltainv, C2z)=C3gammainv,(C2e, C2b)=C3alphainv,(C2e, IC3delta)=IC4yinv,(IC4z , IC3alpha)=C2f,(IC4yinv, C4y)=Is,(IC3deltainv, IC2c)=C4z,(IC2c, IC3betainv)= C4xinv,(IC2c, IC2a)=C3gamma,(C4zinv, Is)=IC4zinv,(C3alphainv, Id)=C3alphainv,( C2d, C4xinv)=C3alphainv,(IC4y, C4x)=IC3alphainv,(IC3alpha, IC2d)=C4x,(IC3beta, IC4x)=C4zinv,(IC3gamma, C4xinv)=IC4zinv,(IC3betainv, IC3gamma)=C2x,(IC3gammainv , C2c)=IC2a,(IC2d, IC2z)=C4yinv,(C4x, C2x)=C4xinv,(C4y, C3alpha)=C4z,(C3beta, IC3gamma)=IC3deltainv,(C2a, C3alphainv)=C4yinv,(C2a, C2c)=C3gammainv,(C2b, IC4z )=IC2y,(Is, C3deltainv)=IC3deltainv,(IC4yinv, C2f)=IC3alphainv,(IC3beta, IC4y)= C2f,(IC3gamma, C4zinv)=IC2d,(IC3alphainv, C3alpha)=Is,(IC3betainv, C4x)=IC2c,( IC2c, C2b)=IC3alpha,(IC2f, IC2b)=C3deltainv,(C4x, IC4zinv)=IC3alphainv,( C3alphainv, C3alpha)=Id,(C2d, IC2e)=IC3betainv,(IC3alpha, IC4y)=C4xinv,( IC3alphainv, Is)=C3alphainv,(IC2d, C3delta)=IC2b,(IC2e, C2z)=IC4xinv,(IC2f, C4x )=IC2z,(Id, C4x)=C4x,(C2y, IC2z)=IC2x,(C3beta, IC3betainv)=Is,(C3betainv, C4x)= C2c,(C3deltainv, C4xinv)=C2c,(C3deltainv, IC2e)=IC4y,(IC4z, C3beta)=IC4xinv,( IC3delta, C3delta)=IC3deltainv,(C3beta, C3gamma)=C3deltainv,(C3gamma, IC2f)= IC2a,(C3gammainv, IC4x)=IC4y,(C2a, C4zinv)=C2y,(C2c, IC3gammainv)=IC2f,(C2f, C2z)=C4x,(IC2x, C2x)=Is,(IC2y, C4x)=IC2f,(IC4yinv, IC4x)=C3betainv,(IC4zinv, C4zinv)=IC2z,(IC2b, IC3deltainv)=C2d,(C4z, C4yinv)=C3betainv,(C4z, IC2a)=IC2y,( C2b, C2d)=C3deltainv,(C2d, C3beta)=C2a,(IC3delta, IC4yinv)=C4x,(IC2c, Id)=IC2c, (IC2c, IC3beta)=C4z,(IC2e, C4yinv)=IC3delta,(C3beta, C3alpha)=C3gammainv,( C3alphainv, C3gammainv)=C3delta,(C2a, IC3gammainv)=IC2c,(C2c, C4x)=C3deltainv,( C2f, IC2d)=IC3delta,(IC4yinv, C3beta)=IC4zinv,(IC4zinv, IC3betainv)=C4yinv,( IC2d, IC3betainv)=C2e,(C2b, IC3alpha)=IC2e,(IC3alpha, IC3gamma)=C3betainv,( IC3delta, IC2b)=C2d,(IC3alphainv, IC3alpha)=Id,(IC2a, C2d)=IC3betainv,(IC2f, C3alphainv)=IC4z,(Id, IC3betainv)=IC3betainv,(C2z, IC3alphainv)=IC3gammainv,( C4zinv, IC3gamma)=IC2e,(C3betainv, C4z)=C2f,(C3deltainv, C2b)=C2f,(IC4x, IC2a)= C3deltainv,(IC4yinv, IC2e)=C3gammainv,(IC3gammainv, IC2f)=C2c,(IC2e, IC4zinv)= C3gammainv,(C2z, C2e)=C4x,(C4yinv, IC4x)=IC3betainv,(C3delta, IC2b)=IC2d,(C2a, C4yinv)=C3alphainv,(C2a, C2a)=Id,(IC2y, C3deltainv)=IC3gammainv,(IC2b, C4yinv)= IC3gammainv,(C2z, IC4yinv)=IC2c,(C3gammainv, IC3beta)=IC2x,(C3gammainv, IC2a)= IC2f,(C2b, IC3alphainv)=IC2c,(C2e, C4xinv)=C2z,(IC4y, C2d)=IC2z,(IC4z, C2b)= IC2x,(IC4z, IC4zinv)=Id,(IC2y, C4yinv)=IC4y,(IC2z, IC4yinv)=C2c,(C4x, IC3alphainv)=IC2a,(C4zinv, IC2b)=IC2y,(C3gamma, C2d)=C4xinv,(C2b, C2b)=Id,(C2c , C4yinv)=C2x,(IC4xinv, C3alphainv)=IC4zinv,(IC4yinv, IC2d)=C2x,(IC3delta, IC3gamma)=C3alphainv,(IC3delta, IC3deltainv)=Id,(IC3deltainv, C4zinv)=IC2e,( IC2b, IC2b)=Id,(IC2d, IC2b)=C3delta,(C3gamma, IC3gammainv)=Is,(C3delta, C4zinv) =C4y,(C3alphainv, IC4xinv)=IC4y,(C3betainv, C2x)=C3deltainv,(C3gammainv, C4xinv )=C2d,(C2d, C2x)=C4y,(IC4z, IC3delta)=C2e,(IC2y, IC2x)=C2z,(IC4yinv, IC2f)= C3alphainv,(IC2e, C3gamma)=IC4y,(C4y, C3delta)=C2a,(C4z, IC3alphainv)=IC4y,( C3delta, IC3beta)=IC3gammainv,(C3gammainv, Id)=C3gammainv,(C2a, Id)=C2a,( IC3alpha, C3delta)=IC3gammainv,(IC3gamma, IC3delta)=C3betainv,(IC2d, Id)=IC2d,( C3alphainv, C2c)=C2b,(C2b, IC2z)=IC2a,(IC2y, IC4z)=C2a,(IC3gammainv, IC3gamma)= Id,(IC3gammainv, IC3gammainv)=C3gamma,(IC3deltainv, IC4y)=C4zinv,(C4y, IC2c)= IC2x,(C4yinv, IC3betainv)=IC2f,(C3betainv, C2y)=C3gammainv,(IC4z, C4y)= IC3gammainv,(IC2x, IC2d)=C4yinv,(IC4zinv, IC4zinv)=C2z,(IC3delta, IC4z)=C2c,( IC3deltainv, C3betainv)=IC3alpha,(IC3deltainv, IC2y)=C3alphainv,(C4z, IC3gamma) =IC4x,(C2y, IC3alpha)=IC3delta,(C4yinv, IC2c)=IC2z,(C3alpha, IC3gamma)= IC3betainv,(C3delta, IC4xinv)=IC4z,(C3delta, IC3deltainv)=Is,(C2c, IC4zinv)= IC3delta,(C2f, Is)=IC2f,(IC3alphainv, C2b)=IC2e,(IC3alphainv, IC4zinv)=C2f,( IC2d, IC3alpha)=C4zinv,(C4y, IC3gamma)=IC4zinv,(C3deltainv, C2e)=C4y,( C3deltainv, IC2d)=IC2b,(C2a, C4x)=C3alpha,(IC4xinv, IC4x)=Id,(IC3beta, IC3alpha )=C3gammainv,(IC3gamma, IC4x)=C2b,(IC2a, IC3alphainv)=C4yinv,(IC2c, IC4yinv)= C2x,(IC2d, C3alpha)=IC4zinv,(C4z, IC2x)=IC2a,(C4yinv, IC3gammainv)=IC4xinv,( C3alphainv, C3beta)=C2y,(IC2x, IC4z)=C2b,(IC3alpha, C3deltainv)=IC2y,( IC3alphainv, IC3delta)=C2x,(IC2a, C2c)=IC3gammainv,(IC2c, C3gamma)=IC2a,(IC2c, C2a)=IC3gamma,(C2z, C4y)=C2d,(C4yinv, IC2f)=IC3alphainv,(C2a, C2z)=C2b,(C2z, IC3deltainv)=IC3betainv,(C3alpha, C4xinv)=C2a,(C3beta, IC2f)=IC4z,(C3gammainv, IC2d)=IC4zinv,(C2d, C2e)=C3betainv,(C2d, IC4xinv)=IC3alphainv,(Is, IC2x)=C2x,( IC3alpha, C4z)=IC2d,(IC2c, C2f)=IC3gammainv,(Id, C2c)=C2c,(C4y, C4y)=C2y,( C4zinv, C4zinv)=C2z,(C3alpha, IC2z)=IC3gamma,(C3deltainv, IC4xinv)=IC2c,( C3deltainv, IC3beta)=IC2z,(C2b, IC3gammainv)=IC4yinv,(C2e, IC2x)=IC2f,(IC2z, C2e)=IC4x,(IC4yinv, IC3beta)=C4zinv,(IC3gammainv, C4zinv)=IC4xinv,(Id, IC4yinv) =IC4yinv,(C4xinv, C3alphainv)=C4zinv,(C4zinv, IC3gammainv)=IC4y,(C3beta, IC2y)= IC3alpha,(C3beta, IC4yinv)=IC4xinv,(C3delta, IC2y)=IC3gamma,(IC4xinv, C4y)= IC3beta,(IC3deltainv, IC3gamma)=C2y,(IC2a, IC3alpha)=C4x,(IC2e, IC3alphainv)= C2b,(Id, IC3gamma)=IC3gamma,(IC4x, IC3gammainv)=C4z,(IC4z, IC2x)=C2a,(IC4yinv, IC2y)=C4y,(IC3beta, C3gammainv)=IC2y,(IC3gamma, IC2f)=C2a,(IC3gammainv, IC4zinv )=C4xinv,(IC3deltainv, IC3gammainv)=C3beta,(Id, IC2x)=IC2x,(C4x, IC2e)=IC2z,( C2x, IC4z)=IC2b,(C4xinv, C2z)=C2e,(C3gamma, IC2z)=IC3alpha,(C3delta, IC2a)= IC4yinv,(IC2y, IC3deltainv)=C3gammainv,(IC4zinv, C3betainv)=IC4yinv,(IC4zinv, IC4xinv)=C3beta,(IC3betainv, C3alpha)=IC2y,(IC2b, IC3betainv)=C4y,(IC2c, IC2e)= C3alphainv,(C2z, C3gammainv)=C3alphainv,(C3deltainv, C2d)=C2b,(C2b, C3betainv)= C4y,(IC4x, IC4y)=C3delta,(IC4z, C3gammainv)=IC2d,(IC2c, IC3alpha)=C2b,(IC2e, C4xinv)=IC2z,(C2x, C2a)=C4zinv,(IC2x, IC4zinv)=C2a,(IC2y, IC2c)=C2d,(IC2z, IC3beta)=C3alpha,(IC2z, IC3gamma)=C3delta,(IC4xinv, C2d)=IC3gamma,(IC4yinv, IC3alpha)=C2a,(IC3gamma, C3gammainv)=Is,(IC3delta, IC3alpha)=C3betainv,( IC3gammainv, IC2a)=C2f,(IC2a, IC3betainv)=C2d,(C2z, IC3betainv)=IC3deltainv,( C3gamma, C4x)=C2b,(C3delta, C2e)=C4zinv,(C3alphainv, IC2e)=IC2c,(C3gammainv, C2x)=C3alphainv,(C2a, C3gamma)=C2f,(C2c, IC2c)=Is,(IC4x, C2x)=IC4xinv,(IC2z, IC4x)=C2e,(IC2z, IC2e)=C4x,(IC4yinv, C4z)=IC3alpha,(IC4yinv, IC4zinv)=C3gamma,( IC4zinv, C3alpha)=IC4xinv,(IC3beta, C2y)=IC3alpha,(IC3beta, C3alphainv)=IC2z,( IC3gammainv, C2f)=IC2c,(IC2d, IC3gammainv)=C4x,(IC2f, C2d)=IC3delta,(C4x, C4y)= C3delta,(C2x, C2x)=Id,(C2y, IC4yinv)=IC4y,(C3alpha, C2b)=C2c,(C3betainv, C2b)= C4xinv,(IC2z, C2c)=IC4yinv,(IC4xinv, IC2f)=C2z,(IC2e, C2a)=IC3betainv,(Id, C3deltainv)=C3deltainv,(C3alphainv, IC2x)=IC3gammainv,(C2a, C2d)=C3betainv,(C2a , IC4yinv)=IC3alphainv,(C2e, IC3betainv)=IC2a,(IC4z, Id)=IC4z,(C3beta, Id)= C3beta,(C3betainv, C3gammainv)=C3alpha,(C3gammainv, C2y)=C3betainv,(C4x, C4yinv )=C3gamma,(C4xinv, IC3gamma)=IC4yinv,(C3gammainv, Is)=IC3gammainv,(C2b, C2f)= C3delta,(C2f, C2d)=C3delta,(Is, IC3gammainv)=C3gammainv,(IC3alpha, IC2e)=C2b,( IC3alphainv, IC2x)=C3gammainv,(IC3deltainv, C3alpha)=IC2x,(C4z, C3deltainv)= C4yinv,(C2e, C3gammainv)=C4zinv,(IC4x, Id)=IC4x,(IC4y, C3gamma)=IC4zinv,( IC3gammainv, IC3alphainv)=C3beta,(IC2d, C2y)=IC2c,(C4xinv, IC4yinv)=IC3alpha,( C4yinv, IC2a)=IC3delta,(C3delta, C3gammainv)=C2z,(IC4x, C4yinv)=IC3gamma,( IC3alpha, IC2x)=C3delta,(IC3betainv, C2a)=IC2e,(C2x, C3gamma)=C3alpha,(C2x, C3betainv)=C3gammainv,(C4xinv, IC2e)=IC2y,(C4zinv, IC3alpha)=IC4xinv,(C2d, Id)= C2d,(C2f, C3beta)=C4yinv,(IC2z, IC2b)=C2a,(IC3gammainv, IC3deltainv)=C3alpha,( IC2b, IC3gamma)=C4xinv,(IC2f, IC4yinv)=C3beta,(C4y, C2z)=C2c,(C4z, C4z)=C2z,( C4z, C2x)=C2a,(C3alphainv, IC4x)=IC2d,(C3deltainv, C3deltainv)=C3delta,(Is, IC4y)=C4y,(IC4z, C3gamma)=IC4x,(IC3delta, IC4xinv)=C4z,(IC2c, C3gammainv)=IC2f, (C2y, IC4zinv)=IC2b,(C2y, IC3alphainv)=IC3betainv,(C3gammainv, C4zinv)=C4xinv,( C3gammainv, C3alphainv)=C3beta,(C2c, IC4y)=IC2z,(IC2z, C4yinv)=IC2c,(IC4yinv, C4xinv)=IC3deltainv,(IC3gamma, C3alpha)=IC3deltainv,(IC2d, C3gamma)=IC4z,(IC2f , IC4x)=C2z,(C4zinv, C4x)=C3gamma,(C3delta, C2f)=C2b,(C3delta, IC2z)=IC3beta,( C2b, IC4y)=IC3betainv,(C2e, IC3gamma)=IC4y,(Is, IC2e)=C2e,(IC2x, C2d)=IC4yinv,( IC4xinv, C2a)=IC3alphainv,(IC2a, C3deltainv)=IC4y,(C4z, C2e)=C3gamma,(C2y, C3beta)=C3gamma,(C4xinv, Is)=IC4xinv,(C4yinv, C3beta)=C4zinv,(C4zinv, IC2z)= IC4z,(C3alphainv, C3betainv)=C3gamma,(C3alphainv, IC2c)=IC2b,(C2b, C2z)=C2a,( C2f, C2c)=C3gamma,(IC4y, IC2x)=C2d,(IC2z, C2d)=IC4y,(IC3alphainv, IC4yinv)= C4zinv,(IC3gammainv, IC3betainv)=C3delta,(C3betainv, IC2f)=IC4y,(C2c, IC3delta) =IC4zinv,(IC2y, C4zinv)=IC2b,(IC4yinv, C2b)=IC3beta,(IC3beta, C4z)=IC4y,( IC3deltainv, IC4x)=C4yinv,(IC2e, IC3alpha)=C2c,(C4x, C2a)=C3deltainv,(C2z, C2c) =C4yinv,(C3delta, IC4yinv)=IC4x,(C2b, IC2y)=IC4z,(C2d, C2b)=C3delta,(IC4zinv, Is)=C4zinv,(IC3gamma, C2x)=IC3beta,(IC3delta, IC3gammainv)=C2z,(IC3alphainv, IC3betainv)=C3gamma,(IC3deltainv, IC3alpha)=C2x,(IC3deltainv, IC3alphainv)= C3gamma,(C4y, IC4x)=IC3alphainv,(C4yinv, C3betainv)=C2f,(C3beta, IC2z)=IC3delta ,(Is, IC3beta)=C3beta,(IC4x, C2a)=IC3deltainv,(IC4x, IC3betainv)=C2b,(IC4y, C4y )=IC2y,(IC4yinv, Id)=IC4yinv,(IC3alpha, IC4z)=C2d,(IC2b, IC2y)=C4z,(IC2d, C2a)= IC3beta,(IC2d, C2e)=IC3betainv,(IC2f, IC3alphainv)=C4z,(C4x, IC3alpha)=IC4yinv, (C2z, IC3delta)=IC3gamma,(C4xinv, C3betainv)=C4z,(C3beta, C3gammainv)=C2y,( C3betainv, C3alphainv)=C3delta,(C3betainv, IC2x)=IC3deltainv,(C2b, C4x)=C3beta, (C2d, C2d)=Id,(C2e, IC3deltainv)=IC4z,(IC4y, Is)=C4y,(IC3alpha, IC3deltainv)= C2y,(IC2e, C3delta)=IC4yinv,(C4z, IC3betainv)=IC2c,(C4yinv, IC3alphainv)=IC4x,( C3alpha, C2c)=C2e,(C2e, C2y)=C4x,(IC3gammainv, C4z)=IC2e,(IC2f, C2z)=IC4x,(C4x , IC3deltainv)=IC4zinv,(C3beta, C2y)=C3alpha,(C3beta, IC2a)=IC2d,(C2e, IC2f)= IC2x,(Is, IC2y)=C2y,(IC4xinv, IC3alpha)=C2d,(IC2f, IC2z)=C4x,(C2y, C4yinv)=C4y, (C2z, C3alpha)=C3beta,(C3betainv, C3delta)=C2z,(C3gammainv, IC4z)=IC2e,(C2b, IC2x)=IC4zinv,(IC4z, C2f)=IC3beta,(IC4z, IC3gammainv)=C2d,(IC2z, C3alpha)= IC3beta,(IC4zinv, C4yinv)=IC3deltainv,(IC3alpha, C2f)=IC4zinv,(IC3beta, IC2e)= C2a,(IC3alphainv, IC4y)=C2a,(IC2a, IC2b)=C2z,(C4x, IC2d)=IC3alpha,(C3beta, IC3delta)=IC3alphainv,(C3delta, C3gamma)=C3alphainv,(C3alphainv, IC3beta)=IC2y, (C2f, IC4xinv)=IC2y,(IC4x, IC2z)=C2f,(IC3beta, IC4yinv)=C4xinv,(IC3betainv, C3deltainv)=IC3gamma,(IC2b, C3delta)=IC2f,(IC2d, C3beta)=IC2a,(C2x, C3alphainv) =C3deltainv,(C4zinv, C2b)=C2y,(C3beta, C3deltainv)=C2x,(C3betainv, C2z)= C3alphainv,(C3gammainv, IC2f)=IC2c,(IC2z, IC2d)=C4y,(IC4xinv, IC3alphainv)= C4zinv,(IC3alphainv, C3alphainv)=IC3alpha,(IC2d, C4xinv)=IC3alphainv,(C4y, IC4xinv)=IC3gammainv,(C4yinv, C3delta)=C4z,(C4zinv, IC3alphainv)=IC2d,(C2d, C2a )=C3beta,(IC2a, C4zinv)=IC2y,(IC2a, IC4yinv)=C3alphainv,(C4z, IC2c)=IC3deltainv ,(C4yinv, C2e)=C3gammainv,(C3alphainv, C3delta)=C2x,(C3gammainv, C3alpha)=C2z,( C3deltainv, Is)=IC3deltainv,(C2a, IC2f)=IC3gamma,(IC3gamma, C2c)=IC2f,( IC3deltainv, C3gammainv)=IC3beta,(C2y, IC2b)=IC4zinv,(C4xinv, IC4z)=IC3gammainv ,(C4yinv, C2z)=C2d,(C3beta, C3alphainv)=C2z,(C3alphainv, IC3gammainv)=IC3delta, (C2b, C3alphainv)=C2c,(IC4z, IC2z)=C4zinv,(IC2e, IC2c)=C3alpha,(Id, IC4z)=IC4z, (C4y, C2y)=C4yinv,(C2z, IC4x)=IC2e,(C2a, IC4y)=IC3deltainv,(C2c, C3betainv)= C4xinv,(C2c, C2a)=C3gamma,(Is, C4z)=IC4z,(IC3beta, C2c)=IC4x,(IC2f, IC3gammainv )=C2a,(C4y, IC3delta)=IC2a,(C4yinv, C2c)=C2z,(C3delta, C3delta)=C3deltainv,(C2b , Id)=C2b,(IC4x, C3beta)=IC4y,(IC2y, C3gamma)=IC3beta,(IC4yinv, IC3delta)=C4z,( IC2d, IC2c)=C2y,(C4x, C3beta)=C4y,(C4y, C2f)=C3betainv,(C4z, IC2d)=IC3alphainv, (C4xinv, C4yinv)=C3alpha,(C3gamma, IC2y)=IC3delta,(IC4yinv, C4x)=IC3betainv,( IC3delta, C3betainv)=IC2x,(C2y, C4z)=C2a,(C3alpha, C3alphainv)=Id,(C2a, C4z)= C2x,(C2d, C3gamma)=C4z,(Is, IC4zinv)=C4zinv,(IC2z, IC3betainv)=C3deltainv,( IC3deltainv, Is)=C3deltainv,(IC2c, C2x)=IC4yinv,(IC2c, C3alpha)=IC2b,(C2a, IC2x )=IC4z,(IC2x, IC3delta)=C3beta,(IC2z, IC3alpha)=C3beta,(IC3gamma, C3deltainv)= IC2z,(IC3gamma, IC2y)=C3delta,(IC3betainv, IC2f)=C4y,(IC2b, IC2x)=C4zinv,(IC2f , IC3alpha)=C4y,(C4x, Id)=C4x,(C4y, C3beta)=C2b,(C3alpha, C3betainv)=C2z,( C3delta, C4y)=C2e,(C3delta, C2b)=C2d,(C3betainv, Id)=C3betainv,(C3betainv, IC3beta)=Is,(IC4x, IC3delta)=C2c,(IC2x, C4yinv)=IC2d,(IC3beta, C3alpha)= IC3gammainv,(IC3gamma, IC3beta)=C3alphainv,(IC3betainv, Id)=IC3betainv,(IC2d, Is)=C2d,(IC2f, Is)=C2f,(IC2f, IC4z)=C3alphainv,(Id, IC2c)=IC2c,(C4z, C3alphainv )=C4y,(C3betainv, C3alpha)=C2y,(C2c, IC2y)=IC2d,(IC3alpha, IC4xinv)=C2a,( IC3alpha, IC4zinv)=C4yinv,(C2x, IC3gammainv)=IC3betainv,(C2e, IC4yinv)=IC3delta ,(IC2x, C3gamma)=IC3alpha,(IC3alphainv, IC4z)=C4x,(IC3gammainv, C4yinv)=IC4z,( IC3gammainv, IC2x)=C3alphainv,(Id, Is)=Is,(C4zinv, C4yinv)=C3deltainv,( C3gammainv, C4x)=C4y,(IC3beta, C4zinv)=IC2c,(IC3alphainv, IC2a)=C4xinv,(Id, IC2d)=IC2d,(C4yinv, Is)=IC4yinv,(Is, IC4yinv)=C4yinv,(IC4x, C2d)=IC3alpha,(IC2y , C2d)=IC2c,(IC3delta, C2e)=IC4zinv,(IC2c, C3alphainv)=IC2e,(Id, IC3delta)= IC3delta,(C3delta, IC2c)=IC4xinv,(C2c, IC2a)=IC3gamma,(IC4y, IC3deltainv)= C4xinv,(IC4y, IC2a)=C3alpha,(IC2y, C3betainv)=IC3alphainv,(IC4xinv, Id)=IC4xinv ,(IC4yinv, IC2a)=C3delta,(IC3alpha, C2a)=IC4y,(IC2f, IC3beta)=C4yinv,(C3alpha, IC2a)=IC4y,(C3alphainv, C4x)=C2d,(IC2x, Id)=IC2x,(IC2y, IC3delta)=C3alpha,( IC4yinv, C2y)=IC4y,(IC3beta, C2b)=IC4yinv,(IC3betainv, IC3betainv)=C3beta,( IC3gammainv, C4x)=IC4y,(IC2c, IC4zinv)=C3delta,(Id, C2x)=C2x,(C4z, C4xinv)= C3alpha,(C2z, C4zinv)=C4z,(C2c, C2e)=C3alphainv,(IC4y, C2a)=IC3alpha,(IC4yinv, C3gamma)=IC2b,(IC4zinv, IC2b)=C2y,(IC3delta, C4xinv)=IC4z,(IC2a, C4xinv)= IC3delta,(C2x, Is)=IC2x,(C3deltainv, IC2f)=IC2d,(C2b, IC2f)=IC3delta,(IC4xinv, C3deltainv)=IC2a,(IC3alphainv, Id)=IC3alphainv,(IC3alphainv, C3beta)=IC2y,(IC2a , C3gammainv)=IC2c,(IC2b, IC2z)=C2a,(C3delta, C4x)=C2a,(IC2y, IC4zinv)=C2b,( IC3alpha, IC3delta)=C3gammainv,(IC3beta, IC3gammainv)=C2y,(IC3alphainv, IC4x)= C2d,(C2z, IC4z)=IC4zinv,(C4zinv, IC2c)=IC3betainv,(Is, IC3delta)=C3delta,(IC2y , IC4y)=C4yinv,(IC2z, IC2y)=C2x,(IC3gamma, IC3gamma)=C3gammainv,(IC2d, C4z)= IC3gamma,(IC2f, C3beta)=IC4yinv,(C4xinv, IC3deltainv)=IC2a,(C3alphainv, C4y)= C2a,(C3betainv, IC3deltainv)=IC3gamma,(C2e, C2a)=C3betainv,(IC2c, IC2d)=C2y,( C4yinv, C4zinv)=C3gamma,(C3alpha, C2y)=C3beta,(C2d, C3deltainv)=C2f,(C2f, C4x)= C2z,(IC4z, IC3alphainv)=C4y,(IC2x, C2y)=IC2z,(IC4xinv, IC2x)=C4x,(IC3alpha, C4y )=IC4xinv,(IC3alphainv, IC2b)=C2e,(C4x, IC3beta)=IC4y,(C4y, IC3beta)=IC2b,(C2x , IC2d)=IC4yinv,(C3alpha, IC2y)=IC3beta,(C2c, IC3gamma)=IC2a,(Is, IC4z)=C4z,( IC2z, C4z)=IC4zinv,(IC2z, C3deltainv)=IC3betainv,(IC4xinv, IC3gamma)=C4yinv,( IC3delta, IC3delta)=C3deltainv,(IC2b, C2d)=IC3deltainv,(C2x, C3gammainv)= C3betainv,(C2y, C4y)=C4yinv,(C3deltainv, C3alphainv)=C3gamma,(IC4y, C2y)= IC4yinv,(IC3betainv, C3betainv)=IC3beta,(IC2a, C3beta)=IC2e,(IC2b, IC2d)= C3deltainv,(C4z, IC2z)=IC4zinv,(C2z, IC3gamma)=IC3delta,(C3gamma, IC3deltainv)= IC2z,(C2f, IC2y)=IC4xinv,(IC4x, C3gamma)=IC2d,(IC3gammainv, IC2d)=C4zinv,(IC2f , IC2y)=C4xinv,(C2f, C3delta)=C2d,(IC2z, IC2f)=C4xinv,(IC3gamma, IC2c)=C2f,( IC3betainv, C4y)=IC4z,(IC3betainv, C2y)=IC3gammainv,(IC2a, IC2z)=C2b,(C2x, IC3gamma)=IC3alpha,(C2y, C4x)=C2f,(C4yinv, C4z)=C3alpha,(C4zinv, IC2a)=IC2x,( C3alpha, C3beta)=C3deltainv,(C3alpha, IC4x)=IC4z,(IC4y, C3alphainv)=IC2f,(IC2y , C4z)=IC2a,(IC4xinv, IC2d)=C3gamma,(IC3gammainv, C2e)=IC4yinv,(IC3deltainv, C4yinv)=IC2a,(IC3deltainv, IC3deltainv)=C3delta,(IC2a, IC3delta)=C4xinv,(Id, IC2y)=IC2y,(C3alpha, Id)=C3alpha,(IC2z, C2f)=IC4xinv,(Id, IC3alphainv)= IC3alphainv,(C3beta, IC4xinv)=IC2b,(C2b, C3delta)=C2f,(C2c, IC4xinv)=IC3betainv ,(Is, IC2f)=C2f,(IC4xinv, C2e)=IC2y,(IC4zinv, C2f)=IC3alpha,(IC3gamma, C4yinv)= IC2e,(C4x, C3deltainv)=C4zinv,(C4zinv, IC4xinv)=IC3beta,(C2a, C3beta)=C2e,(C2f , C2y)=C4xinv,(C2f, IC3betainv)=IC4zinv,(IC2x, C2b)=IC4z,(IC2y, IC3gammainv)= C3deltainv,(IC3betainv, IC3beta)=Id,(IC3betainv, IC3gammainv)=C3alpha,( IC3gammainv, IC4xinv)=C2d,(C4xinv, IC2z)=IC2e,(IC3alphainv, IC2y)=C3deltainv,( IC2c, C4zinv)=IC3delta,(IC2c, IC4x)=C3deltainv,(IC2d, C3deltainv)=IC2f,(C3alpha , IC2e)=IC2b,(C3betainv, IC3gamma)=IC2x,(C3gammainv, C2e)=C4yinv,(IC4zinv, C4xinv)=IC3beta,(IC3delta, C4x)=IC2a,(IC3gammainv, IC2z)=C3deltainv,(IC2c, C3delta)=IC4zinv,(Id, C4y)=C4y,(C4z, C2f)=C3beta,(C2x, IC3betainv)=IC3gammainv, (C2y, IC2c)=IC2d,(C4xinv, C2a)=C3alphainv,(C3delta, C3beta)=C3gammainv,( C3deltainv, IC2c)=IC4z,(IC3beta, C2f)=IC4z,(IC3alphainv, IC2f)=C4yinv,(C4z, C3gammainv)=C2d,(C2z, C2d)=C4y,(C3gammainv, C3beta)=C2x,(C2c, IC2b)=IC3alpha,( IC2z, C4y)=IC2d,(IC4xinv, C4xinv)=IC2x,(IC4yinv, C3alpha)=IC2a,(IC2c, C3betainv )=IC4xinv,(IC2d, IC4z)=C3gamma,(IC2f, Id)=IC2f,(IC2f, C3gammainv)=IC2a,(C4zinv , IC4yinv)=IC3deltainv,(C3beta, C2c)=C4x,(C3alphainv, IC2z)=IC3betainv,(C2b, C4xinv)=C3gamma,(Is, C2y)=IC2y,(IC4y, IC4xinv)=C3gammainv,(IC4z, C2x)=IC2a,( IC2z, Id)=IC2z,(IC4xinv, IC3delta)=C4y,(IC3delta, IC2f)=C2b,(IC3deltainv, C2z)= IC3gammainv,(IC2b, C2x)=IC4zinv,(IC2e, IC3gamma)=C4y,(C2z, IC3beta)=IC3alpha,( C4yinv, C2a)=C3delta,(IC3delta, IC4zinv)=C4y,(C2z, C2a)=C2b,(C3gamma, C2e)=C4z, (C2c, C3alphainv)=C2e,(C2d, C4zinv)=C3alpha,(C2f, C4y)=C3alpha,(IC2z, C3gamma)= IC3delta,(IC3betainv, C3gamma)=IC2x,(IC3betainv, IC4yinv)=C2b,(IC3gammainv, IC2e)=C4yinv,(C4x, C3alphainv)=C2a,(C3beta, IC4y)=IC2f,(C2e, C2e)=Id,(C2f, C4z) =C3alphainv,(Is, IC3deltainv)=C3deltainv,(IC4x, IC4xinv)=Id,(IC4z, C3deltainv)= IC4yinv,(IC4z, IC2e)=C3gamma,(IC2y, Is)=C2y,(IC3beta, Id)=IC3beta,(IC3delta, C4z)=IC2c,(IC3alphainv, C4z)=IC4x,(IC3deltainv, C2c)=IC4z,(C4y, C2b)=C3gamma,( C4z, C4y)=C3gammainv,(C2y, C3alpha)=C3delta,(C2z, IC4zinv)=IC4z,(C4xinv, C4xinv )=C2x,(C2a, IC2a)=Is,(C2b, IC4yinv)=IC3gammainv,(C2d, IC4zinv)=IC3alpha,(IC2y, C2x)=IC2z,(IC3gamma, IC4zinv)=C2d,(IC3alphainv, IC2z)=C3betainv,(IC3betainv, C4zinv)=IC4x,(IC2c, IC2f)=C3gammainv,(Id, C4zinv)=C4zinv,(Id, IC2b)=IC2b,( C3beta, C3delta)=C3alphainv,(C3betainv, IC3gammainv)=IC3alpha,(C2b, IC2b)=Is,( IC2z, C2y)=IC2x,(IC2z, C3gammainv)=IC3alphainv,(IC3gammainv, IC2b)=C4x,( IC3deltainv, C4y)=IC4zinv,(C4x, IC2a)=IC3deltainv,(C4y, C2e)=C3deltainv,(C2e, IC2z)=IC4xinv,(IC2y, IC4x)=C2f,(IC3beta, C3betainv)=Is,(IC3alphainv, C3delta)= IC2x,(IC3alphainv, C2a)=IC4xinv,(IC2f, C3betainv)=IC4zinv,(Id, IC4xinv)=IC4xinv ,(C4x, IC4yinv)=IC3gamma,(C3alphainv, IC3gamma)=IC2z,(C3betainv, IC4x)=IC2c,( C2a, IC2b)=IC2z,(IC4y, IC3betainv)=C4x,(IC2y, C3delta)=IC3alpha,(IC4yinv, C2d)= IC2x,(IC2d, IC4zinv)=C3alpha,(Id, IC2a)=IC2a,(C4x, C2z)=C2f,(C4xinv, C4y)= C3beta,(C4xinv, IC3alpha)=IC2d,(C3alpha, IC2x)=IC3delta,(C3gamma, C2a)=C2c,( C3delta, IC3gammainv)=IC2z,(C2f, C3gamma)=C2c,(IC3alphainv, C4zinv)=IC2f,(C2y, C2x)=C2z,(C2z, IC2f)=IC4xinv,(C2b, C3deltainv)=C2d,(Is, C4zinv)=IC4zinv,( IC3delta, IC2d)=C2f,(IC3alphainv, C2y)=IC3deltainv,(IC3deltainv, IC2f)=C2d,( IC2e, IC4x)=C2y,(C4x, IC4xinv)=Is,(C2z, C2x)=C2y,(C3beta, C2x)=C3gamma,( C3gammainv, IC2x)=IC3alphainv,(C2d, C2c)=C2y,(C2d, IC3deltainv)=IC2f,(IC4y, C2x )=IC2d,(IC3beta, C4x)=IC4zinv,(IC3betainv, C2d)=IC2a,(IC2e, IC2b)=C3alphainv,( IC2e, IC2d)=C3beta,(IC2e, IC2f)=C2x,(Id, C3alpha)=C3alpha,(C2z, C2b)=C2a,( C4zinv, Id)=C4zinv,(C4zinv, IC2e)=IC3delta,(C3alpha, C2f)=C4zinv,(C3gammainv, C2b)=C4x,(C2a, C3deltainv)=C4y,(IC4x, C2f)=IC2y,(IC4z, C2y)=IC2b,(IC2x, IC2y)= C2z,(IC2y, IC2e)=C4xinv,(IC2a, C3delta)=IC4xinv,(IC2f, C3deltainv)=IC2b,(C2y, IC2f)=IC4x,(C2c, C2z)=C4y,(C2d, IC2b)=IC3delta,(C2e, C3delta)=C4yinv,(IC4y, C4zinv)=IC3beta,(IC4y, C2e)=IC3deltainv,(IC2x, C3deltainv)=IC3alphainv,(IC4xinv , IC4y)=C3beta,(IC3beta, IC2d)=C2e,(IC3gamma, C4x)=IC2b,(IC2a, IC2e)=C3beta,( C3alpha, C4zinv)=C4yinv,(C3betainv, C2a)=C2e,(C2e, C2d)=C3beta,(IC2c, IC2c)=Id, (C4y, Id)=C4y,(C4z, IC4y)=IC3gammainv,(IC4x, IC2e)=C2z,(IC2z, C3delta)=IC3gamma ,(IC4xinv, IC4z)=C3gammainv,(IC2b, C2e)=IC3alpha,(C4y, IC2a)=IC3alpha,(C2x, C3alpha)=C3gamma,(C2y, C2d)=C2c,(C4yinv, C4x)=C3betainv,(C3gamma, C4xinv)= C4zinv,(C3gamma, IC2b)=IC4y,(C3alphainv, IC2f)=IC4yinv,(C3betainv, C3gamma)=C2x ,(C3betainv, IC4z)=IC2f,(C3betainv, IC2z)=IC3alphainv,(C3betainv, IC4yinv)=IC2b ,(Is, C2x)=IC2x,(IC4x, C4zinv)=IC3alphainv,(IC4x, IC3gamma)=C2d,(IC4y, IC3alphainv)=C2f,(IC4z, IC2y)=C2b,(IC2x, IC2a)=C4zinv,(C4x, IC2b)=IC3gammainv,( C3delta, C4yinv)=C4x,(C3alphainv, IC4zinv)=IC2f,(IC4x, IC2f)=C2y,(IC4z, IC2b)= C2x,(IC4xinv, IC2c)=C3delta,(IC4zinv, C2b)=IC2y,(IC4zinv, IC2c)=C3betainv,( IC3gammainv, C2x)=IC3alphainv,(IC2e, C3alpha)=IC2c,(IC2f, IC2a)=C3gammainv,(Id , IC2f)=IC2f,(C4y, C4zinv)=C3beta,(C3alpha, C4x)=C4z,(C3delta, C3alpha)= C3betainv,(C2c, C2b)=C3alpha,(IC3alphainv, IC3deltainv)=C3beta,(IC2d, IC3beta)= C2a,(C4y, IC2d)=IC2z,(C2y, C3alphainv)=C3betainv,(C2y, C2e)=C4xinv,(C3alpha, IC3alpha)=IC3alphainv,(C3beta, C2f)=C4z,(IC4z, IC4xinv)=C3alpha,(IC2b, IC4zinv) =C2x]): # The table of inverse operations. If A is an element of O_h, # then ginv[A] is the inverse of A, which is also a group element. ginv := table([(C2z)=C2z,(C4xinv)=C4x,(C4yinv)=C4y,(C4zinv)=C4z,(C3alpha)= C3alphainv,(C3beta)=C3betainv,(C3gamma)=C3gammainv,(C3delta)=C3deltainv,(C2a)= C2a,(C2b)=C2b,(C2c)=C2c,(C2d)=C2d,(C3betainv)=C3beta,(C2e)=C2e,(C2f)=C2f,(Is)= Is,(C3gammainv)=C3gamma,(IC4x)=IC4xinv,(C3deltainv)=C3delta,(IC4y)=IC4yinv,( IC4z)=IC4zinv,(IC3alpha)=IC3alphainv,(IC2x)=IC2x,(IC3gamma)=IC3gammainv,(IC2y)= IC2y,(IC2z)=IC2z,(IC3delta)=IC3deltainv,(IC4xinv)=IC4x,(IC3alphainv)=IC3alpha,( IC4yinv)=IC4y,(IC3betainv)=IC3beta,(IC2e)=IC2e,(IC3gammainv)=IC3gamma,(IC2f)= IC2f,(IC3deltainv)=IC3delta,(IC4zinv)=IC4z,(IC3beta)=IC3betainv,(C3alphainv)= C3alpha,(C2y)=C2y,(C2x)=C2x,(C4y)=C4yinv,(C4z)=C4zinv,(IC2a)=IC2a,(C4x)=C4xinv, (IC2b)=IC2b,(Id)=Id,(IC2c)=IC2c,(IC2d)=IC2d]): # Now for some utility routines which the end user # should not need. Being utility routines, very little # argument checking is done. ############################################ # # Useful utility routine for checking procedure arguments. # This routine returns TRUE if "setofelements" is a set # of O_h group elements. # ############################################ check_elements:=proc(setofelements) local temp: global Oh: if not type(setofelements,set) then RETURN(false): fi: temp:=setofelements intersect Oh; if nops(temp)=nops(setofelements) then RETURN(true): fi: RETURN(false): end: ############################################ # # Given the elements of a group (setofelements), this # routine returns a set of the classes of the group. # Each element of the set returned is a set of group # elements. # ############################################ determine_classes:=proc(setofelements) local classes,newclass,leftover,anelement,aa: global gprod,ginv: if not check_elements(setofelements) then ERROR(`bad arguments in determine_classes`): fi: leftover:=setofelements: classes:={}: while (leftover<>{}) do anelement:=leftover[1]; newclass:={anelement}; for aa in setofelements do newclass:=newclass union {gprod[gprod[aa,anelement],ginv[aa]]}: od: classes:=classes union {newclass}; leftover:=leftover minus newclass; od: RETURN(classes): end: ############################################## # # This routine checks to make sure that the elements # in the set "setofelements" form a subgroup of O_h. # ############################################## check_if_Oh_subgroup:=proc(setofelements) global gprod,ginv,Oh,Id: local aa,bb,cc: if not check_elements(setofelements) then RETURN(false): fi: if not member(Id,setofelements) then RETURN(false): fi: for aa in setofelements do for bb in setofelements do cc:=gprod[aa,bb]: if not member(cc,setofelements) then RETURN(false): fi: od: od: RETURN(true): end: ############################################ # # Given a set of elements which comprise a subgroup # of O_h, this routine returns a set of the right cosets # of the subgroup in O_h. # ############################################ determine_right_cosets_of_Oh:=proc(Oh_subgroup) local cosets,newcoset,leftover,anelement,aa: global gprod,ginv,Oh: if not check_if_Oh_subgroup(Oh_subgroup) then ERROR(`argument of determine_right_cosets is not an O_h subgroup`): fi: leftover:=Oh: cosets:={}: while (leftover<>{}) do anelement:=leftover[1]; newcoset:={anelement}; for aa in Oh_subgroup do newcoset:=newcoset union {gprod[aa,anelement]}: od: cosets:=cosets union {newcoset}; leftover:=leftover minus newcoset; od: RETURN(cosets): end: ############################################### # # This procedure returns true or false depending # on whether or not "character" is a valid character # vector for O_h. A list of ten integers is considered # a valid character vector (corresponding to the ten # classes of O_h). # ############################################### valid_Oh_character:=proc(character) if not type(character,list) then RETURN(false): elif nops(character)<>10 then RETURN(false): elif convert(map(type,character,integer),set)<>{true} then RETURN(false): else RETURN(true): fi: end: ############################################### # # Given an element of some group, this routine returns # the index of the class to which it belongs. # The class indices are determined by their ordering # in the variable "classes". # ############################################### class_index:=proc(element,classes) local nclass,i: nclass:=nops(classes): for i from 1 to nclass do if member(element,classes[i]) then RETURN(i): fi: od: ERROR(`index not found!`): end: # Now for the routines which will be used by the # end user. These routines have strict argument # checking. ############################################### # # Given two character vectors for two representations A and B, # this routine returns the character for the direct # product representation A X B. # ################################################ direct_product_of_reps:=proc(characterA,characterB) local i: if not type(characterA,list) then ERROR(`first argument must be list of characters`): fi: if convert(map(type,characterA,integer),set)<>{true} then ERROR(`first argument must be list of characters`): fi: if not type(characterB,list) then ERROR(`second argument must be list of characters`): fi: if convert(map(type,characterB,integer),set)<>{true} then ERROR(`second argument must be list of characters`): fi: if nops(characterA)<>nops(characterB) then ERROR(`both arguments must be character vectors for same group`): fi: [seq(characterA[i]*characterB[i],i=1..nops(characterA))]: end: ############################################### # # Given the characters for a reducible representation # of O_h (a list of integers corresponding to the classes # in the global variable "Oh_classes"), this procedure # returns a list of the number of times each irrep of O_h # occurs in the representation. The ordering of the irreps # in the list returned is # [A1g,A2g,Eg,T1g,T2g,A1u,A2u,Eu,T1u,T2u] # ################################################ decompose_into_Oh_irreps:=proc(Oh_character) local i,j,decomp: global Oh_irrep_characters,Oh_irreps,Oh_class_dims: if not valid_Oh_character(Oh_character) then ERROR(`not a valid O_h character vector`): fi: decomp:=[]: for j from 1 to 10 do decomp:=[op(decomp),convert([seq(Oh_class_dims[i] *Oh_irrep_characters[Oh_irreps[j]][i] *Oh_character[i],i=1..10)],`+`)/48]: od: RETURN(decomp): end: ############################################### # # Given two characters for representations of some group # and the associated class dimensions, this routine # returns the dot product of the classes (dividing by # the number of group elements). # ################################################ character_dot_prod:=proc(characterA,characterB,classdims) local i: # check arguments if not (type(characterA,list) and type(characterB,list) and type(classdims,list)) then ERROR(`invalid characters`): fi: if convert(map(type,characterA,integer),set)<>{true} then ERROR(`first argument must be list of characters`): fi: if convert(map(type,characterB,integer),set)<>{true} then ERROR(`second argument must be list of characters`): fi: if convert(map(type,classdims,nonnegint),set)<>{true} then ERROR(`3rd argument must be list of class dimensions`): fi: if (nops(characterA)<>nops(characterB)) or (nops(characterA)<>nops(characterB)) then ERROR(`arguments do not properly match`): fi: convert([seq(classdims[i]*characterA[i]*characterB[i], i=1..nops(characterA))],`+`)/convert(classdims,`+`): end: ########################################################## # # Given the character vector for a rep of Oh, input in # "Oh_character", and a list of classes (each of which is # a set of group elements) of some subgroup of Oh, this # procedure returns the character vector for the subduction # of Oh_character into the Oh_subgroup. The ordering of # characters in the returned list will match the class # ordering in "Oh_subgroup_classes." # ########################################################## subduce:=proc(Oh_character,Oh_subgroup_classes) local Oh_subgroup,check,newcharacter,xx,i: global Oh_class_index: # first, check the arguments if not valid_Oh_character(Oh_character) then ERROR(`first argument must be valid Oh character vector`): fi: if not type(Oh_subgroup_classes,list) then ERROR(`first argument must be a list of class sets`): fi: if convert(map(type,Oh_subgroup_classes,set),set)<>{true} then ERROR(`first argument is not valid list of class sets`): fi: Oh_subgroup:=convert(map(op,Oh_subgroup_classes),set); if not check_if_Oh_subgroup(Oh_subgroup) then ERROR(`first argument does not correspond to subgroup of O_h`): fi: check:=determine_classes(Oh_subgroup): if check<>convert(Oh_subgroup_classes,set) then ERROR(`first argument is not valid list of classes`): fi: # now perform subduction -- first, take one element from # each class in Oh_subgroup_classes; for each element, locate # which class it occurs in using global variable Oh_class_index; # then take characters from Oh_character corresponding to these # classes newcharacter:=map(proc(xx) op(1,xx) end,Oh_subgroup_classes): [seq(Oh_character[Oh_class_index[newcharacter[i]]],i=1..nops(newcharacter))] end: ############################################### # # Given the elements of a subgroup of Oh grouped into classes # and the character corresponding to a representation of # this subgroup, this routine induces a representation of # O_h. It returns the character of the induced representation. # "Oh_subgroup_classes" is a list of sets, each set containing # the elements of the classes of the O_h subgroup. # "subgroup_character" is the character vector for any rep of # the subgroup, and the ordering of the characters in this # vector must correspond to the class ordering in # "Oh_subgroup_classes". The ordering of the returned # character vector of O_h matches the ordering in Oh_classes. # ############################################### induce_Oh_representation:=proc(Oh_subgroup_classes,subgroup_character) local rcosets,xx,Oh_subgroup,Oh_elements, newcharacter,G,Gj,newelement,i: global Oh_classes,gprod,ginv: # argument checks if not type(Oh_subgroup_classes,list) then ERROR(`first argument must be a list of class sets`): fi: if not type(subgroup_character,list) then ERROR(`second argument must be a list of characters`): fi: if nops(Oh_subgroup_classes)<>nops(subgroup_character) then ERROR(`mismatch in the two arguments`): fi: if convert(map(type,subgroup_character,integer),set)<>{true} then ERROR(`second argument is not valid character vector`): fi: if convert(map(type,Oh_subgroup_classes,set),set)<>{true} then ERROR(`first argument is not valid list of class sets`): fi: Oh_subgroup:=convert(map(op,Oh_subgroup_classes),set); if not check_if_Oh_subgroup(Oh_subgroup) then ERROR(`first argument does not correspond to subgroup of O_h`): fi: rcosets:=determine_classes(Oh_subgroup): if rcosets<>convert(Oh_subgroup_classes,set) then ERROR(`first argument is not valid list of classes`): fi: # Use the method of induced representations (see J.F.Cornwell, # Group Theory in Physics, Vol I, p.118). # First, determine a set of right coset representatives. rcosets:=determine_right_cosets_of_Oh(Oh_subgroup): rcosets:=map(proc(xx) op(1,xx) end, rcosets): # Now pick a representative Oh group element from each class of Oh. Oh_elements:=map(proc(xx) op(1,xx) end, Oh_classes): # Now loop over the classes of Oh to make the induced character. newcharacter:=[]: for G in Oh_elements do xx:=[]: for Gj in rcosets do newelement:=gprod[gprod[Gj,G],ginv[Gj]]: # Eq. 5.50 in Cornwell if member(newelement,Oh_subgroup) then xx:=[op(xx),newelement]: fi: od: if nops(xx)=0 then newcharacter:=[op(newcharacter),0]: else xx:=map(class_index,xx,Oh_subgroup_classes): # class indices xx:=convert([seq(subgroup_character[xx[i]],i=1..nops(xx))],`+`): newcharacter:=[op(newcharacter),xx]: fi: od: RETURN(newcharacter): end: # occurrence should be a non-negative integer; # if zero, this routine returns NULL; otherwise, # the label is returned. keep:=proc(irreplabel,occurrence) if not type(occurrence,nonnegint) then ERROR(`2nd argument should be non-negative integer`): fi: if occurrence=0 then RETURN(NULL): else RETURN(irreplabel): fi: end: extract_info:=proc(irreplabel) local b,n,res: b:=substring(irreplabel,1..1): n:=2: if b<>E then b:=substring(irreplabel,1..2): n:=3: fi: if substring(irreplabel,n..n)=p then res:=cat(b,`g`): else res:=cat(b,`u`): fi: if substring(irreplabel,n+1..n+1)=p then RETURN([res,1]): else RETURN([res,-1]): fi: end: extract_parity:=proc(irreplabel) local b,n: b:=substring(irreplabel,1..1): n:=2: if b<>E then b:=substring(irreplabel,1..2): n:=3: fi: if substring(irreplabel,n..n)='p' then RETURN(1): else RETURN(-1): fi: end: # error propagation routines mult:=proc(a,b) [a[1]*b[1],abs(a[1]*b[1])*sqrt((a[2]/a[1])^2+(b[2]/b[1])^2)]: end: addup:=proc(a,b) [a[1]+b[1],sqrt(a[2]^2+b[2]^2)]: end: pow:=proc(a,nn) [a[1]^nn, abs(nn*a[1]^(nn-1)*a[2])]: end: sqr:=proc(a) [a[1]^2, abs(2*a[1]*a[2])]: end: sqroot:=proc(a) [sqrt(a[1]),abs(a[2]/sqrt(a[1])/2)]: end: energy:=proc(mass1,mass2,p_vector,xi,latt_sites) local pp: pp:=[evalf((2*Pi/(latt_sites*xi))^2 *(p_vector[1]^2+p_vector[2]^2+p_vector[3]^2)),0]: addup(sqroot(addup(sqr(mass1),pp)),sqroot(addup(sqr(mass2),pp))): end: identical_energy:=proc(mass1,p_vector,xi,latt_sites) local pp: pp:=[evalf((2*Pi/(latt_sites*xi))^2 *(p_vector[1]^2+p_vector[2]^2+p_vector[3]^2)),0]: pp:=sqroot(addup(sqr(mass1),pp)): [2*pp[1],2*pp[2]]: end: sort_order:=proc(a,b) evalb(a[3][1]0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ########################################################### # # Momentum vector p = ( 0, 0, n ) n != 0 # # Two different glueballs --- C_4v symmetry subgroup # ########################################################### p_vector := [ 0, 0, n ]: subgroup_name:=C4v: subgroup:={Id, C4z, C2z, C4zinv, IC2x, IC2y, IC2a, IC2b}: subgroup_classes:=[ {Id}, {C2z}, {C4z, C4zinv}, {IC2x, IC2y}, {IC2b, IC2a} ]: subgroup_class_dims:=map(nops,subgroup_classes): subgroup_irreps:=[A1,B1,A2,B2,E]: subgroup_irrep_characters[A1]:=[ 1, 1, 1, 1, 1 ]: subgroup_irrep_characters[B1]:=[ 1, 1,-1, 1,-1 ]: subgroup_irrep_characters[A2]:=[ 1, 1, 1,-1,-1 ]: subgroup_irrep_characters[B2]:=[ 1, 1,-1,-1, 1 ]: subgroup_irrep_characters[E ]:=[ 2,-2, 0, 0, 0 ]: print(` `): print(` *************************************************`): print(` `): print(` Subduction of Oh irrep glueballs at rest into`): print(` irreps for a propagating glueball having`): print(` momentum p, where`): print(` p = `,p_vector): print(` `): print(` Symmetry group of propagating glueball is `,subgroup_name): print(` Propagating glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of the subductions:`): print(` `): print(` each row = Oh irrep then occurrences in `,subgroup_irreps): print(` `): for Oh_irrep in Oh_irreps do subd_character:=subduce(Oh_irrep_characters[Oh_irrep],subgroup_classes); decomp:=[seq(character_dot_prod(subd_character, subgroup_irrep_characters[subgroup_irreps[i]], subgroup_class_dims), i=1..nops(subgroup_irreps))]: print(Oh_irrep,decomp): subd[Oh_irrep]:={seq(keep(subgroup_irreps[i],decomp[i]), i=1..nops(subgroup_irreps))}: od: # loop over all of the two-glueball combinations. # Note: since the two glueballs are distinct, the two-glueball # state is a direct product of the single glueball states. # In cases where the irrepA=irrepB, we assume that the two # glueballs have opposite charge-conjugation--parity in # order that they can still be distinguished. print(` `): print(` *************************************************`): print(` `): print(` Two glueball state:`): print(` glueball A has momentum p and C-parity CA,`): print(` glueball B has momentum -p and C-parity CB,`): print(` glueball A and B are distinct,`): print(` C-parity of two-glueball state is CA*CB`): print(` momentum vector p = `,p_vector): print(` `): print(` Symmetry group is `,subgroup_name): print(` Single glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of these states into Oh irreps:`): print(` `): print(` Oh irrep orderings: `,Oh_irreps): print(` `): for irrepA in subgroup_irreps do for irrepB in subgroup_irreps do character:=direct_product_of_reps(subgroup_irrep_characters[irrepA], subgroup_irrep_characters[irrepB]): induced_character:=induce_Oh_representation(subgroup_classes,character): decomp:=decompose_into_Oh_irreps(induced_character): print(irrepA,irrepB,decomp): ind_rep[irrepA,irrepB]:=decomp; od: od: for i from 1 to nops(setofglueballs) do gA:=setofglueballs[i]: gAOh:=extract_info(gA): gAC:=gAOh[2]: gAOh:=gAOh[1]: gAsubd:=subd[gAOh]: for j from i to nops(setofglueballs) do gB:=setofglueballs[j]: gBOh:=extract_info(gB): gBC:=gBOh[2]: gBOh:=gBOh[1]: gBsubd:=subd[gBOh]: newstate:=[{gA,gB},p_vector]: for xx in gAsubd do for yy in gBsubd do decomp:=ind_rep[xx,yy]: for l from 1 to nops(decomp) do if decomp[l]<>0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ########################################################### # # Momentum vector p = ( 0, n, n ) n != 0 # # Two different glueballs --- C_2v symmetry subgroup # ########################################################### p_vector := [ 0, n, n ]: subgroup_name:=C2v: subgroup:={IC2x, IC2f, Id, C2e}: subgroup_classes:=[{Id}, {C2e}, {IC2x}, {IC2f}]: subgroup_class_dims:=map(nops,subgroup_classes): subgroup_irreps:=[A1,A2,B1,B2]: subgroup_irrep_characters[A1]:=[ 1, 1, 1, 1 ]: subgroup_irrep_characters[A2]:=[ 1, 1,-1,-1 ]: subgroup_irrep_characters[B1]:=[ 1,-1,-1, 1 ]: subgroup_irrep_characters[B2]:=[ 1,-1, 1,-1 ]: print(` `): print(` *************************************************`): print(` `): print(` Subduction of Oh irrep glueballs at rest into`): print(` irreps for a propagating glueball having`): print(` momentum p, where`): print(` p = `,p_vector): print(` `): print(` Symmetry group of propagating glueball is `,subgroup_name): print(` Propagating glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of the subductions:`): print(` `): print(` each row = Oh irrep then occurrences in `,subgroup_irreps): print(` `): for Oh_irrep in Oh_irreps do subd_character:=subduce(Oh_irrep_characters[Oh_irrep],subgroup_classes); decomp:=[seq(character_dot_prod(subd_character, subgroup_irrep_characters[subgroup_irreps[i]], subgroup_class_dims), i=1..nops(subgroup_irreps))]: subd[Oh_irrep]:={seq(keep(subgroup_irreps[i],decomp[i]), i=1..nops(subgroup_irreps))}: print(Oh_irrep,decomp): od: # loop over all of the two-glueball combinations. # Note: since the two glueballs are distinct, the two-glueball # state is a direct product of the single glueball states. # In cases where the irrepA=irrepB, we assume that the two # glueballs have opposite charge-conjugation--parity in # order that they can still be distinguished. print(` `): print(` *************************************************`): print(` `): print(` Two glueball state:`): print(` glueball A has momentum p and C-parity CA,`): print(` glueball B has momentum -p and C-parity CB,`): print(` glueball A and B are distinct,`): print(` C-parity of two-glueball state is CA*CB`): print(` momentum vector p = `,p_vector): print(` `): print(` Symmetry group is `,subgroup_name): print(` Single glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of these states into Oh irreps:`): print(` `): print(` Oh irrep orderings: `,Oh_irreps): print(` `): for irrepA in subgroup_irreps do for irrepB in subgroup_irreps do character:=direct_product_of_reps(subgroup_irrep_characters[irrepA], subgroup_irrep_characters[irrepB]): induced_character:=induce_Oh_representation(subgroup_classes,character): decomp:=decompose_into_Oh_irreps(induced_character): print(irrepA,irrepB,decomp): ind_rep[irrepA,irrepB]:=decomp; od: od: for i from 1 to nops(setofglueballs) do gA:=setofglueballs[i]: gAOh:=extract_info(gA): gAC:=gAOh[2]: gAOh:=gAOh[1]: gAsubd:=subd[gAOh]: for j from i to nops(setofglueballs) do gB:=setofglueballs[j]: gBOh:=extract_info(gB): gBC:=gBOh[2]: gBOh:=gBOh[1]: gBsubd:=subd[gBOh]: newstate:=[{gA,gB},p_vector]: for xx in gAsubd do for yy in gBsubd do decomp:=ind_rep[xx,yy]: for l from 1 to nops(decomp) do if decomp[l]<>0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ########################################################### # # Momentum vector p = ( n, n, n ) n != 0 # # Two different glueballs --- C_3v symmetry subgroup # ########################################################### p_vector := [ n, n, n ]: subgroup_name:=C3v: subgroup:={C3delta, IC2b, IC2d, IC2f, C3deltainv, Id}: subgroup_classes:=[{Id}, {C3deltainv, C3delta}, {IC2d, IC2f, IC2b}]: subgroup_class_dims:=map(nops,subgroup_classes): subgroup_irreps:=[A1,A2,E]: subgroup_irrep_characters[A1]:=[ 1, 1, 1 ]: subgroup_irrep_characters[A2]:=[ 1, 1,-1 ]: subgroup_irrep_characters[E ]:=[ 2,-1, 0 ]: print(` `): print(` *************************************************`): print(` `): print(` Subduction of Oh irrep glueballs at rest into`): print(` irreps for a propagating glueball having`): print(` momentum p, where`): print(` p = `,p_vector): print(` `): print(` Symmetry group of propagating glueball is `,subgroup_name): print(` Propagating glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of the subductions:`): print(` `): print(` each row = Oh irrep then occurrences in `,subgroup_irreps): print(` `): for Oh_irrep in Oh_irreps do subd_character:=subduce(Oh_irrep_characters[Oh_irrep],subgroup_classes); decomp:=[seq(character_dot_prod(subd_character, subgroup_irrep_characters[subgroup_irreps[i]], subgroup_class_dims), i=1..nops(subgroup_irreps))]: subd[Oh_irrep]:={seq(keep(subgroup_irreps[i],decomp[i]), i=1..nops(subgroup_irreps))}: print(Oh_irrep,decomp): od: # loop over all of the two-glueball combinations. # Note: since the two glueballs are distinct, the two-glueball # state is a direct product of the single glueball states. # In cases where the irrepA=irrepB, we assume that the two # glueballs have opposite charge-conjugation--parity in # order that they can still be distinguished. print(` `): print(` *************************************************`): print(` `): print(` Two glueball state:`): print(` glueball A has momentum p and C-parity CA,`): print(` glueball B has momentum -p and C-parity CB,`): print(` glueball A and B are distinct,`): print(` C-parity of two-glueball state is CA*CB`): print(` momentum vector p = `,p_vector): print(` `): print(` Symmetry group is `,subgroup_name): print(` Single glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of these states into Oh irreps:`): print(` `): print(` Oh irrep orderings: `,Oh_irreps): print(` `): for irrepA in subgroup_irreps do for irrepB in subgroup_irreps do character:=direct_product_of_reps(subgroup_irrep_characters[irrepA], subgroup_irrep_characters[irrepB]): induced_character:=induce_Oh_representation(subgroup_classes,character): decomp:=decompose_into_Oh_irreps(induced_character): print(irrepA,irrepB,decomp): ind_rep[irrepA,irrepB]:=decomp; od: od: for i from 1 to nops(setofglueballs) do gA:=setofglueballs[i]: gAOh:=extract_info(gA): gAC:=gAOh[2]: gAOh:=gAOh[1]: gAsubd:=subd[gAOh]: for j from i to nops(setofglueballs) do gB:=setofglueballs[j]: gBOh:=extract_info(gB): gBC:=gBOh[2]: gBOh:=gBOh[1]: gBsubd:=subd[gBOh]: newstate:=[{gA,gB},p_vector]: for xx in gAsubd do for yy in gBsubd do decomp:=ind_rep[xx,yy]: for l from 1 to nops(decomp) do if decomp[l]<>0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ########################################################### # # Momentum vector p = ( 0, m, n ) m != n, m != 0, n != 0 # # Two different glueballs --- C_s symmetry subgroup # ########################################################### p_vector := [ 0, m, n ]: subgroup_name:=Cs: subgroup:={IC2x, Id}: subgroup_classes:=[{Id}, {IC2x}]: subgroup_class_dims:=map(nops,subgroup_classes): subgroup_irreps:=[Ap,App]: subgroup_irrep_characters[Ap ]:=[ 1, 1 ]: subgroup_irrep_characters[App]:=[ 1,-1 ]: print(` `): print(` *************************************************`): print(` `): print(` Subduction of Oh irrep glueballs at rest into`): print(` irreps for a propagating glueball having`): print(` momentum p, where`): print(` p = `,p_vector): print(` `): print(` Symmetry group of propagating glueball is `,subgroup_name): print(` Propagating glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of the subductions:`): print(` `): print(` each row = Oh irrep then occurrences in `,subgroup_irreps): print(` `): for Oh_irrep in Oh_irreps do subd_character:=subduce(Oh_irrep_characters[Oh_irrep],subgroup_classes); decomp:=[seq(character_dot_prod(subd_character, subgroup_irrep_characters[subgroup_irreps[i]], subgroup_class_dims), i=1..nops(subgroup_irreps))]: subd[Oh_irrep]:={seq(keep(subgroup_irreps[i],decomp[i]), i=1..nops(subgroup_irreps))}: print(Oh_irrep,decomp): od: # loop over all of the two-glueball combinations. # Note: since the two glueballs are distinct, the two-glueball # state is a direct product of the single glueball states. # In cases where the irrepA=irrepB, we assume that the two # glueballs have opposite charge-conjugation--parity in # order that they can still be distinguished. print(` `): print(` *************************************************`): print(` `): print(` Two glueball state:`): print(` glueball A has momentum p and C-parity CA,`): print(` glueball B has momentum -p and C-parity CB,`): print(` glueball A and B are distinct,`): print(` C-parity of two-glueball state is CA*CB`): print(` momentum vector p = `,p_vector): print(` `): print(` Symmetry group is `,subgroup_name): print(` Single glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of these states into Oh irreps:`): print(` `): print(` Oh irrep orderings: `,Oh_irreps): print(` `): for irrepA in subgroup_irreps do for irrepB in subgroup_irreps do character:=direct_product_of_reps(subgroup_irrep_characters[irrepA], subgroup_irrep_characters[irrepB]): induced_character:=induce_Oh_representation(subgroup_classes,character): decomp:=decompose_into_Oh_irreps(induced_character): print(irrepA,irrepB,decomp): ind_rep[irrepA,irrepB]:=decomp; od: od: for i from 1 to nops(setofglueballs) do gA:=setofglueballs[i]: gAOh:=extract_info(gA): gAC:=gAOh[2]: gAOh:=gAOh[1]: gAsubd:=subd[gAOh]: for j from i to nops(setofglueballs) do gB:=setofglueballs[j]: gBOh:=extract_info(gB): gBC:=gBOh[2]: gBOh:=gBOh[1]: gBsubd:=subd[gBOh]: newstate:=[{gA,gB},p_vector]: for xx in gAsubd do for yy in gBsubd do decomp:=ind_rep[xx,yy]: for l from 1 to nops(decomp) do if decomp[l]<>0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ########################################################### # # Momentum vector p = ( m, m, n ) m != n, m != 0, n != 0 # # Two different glueballs --- C_s symmetry subgroup # ########################################################### p_vector := [ m, m, n ]: subgroup_name:=Cs: subgroup:={IC2b, Id}: subgroup_classes:=[{Id}, {IC2b}]: subgroup_class_dims:=map(nops,subgroup_classes): subgroup_irreps:=[Ap,App]: subgroup_irrep_characters[Ap ]:=[ 1, 1 ]: subgroup_irrep_characters[App]:=[ 1,-1 ]: print(` `): print(` *************************************************`): print(` `): print(` Subduction of Oh irrep glueballs at rest into`): print(` irreps for a propagating glueball having`): print(` momentum p, where`): print(` p = `,p_vector): print(` `): print(` Symmetry group of propagating glueball is `,subgroup_name): print(` Propagating glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of the subductions:`): print(` `): print(` each row = Oh irrep then occurrences in `,subgroup_irreps): print(` `): for Oh_irrep in Oh_irreps do subd_character:=subduce(Oh_irrep_characters[Oh_irrep],subgroup_classes); decomp:=[seq(character_dot_prod(subd_character, subgroup_irrep_characters[subgroup_irreps[i]], subgroup_class_dims), i=1..nops(subgroup_irreps))]: subd[Oh_irrep]:={seq(keep(subgroup_irreps[i],decomp[i]), i=1..nops(subgroup_irreps))}: print(Oh_irrep,decomp): od: # loop over all of the two-glueball combinations. # Note: since the two glueballs are distinct, the two-glueball # state is a direct product of the single glueball states. # In cases where the irrepA=irrepB, we assume that the two # glueballs have opposite charge-conjugation--parity in # order that they can still be distinguished. print(` `): print(` *************************************************`): print(` `): print(` Two glueball state:`): print(` glueball A has momentum p and C-parity CA,`): print(` glueball B has momentum -p and C-parity CB,`): print(` glueball A and B are distinct,`): print(` C-parity of two-glueball state is CA*CB`): print(` momentum vector p = `,p_vector): print(` `): print(` Symmetry group is `,subgroup_name): print(` Single glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of these states into Oh irreps:`): print(` `): print(` Oh irrep orderings: `,Oh_irreps): print(` `): for irrepA in subgroup_irreps do for irrepB in subgroup_irreps do character:=direct_product_of_reps(subgroup_irrep_characters[irrepA], subgroup_irrep_characters[irrepB]): induced_character:=induce_Oh_representation(subgroup_classes,character): decomp:=decompose_into_Oh_irreps(induced_character): print(irrepA,irrepB,decomp): ind_rep[irrepA,irrepB]:=decomp; od: od: for i from 1 to nops(setofglueballs) do gA:=setofglueballs[i]: gAOh:=extract_info(gA): gAC:=gAOh[2]: gAOh:=gAOh[1]: gAsubd:=subd[gAOh]: for j from i to nops(setofglueballs) do gB:=setofglueballs[j]: gBOh:=extract_info(gB): gBC:=gBOh[2]: gBOh:=gBOh[1]: gBsubd:=subd[gBOh]: newstate:=[{gA,gB},p_vector]: for xx in gAsubd do for yy in gBsubd do decomp:=ind_rep[xx,yy]: for l from 1 to nops(decomp) do if decomp[l]<>0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ########################################################### # # Momentum vector p = ( l, m, n) l,m,n all non-zero and different # # Two different glueballs --- C_1 symmetry subgroup # ########################################################### p_vector := [ r, m, n ]: subgroup_name:=C1: subgroup:={Id}: subgroup_classes:=[ {Id} ]: subgroup_class_dims:=map(nops,subgroup_classes): subgroup_irreps:=[A]: subgroup_irrep_characters[A]:=[ 1 ]: print(` `): print(` *************************************************`): print(` `): print(` Subduction of Oh irrep glueballs at rest into`): print(` irreps for a propagating glueball having`): print(` momentum p, where`): print(` p = `,p_vector): print(` `): print(` Symmetry group of propagating glueball is `,subgroup_name): print(` Propagating glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of the subductions:`): print(` `): print(` each row = Oh irrep then occurrences in `,subgroup_irreps): print(` `): for Oh_irrep in Oh_irreps do subd_character:=subduce(Oh_irrep_characters[Oh_irrep],subgroup_classes); decomp:=[seq(character_dot_prod(subd_character, subgroup_irrep_characters[subgroup_irreps[i]], subgroup_class_dims), i=1..nops(subgroup_irreps))]: subd[Oh_irrep]:={seq(keep(subgroup_irreps[i],decomp[i]), i=1..nops(subgroup_irreps))}: print(Oh_irrep,decomp): od: # loop over all of the two-glueball combinations. # Note: since the two glueballs are distinct, the two-glueball # state is a direct product of the single glueball states. # In cases where the irrepA=irrepB, we assume that the two # glueballs have opposite charge-conjugation--parity in # order that they can still be distinguished. print(` `): print(` *************************************************`): print(` `): print(` Two glueball state:`): print(` glueball A has momentum p and C-parity CA,`): print(` glueball B has momentum -p and C-parity CB,`): print(` glueball A and B are distinct,`): print(` C-parity of two-glueball state is CA*CB`): print(` momentum vector p = `,p_vector): print(` `): print(` Symmetry group is `,subgroup_name): print(` Single glueball states labelled by irreps `,subgroup_irreps): print(` `): print(` Decomposition of these states into Oh irreps:`): print(` `): print(` Oh irrep orderings: `,Oh_irreps): print(` `): for irrepA in subgroup_irreps do for irrepB in subgroup_irreps do character:=direct_product_of_reps(subgroup_irrep_characters[irrepA], subgroup_irrep_characters[irrepB]): induced_character:=induce_Oh_representation(subgroup_classes,character): decomp:=decompose_into_Oh_irreps(induced_character): print(irrepA,irrepB,decomp): ind_rep[irrepA,irrepB]:=decomp; od: od: for i from 1 to nops(setofglueballs) do gA:=setofglueballs[i]: gAOh:=extract_info(gA): gAC:=gAOh[2]: gAOh:=gAOh[1]: gAsubd:=subd[gAOh]: for j from i to nops(setofglueballs) do gB:=setofglueballs[j]: gBOh:=extract_info(gB): gBC:=gBOh[2]: gBOh:=gBOh[1]: gBsubd:=subd[gBOh]: newstate:=[{gA,gB},p_vector]: for xx in gAsubd do for yy in gBsubd do decomp:=ind_rep[xx,yy]: for l from 1 to nops(decomp) do if decomp[l]<>0 then jpc:=[Oh_irreps[l],gAC*gBC]: twoglueballstates[jpc]:=twoglueballstates[jpc] union {newstate}: fi: od: od: od: od:od: subd:='subd': ind_rep:='ind_rep': ############################################################## # # Now substitute in some numbers and screen out the # two-glueball states which lie higher than some maximum # energy value. # ############################################################## mass[A1pp] :=[ 0.288, 0.004]: mass[A1ppS]:=[ 0.511, 0.007]: mass[A2pp] :=[ 0.713, 0.014]: mass[Epp] :=[ 0.472, 0.004]: mass[EppS] :=[ 0.652, 0.008]: mass[T1pp] :=[ 0.728, 0.008]: mass[T1ppS]:=[ 0.823, 0.013]: mass[T2pp] :=[ 0.477, 0.003]: mass[T2ppS]:=[ 0.660, 0.006]: mass[A1mp] :=[ 0.522, 0.007]: mass[A1mpS]:=[ 0.72, 0.02 ]: mass[A2mp] :=[ 0.93, 0.01 ]: mass[Emp] :=[ 0.611, 0.007]: mass[EmpS] :=[ 0.782, 0.013]: mass[T1mp] :=[ 0.82, 0.01 ]: mass[T2mp] :=[ 0.619, 0.003]: mass[T2mpS]:=[ 0.771, 0.009]: mass[A1pm] :=[ 0.94, 0.01 ]: mass[A2pm] :=[ 0.700, 0.014]: mass[Epm] :=[ 0.82, 0.01 ]: mass[T1pm] :=[ 0.590, 0.005]: mass[T1pmS]:=[ 0.73, 0.02 ]: mass[T2pm] :=[ 0.701, 0.007]: mass[T2pmS]:=[ 0.83, 0.01 ]: mass[A1mm] :=[ 0.98, 0.01 ]: mass[A2mm] :=[ 0.82, 0.02 ]: mass[Emm] :=[ 0.78, 0.01 ]: mass[T1mm] :=[ 0.760, 0.009]: mass[T2mm] :=[ 0.77, 0.01 ]: lattice_extent:=10: # 10^3 lattice as_over_at:=5: maximum_energy:=1.0: for jpc in JPC do keepstates:={}: allstates:=twoglueballstates[OhC[jpc]]: for ss in allstates do p_vector:=ss[2]: vars:=indets(p_vector): pmax:=iquo(lattice_extent,2): indsubs:={}: if nops(vars)>0 then indsubs:={seq({i},i=1..pmax)}: fi: for j from 2 to nops(vars) do newinds:=map(proc(xx) seq({op(xx),i},i=1..pmax): end,indsubs): indsubs:=map(proc(xx) if nops(xx)=j then xx else NULL fi end,newinds): od: if nops(vars)=0 then pvs:={p_vector}: else pvs:={}: for xx in indsubs do pvs:=pvs union {subs(seq(vars[j]=xx[j],j=1..nops(vars)),p_vector)}: od: fi: pvs:=map(sort,pvs): # ordering of p-components immaterial gA:=ss[1]: if nops(gA)=2 then gB:=gA[2]: gA:=gA[1]: for pv in pvs do eng:=energy(mass[gA],mass[gB],pv,as_over_at,lattice_extent); if (eng[1]-eng[2]<=maximum_energy) then keepstates:=keepstates union {[{gA,gB},pv,eng]}: fi: od: else gA:=gA[1]: for pv in pvs do eng:=identical_energy(mass[gA],pv,as_over_at,lattice_extent); if (eng[1]-eng[2]<=maximum_energy) then keepstates:=keepstates union {[{gA},pv,eng]}: fi: od: fi: od: twoglueballstates[jpc]:=keepstates; od: # Now order the elements of interest. # Next, we apply the identical glueball filter---for states # made of two identical A1pp glueballs, we eliminate all negative # parity states. For other identical glueballs, we would rather # not worry about them, so we eliminate them and all higher lying # states. (They are suitably high lying anyways.) for jpc in JPC do temp:=sort(convert(twoglueballstates[jpc],list),sort_order): temp2:=[]: for i from 1 to nops(temp) do ss:=temp[i][1]: if nops(ss)=1 then if op(ss)<>A1pp then temp2:=[op(temp2),i]: fi: fi: od: if nops(temp2)>0 then temp:=[op(1..temp2[1]-1,temp)]: fi: if extract_parity(jpc)=-1 then temp2:=[]: for i from 1 to nops(temp) do ss:=temp[i][1]: if nops(ss)=1 then if op(ss)=A1pp then temp2:=[op(temp2),i]: fi: fi: od: if nops(temp2)>0 then for i from nops(temp2) by -1 to 1 do temp:=subsop(temp2[i]=NULL,temp): od: fi: fi: two_glueball_states[jpc]:=temp: od: