{VERSION 6 0 "IBM INTEL NT" "6.0" }
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{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
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11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 
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0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 326 "This worksheet initialize
s remember tables for certain procedures in the package 'SP'.\n\nIn or
der to insert the SP package into a Maple archive, this worksheet need
s to be executed first. Procedures in this worksheet belong to the SP \+
package and their remember tables are initialized here.\n\n\nCookevill
e, TN, December 20, 2007\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "restart:\nlibname[1];\nsavelibname;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q7C:\\Maple11/Cliffordlib6\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q7C:\\Maple11/Cliffordlib6\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 577 "SymmetricGroup:=proc(n::posint) local II,m,i,L,G;
\noption remember;\nII:=linalg:-diag(1$n):\nL:=combinat:-permute(n);\n
G:=[]:\nfor m in L do\nG:=[op(G),linalg:-augment(seq(linalg:-col(II,m[
i]),i=1..n))];\nend do;\nreturn G;\nend proc:\nSymmetricGroup(2):=[mat
rix([[1, 0], [0, 1]]), matrix([[0, 1], [1, 0]])];\nSymmetricGroup(3):=
[matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]), matrix([[1, 0, 0], [0, 0, \+
1], [0, 1, 0]]), matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]]), matrix([[0
, 0, 1], [1, 0, 0], [0, 1, 0]]), matrix([[0, 1, 0], [0, 0, 1], [1, 0, \+
0]]), matrix([[0, 0, 1], [0, 1, 0], [1, 0, 0]])];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%/SymmetricGroupG6#\"\"#7$K%'matrixG6#7$7$\"\"\"\"\"!
7$F/F.Q(pprint06\"KF*6#7$F0F-Q(pprint1F2" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>-%/SymmetricGroupG6#\"\"$7(K%'matrixG6#7%7%\"\"\"\"\"!F/7%F/F.F
/7%F/F/F.Q(pprint06\"KF*6#7%F-F1F0Q(pprint1F3KF*6#7%F0F-F1Q(pprint2F3K
F*6#7%F1F-F0Q(pprint3F3KF*6#7%F0F1F-Q(pprint4F3KF*6#7%F1F0F-Q(pprint5F
3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "_SymmetricGroup_rem_ta
ble:=op(4,eval(SymmetricGroup));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
:_SymmetricGroup_rem_tableGK%&TABLEG6#7$/\"\"#7$K%'matrixG6#7$7$\"\"\"
\"\"!7$F2F1Q(pprint06\"KF-6#7$F3F0Q(pprint1F5/\"\"$7(KF-6#7%7%F1F2F27%
F2F1F27%F2F2F1Q(pprint2F5KF-6#7%F@FBFAQ(pprint3F5KF-6#7%FAF@FBQ(pprint
4F5KF-6#7%FBF@FAQ(pprint5F5KF-6#7%FAFBF@Q(pprint6F5KF-6#7%FBFAF@Q(ppri
nt7F5Q(pprint8F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "savelib
name;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7C:\\Maple11/Cliffordlib6\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "if not _SymmetricGroup_
rem_table=NULL then savelib(_SymmetricGroup_rem_table) end if;\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "############################
###############################################" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 530 
"AlternatingGroup:=proc(n::posint) local II,m,i,L,Leven,G;\noption rem
ember;\nII:=linalg:-diag(1$n):\nL:=combinat:-permute(n);\nLeven:=[]:\n
for m in L do\n    if SP:-permsign(m)=1 then Leven:=[op(Leven),m] end \+
if;\nend do:\nG:=[]:\nfor m in Leven do\nG:=[op(G),linalg:-augment(seq
(linalg:-col(II,m[i]),i=1..n))];\nend do;\nreturn G;\nend proc:\nAlter
natingGroup(2):=[matrix([[1, 0], [0, 1]])];\nAlternatingGroup(3):=[mat
rix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]), matrix([[0, 0, 1], [1, 0, 0], \+
[0, 1, 0]]), matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>-%1AlternatingGroupG6#\"\"#7#K%'matrixG6#7$7$\"
\"\"\"\"!7$F/F.Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%1Alt
ernatingGroupG6#\"\"$7%K%'matrixG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/F.Q(
pprint06\"KF*6#7%F1F-F0Q(pprint1F3KF*6#7%F0F1F-Q(pprint2F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "_AlternatingGroup_rem_table:=op(4,
eval(AlternatingGroup)):\nif not _AlternatingGroup_rem_table=NULL then
 savelib(_AlternatingGroup_rem_table) end if;\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "
######################################################################
######" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 397 "Reynolds:=proc(f::polynom, varslist::lis
t(\{symbol,indexed\})) local G,m:\noption remember;\nG:=op(procname);
\nif not type(G,list(\{name,symbol,matrix,algebraic\})) then \n   erro
r `index G must be of type list(\{name,symbol,matrix,algebraic\})` end
 if;\nm:=nops(G):\nadd(1/m*SP:-MatrixAction(G[i],f,varslist),i=1..m);
\nend proc:\nf := x*y+x*z+y*z:\nS3:=SymmetricGroup(3):\nReynolds[S3](f
,[x,y,z]):=x*y+x*z+y*z:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
117 "_Reynolds_rem_table:=op(4,eval(Reynolds)):\nif not _Reynolds_rem_
table=NULL then savelib(_Reynolds_rem_table) end if;\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "###################################
########################################" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2264 "Finit
eGroups:=proc(G::symbol) local k,A,A1,A2,A12,A21,_C2,_C4,_C8,_D6,_D8,_
V4:\noption remember;\n###############################################
###\nif G='C2' then \n   A:=matrix(2,2,[-1,0,0,-1]);\n   _C2:=map(eval
m,[linalg:-diag(1$2),A]);\n   return _C2;\n###########################
#######################\nelif G='C4' then\n   A:=matrix(2,2,[0,-1,1,0]
):\n   _C4:=[linalg:-diag(1$2),seq(evalm(A^k),k=1..3)]:\n   return _C4
;\n##################################################\nelif G='C8' the
n\n   A[1]:=matrix(3,3,[1,0,0,0,1,0,0,0,1]):\n   A[2]:=matrix(3,3,[1,0
,0,0,1,0,0,0,-1]):\n   A[3]:=matrix(3,3,[1,0,0,0,-1,0,0,0,1]):\n   A[4
]:=matrix(3,3,[1,0,0,0,-1,0,0,0,-1]):\n   A[5]:=matrix(3,3,[-1,0,0,0,1
,0,0,0,1]):\n   A[6]:=matrix(3,3,[-1,0,0,0,1,0,0,0,-1]):\n   A[7]:=mat
rix(3,3,[-1,0,0,0,-1,0,0,0,1]):\n   A[8]:=matrix(3,3,[-1,0,0,0,-1,0,0,
0,-1]):\n   _C8:=map(evalm,[seq(A[i],i=1..8)]);\n   return _C8;\n#####
#############################################\nelif G='D6' then\n     \+
A1,A2:=linalg:-diag(1,-1,-1),matrix(3,3,[1/2,-sqrt(3)/2,0,sqrt(3)/2,1/
2,0,0,0,1]);\n     _D6:=map(evalm,[linalg:-diag(1,1,1),A2,A2^2,A2^3,A2
^4,A2^5,A1,A1 &* A2, A1 &* A2^2, \n                                   \+
      A1 &* A2^3,A1 &* A2^4, A1 &* A2^5]);\n     return _D6;\n########
##########################################\nelif G='D8' then\n     A1,
A2:=matrix(2,2,[1/sqrt(2),1/sqrt(2),1/sqrt(2),-1/sqrt(2)]),matrix(2,2,
[1,0,0,-1]):\n     A12:=evalm(A1 &* A2):\n     A21:=evalm(A2 &* A1):\n
     _D8:=map(evalm,[linalg:-diag(1,1),A12^6 &* A2 &* A12^2,A12^6 &* A
2, A12^6,A12 &* A1,A2,\n                                       A12^4,A
12^2,A1,A12,A12^3,A12^5,A12^7,A2 &* A12,\n                            \+
           A12^6 &* A2 &* A12^2 &* A12,A12^6 &* A2 &* A12^2 &* A12^7])
;\n     return _D8;\n#################################################
#\nelif G='V4' then\n     A1,A2:=matrix(2,2,[1,0,0,-1]),matrix(2,2,[-1
,0,0,1]):\n     _V4:=map(evalm,[linalg:-diag(1$2),A1,A2,A1 &* A2]);\n \+
    return _V4;\n\n \n################################################
##\nelse \nprintf(\"Group %s is not known to FiniteGroups. You can mak
e it known by assigning it to FiniteGroups('%s').\\n\",G,G);\nend if;
\n\nend proc:\n\nFiniteGroups('C4'):=[matrix([[1, 0], [0, 1]]), matrix
([[0, -1], [1, 0]]), matrix([[-1, 0], [0, -1]]), matrix([[0, 1], [-1, \+
0]])]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "_FiniteGroups_re
m_table:=op(4,eval(FiniteGroups)):\nif not _FiniteGroups_rem_table=NUL
L then savelib(_FiniteGroups_rem_table) end if;\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "############################################
#######" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1369 "generateGinvariants:=proc(vars::l
ist(monomial)) local G,ord,nvars,T,i,j,INV,\nm,n,s,alpha,mons,Rmons,rd
im,cdim,size;\noption remember:\n#####################################
#####################\nG:=op(procname);\nif not type(G,list(\{name,sym
bol,matrix,algebraic\})) then \n   error `index G must be of type list
(\{name,symbol,matrix,algebraic\})` end if;\nrdim,cdim:=convert(map(li
nalg:-rowdim,G),set),convert(map(linalg:-rowdim,G),set);\nif nops(rdim
) <> 1 or nops(cdim) <> 1 or rdim <> cdim then\n   error `matrices in \+
the list G must be all square of the same size`\nend if:\n############
################################################\nsize:=op(rdim):\nnva
rs:=nops(vars):\nif nvars<>size then error `number of indeterminates m
ust equal the size of group G matrices` end if:\n#####################
#######################################\nord:=nops(G):\nT:=combinat:-c
artprod([seq([seq(i,i=0..ord)],j=1..nvars)]):\nINV:=[]:\nwhile not T[f
inished] do\n      m:=T[nextvalue]():\n      s:=add(i,i=m):\n      if \+
s<=ord and s>0 then INV:=[op(INV),m] end if: \nend do;\nmons:=[seq(mul
(vars[i]^alpha[i],i=1..nvars),alpha=INV)];\nRmons:=map(SP:-Reynolds[G]
,mons,vars);\nRmons:=convert(convert(Rmons,set),list);\nRmons:=remove(
type,Rmons,numeric);\n#return Rmons;\nRmons:=[seq(mm/content(mm),mm=Rm
ons)]:\nreturn Rmons; \nend proc:\n\nS2:=SymmetricGroup(2):\ngenerateG
invariants[S2]([x,y]):=[x*y, y+x, x^2+y^2]:\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 161 "_generateGinvariants_rem_table:=op(4,eval(gener
ateGinvariants)):\nif not _generateGinvariants_rem_table=NULL then sav
elib(_generateGinvariants_rem_table) end if;\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "
######################################################################
######" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart:with(Libr
aryTools);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72%1ActivationModuleG%1A
ddFromDirectoryG%'AuthorG%'BrowseG%3BuildFromDirectoryG%/ConvertVersio
nG%'CreateG%'DeleteG%,FindLibraryG%,PrefixMatchG%)PriorityG%%SaveG%-Sh
owContentsG%*TimestampG%4UpdateFromDirectoryG%*WriteModeG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ShowContents(libname[1]);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#787&Q*Bigebra.m6\"7(\"%2?\"#7\"#?\"#:\"#<F*
\"'/da\"%Z77&Q+Clifford.mF&7(F(F)F*\"#;F*\"#S\"'3dx\"%JC7&Q*Cliplus.mF
&7(F(F)F*F+\"#>\"#\\\"'Ywc\"$x$7&Q/code_support.mF&7(F(F)F*F+\"#X\"#J
\"'K5f\"$w%7&Q)Define.mF&7(F(F)F*F+\"#9\"#d\"'Q.a\"$$[7&Q&GTP.mF&7(F(F
)F*F+\"#`\"#@\"'%Q8'\"$\"G7&Q&GfG.mF&7(F(F)F*F+\"#^\"\"*\"'&e,'\"$:(7&
Q>_AlternatingGroup_rem_table.mF&7(F(F)F*F2\"#E\"#M\"''f#*)FQ7&Q6_Reyn
olds_rem_table.mF&7(F(F)F*F2Ffn\"#O\"'xG*)\"$L%7&Q<_SymmetricGroup_rem
_table.mF&7(F(F)F*F2Ffn\"#K\"'^@*)\"$X%7&QA_generateGinvariants_rem_ta
ble.mF&7(F(F)F*F2FfnF3\"')\\$*)\"$#>7&Q<_implicitbezierpolynomial.mF&7
(F(F)F*F2\"\"!FO\"'dkk\"&@7\"7&Q+matcompL.mF&7(F(F)F*F+\"#8FG\"'D#[$\"
&=X(7&Q+matcompR.mF&Fbp\"'VFU\"&^M(7&Q+matquatL.mF&Fbp\"'%>'\\\"&9@#7&
Q+matquatR.mF&Fbp\"'3$=&\"&I?#7&Q+matrealL.mF&Fbp\"'na6\"'9k67&Q+Octon
ion.mF&7(F(F)F*F+FH\"\"%\"'J7i\"$P%7&Q,RJgrobner.mF&7(F(F)F*F+\"#f\"#]
\"'y#G'\"%]57&Q+SchurFkt.mF&7(F(F)F*F2\"#6\"#F\"'$p;(\"%DQ7&Q:_FiniteG
roups_rem_table.mF&7(F(F)F*F2Ffn\"#Q\"'5L*)\"$)=7&Q+matrealR.mF&Fbp\"'
\")=B\"'Wj6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "with(code_su
pport);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%inModule~code_cupport~ver.~1.03~for~CLIFFORD~et~al.~fo
r~Maple~11G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%apCopyright~(c)~2002-2
008~by~Rafal~Ablamowicz~and~Bertfried~Fauser.~All~rights~reserved.G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@Last~revised:~December~20,~2007G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#7-%/NamesInLibraryG%1change_helpfilesG%,change_nameG%*copy_fileG%)ge
t_TEXTG%(get_dirG%1insert_helppagesG%)makeLISTG%+modifyLISTG%0replace_
in_fileG%&splitG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "NamesIn
Library(libname[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "68Q*Bigebra.m6\"
Q+Clifford.mF$Q*Cliplus.mF$Q)Define.mF$Q&GTP.mF$Q&GfG.mF$Q+Octonion.mF
$Q,RJgrobner.mF$Q+SchurFkt.mF$Q>_AlternatingGroup_rem_table.mF$Q:_Fini
teGroups_rem_table.mF$Q6_Reynolds_rem_table.mF$Q<_SymmetricGroup_rem_t
able.mF$QA_generateGinvariants_rem_table.mF$Q<_implicitbezierpolynomia
l.mF$Q/code_support.mF$Q+matcompL.mF$Q+matcompR.mF$Q+matquatL.mF$Q+mat
quatR.mF$Q+matrealL.mF$Q+matrealR.mF$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Cookeville, December 20, 2007" }}}
}{MARK "30 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }