{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 
1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 
-1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 
2 2 0 1 }{PSTYLE "Maple Output" -1 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 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map
le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 
1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "This is package SP for sym
metric functions. \n\n" }{TEXT 256 303 "In order to execute this works
heet amd insert this new package into a Maple archive, worksheet SPini
tialization_06_M11.mws needs to be executed first or the Maple archive
 will become corrupted.\nThe worksheet SPinitialization_06_M11.mws ini
tializes remember tables of certain procedures in this package.\n" }
{TEXT -1 30 "\nCookeville, December 20, 2007" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 560 "r
estart:\nSP:=module()\n###################################\nexport SPv
ersion, Sigma, isSymmetric, a_polynom, Schur_polynom, hpolynom, gpolyn
om, powersum,sigma_to_powersum, \nMatrixAction, isGinvariant, Reynolds
, isContained, SymmetricGroup, AlternatingGroup, permsign, \ngenerateG
invariants, reduceGinvariants, load_remember_table, SyzygyIdeal, Dpoly
nom, ispartition,\ncreate_partitions, Hilbert_series, Molien_series, m
axmindegree,FiniteGroups, \nModuleLoad, ModuleUnload;  \n#############
######################\noption package, load=ModuleLoad , unload=Modul
eUnload;" }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1299 "SPve
rsion:= proc()\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz a
nd Bertfried Fauser. All rights reserved.`;\ndescription `Revised: Dec
ember 20, 2007`;\n####################################################
###################\nprint(`++++++++++++++++++++++++++++++++++++++++++
+`);\nprint(`SP - A Maple 11 Small Package for Symmetric Polynomials`)
; \nprint(`Last revised: December 20, 2007 (Source file: SP_06_M11.mws
)`);\nprint(`Copyright 2007-2008 by Rafal Ablamowicz(*) and Bertfried \+
Fauser(**)`);\nprint(``);\nprint(`(*)  Department of Mathematics, Box \+
5054`);\nprint(`     Tennessee Technological University, Cookeville, T
N 38505`);\nprint(`     tel: USA (931) 372-3622, fax: USA (931) 372-63
53`);\nprint(`     rablamowicz@tntech.edu`);\nprint(`     http://math.
tntech.edu/rafal/`);\nprint(`(**) University of Konstanz`);\nprint(`  \+
   Fachbereich Physik, Fach M678`);\nprint(`     Universit\344tsstrass
e 10, D-78457 Konstanz, Germany`);\nprint(`     Phone: +49 (0)7531 693
491`);\nprint(`     E-mail: Bertfried.Fauser@uni-konstanz.de`);\nprint
(`     http://clifford.physik.uni-konstanz.de/~fauser/`);\nprint(``);
\nprint(`++++++++++++This is Symmetric Polynomials Package (SP) for Ma
ple 11 version 06++++++++++++`);\nend proc:\n\n#######################
##############################################################\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 618 "Sigma:=proc() local n,k,L,i:\nopti
ons `Copyright (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser.
 All rights reserved.`;\n#############################################
##########################\nk:=op(procname);\nn:=nargs:\nif n=0 then e
rror `at least one indeterminate is needed as argument` end if:\nif no
t type(k,posint) then error `index must be of type posint` end if:\nif
 k>n then error `index must be no larger than the number of arguments`
 end if:\nL:=combinat:-choose([args],k);\nadd(`*`(op(L[i])),i=1..nops(
L)); \nend proc:\n\n##################################################
###################################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 998 "isSymmetric:=proc(f::polynom) local vars,n,xi,yi,k,T,i,F,GB,r,f
lag,sigma,_y;\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz an
d Bertfried Fauser. All rights reserved.`;\n##########################
#############################################\nvars:=sort(convert(inde
ts(f),list)):\nn:=nops(vars);\n#######################################
#################################\n#### if n<=3 we will use Maple but \+
if n>=4 then we will use FGb package\n################################
########################################\nxi:=op(vars):\nyi:=seq(_y[k]
,k=1..n);\nT:=plex(xi,yi);\nif n<=3 then \n   F:=[seq(SP:-Sigma[i](xi)
 - _y[i],i=1..n)];\n   GB:=Groebner:-Basis(F,T);\n   else\n   GB:=FGb:
-fgb_gbasis([seq(SP:-Sigma[i](xi)-_y[i],i=1..n)],0,[xi,yi],[]);\nend i
f:\nr:=Groebner:-NormalForm(f,GB,T);\nflag:=`subset`(indets(r),\{yi\})
;\nif not flag or nargs=1 then return flag end if;\nsubs(\{seq(_y[i]=s
igma[i],i=1..n)\},r);\nend proc:\n\n##################################
###################################################\n" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 765 "hpolynom:=proc() local s,i,L,M,j,k,P:\noptions \+
`Copyright (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All
 rights reserved.`;\n#################################################
######################\ns:=nargs:\nif not type(s,posint) then error `n
umber of arguments must be of type posint` end if:\ni:=op(procname);\n
if not type(i,nonnegint) then error `index must be of type nonnegint` \+
end if:\nif i=0 then return 1 end if:\nL:=[]:\nP:=combinat:-partition(
i):\nfor j from 1 to nops(P) do\n    if nops(P[j]) <= s then \n       \+
L:=[op(L),[op(P[j]),0$(s-nops(P[j]))]];\n    end if:\nend do;\nM:=map(
op@combinat:-permute,L);\nadd(product(args[k]^M[j][k],k=1..s),j=1..nop
s(M));\nend proc:\n\n#################################################
####################################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 665 "gpolynom:=proc() local i,j,k,n,t;\noptions `Copyright (c) 200
7-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`
;\n###################################################################
####\nn:=nargs:\nif not type(n,posint) then error `number of arguments
 must be of type posint` end if:\nk:=op(procname):\nif not (type(k,pos
int) and k <=n) then \n   error `index must be of type posint and it m
ust be no larger than number of arguments`\nend if:\nSP:-hpolynom[k](s
eq(args[k+i],i=0..(n-k)))+add((-1)^j*SP:-hpolynom[k-j](seq(args[k+i],i
=0..(n-k)))*y[j],j=1..k);\nend proc:\n\n##############################
#######################################################\n" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 487 "powersum:=proc() local i,k,n;\noptions `Cop
yright (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rig
hts reserved.`;\n#####################################################
##################\nk:=op(procname):\nif not k>=1 then error `index mu
st be of type posint` end if;\nn:=nargs:\nif n=0 then error `at least \+
one indeterminate is needed as argument` end if: \nadd(args[i]^k,i=1..
n);\nend proc:\n\n####################################################
#################################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
746 "sigma_to_powersum:=proc(ss) local n,s,k,p;\noptions `Copyright (c
) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights reser
ved.`;\n##############################################################
#########\nn:=nargs:\nif not type(n,posint) then error `number of argu
ments must be of type posint` end if:\nk:=op(procname);\nif not type(k
,posint) then error `index must be of type posint` end if:\nif k>n the
n error `index must be no greater than the number of variables in the \+
sequence` end if: \nif k=1 then return s[1] else\n   p:=(-1)^(k-1)*1/k
*(s[k] +add((-1)^j*procname[j](args)*s[k-j],j=1..(k-1))); \nend if;\nr
eturn expand(convert(p,`global`));\nend proc:\n\n#####################
################################################################\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 678 "a_polynom:=proc() local n,i,j,r,la
mbda:\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz and Bertfr
ied Fauser. All rights reserved.`;\n##################################
#####################################\nlambda:=op(1,procname):\nif not
 SP:-ispartition(lambda) then \n   error `index must be a partition en
tered as a list` \nend if:\nn:=nargs:\nif n<>nops(lambda) then \n   er
ror `length of the partition lambda must equal the number of arguments
` \nend if;\n\nfor j from 1 to n do\nr[j]:=seq(args[i]^(lambda[j]+n-j)
,i=1..n);\nend do:\nreturn linalg:-det(matrix(n,n,[seq(r[i],i=1..n)]))
;\nend proc:\n\n######################################################
###############################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
393 "Schur_polynom:=proc() local lambda:\noptions `Copyright (c) 2007-
2008 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;
\n####################################################################
###\nlambda:=op(1,procname);\nexpand(simplify(SP:-a_polynom[lambda](ar
gs)/SP:-Dpolynom(args)));\nend proc:\n\n##############################
#######################################################\n" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 1300 "MatrixAction:=proc(A::\{matrix,matrix^nonn
egint\}, f::polynom, varslist::list(\{symbol,indexed\})) local m,n,var
s,_X,ringvars,findinlist,p:\noptions `Copyright (c) 2007-2008 by Rafal
 Ablamowicz and Bertfried Fauser. All rights reserved.`;\n############
###########################################################\nfindinlis
t:=proc() local i,flag:\nflag:=false:\nif not member(args[1],args[2]) \+
then return flag end if:\nfor i from 1 to nops(args[2]) while not flag
 do\n    if args[1]=args[2][i] then flag:=true end if;\nend do:\nretur
n i-1;\nend proc:\n###################################################
######\nm,n:=linalg:-rowdim(A),linalg:-coldim(A):\nif m<>n then \n    \+
    error `matrix A needs to be square` \nend if:\nif nops(varslist) <
> m then \n                       error `size of the matrix A must equ
al the length of the variable list` \nend if:\nif not `subset`(indets(
f),convert(varslist,set)) then \n   error `variables in polynomial f m
ust be a subset of the set of the ring indeterminates`\nend if;\nvars:
=sort([op(indets(f))]):\nringvars:=sort(varslist):\n_X:=evalm(A &* mat
rix(m,1,ringvars));\n_X:=[seq(_X[i,1],i=1..m)];\np:=expand(subs(\{seq(
vars[i]=_X[findinlist(vars[i],ringvars)],i=1..nops(vars))\},f));\n#ret
urn p/icontent(p);\nreturn p;\nend proc:\n\n##########################
##################################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 1555 "isGinvariant:=proc(f::\{polynom,monomial\},varslist::list(\{sy
mbol,indexed\})) local G,rdim,cdim,size,vars,nvars,flag,A:\noptions `C
opyright (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All r
ights reserved.`;\n###################################################
####################\nG:=op(procname);\nif not type(G,list(\{name,symb
ol,matrix,algebraic\})) then \n   error `index G must be of type list(
\{name,symbol,matrix,algebraic\})` \nend if;\nrdim,cdim:=convert(map(l
inalg:-rowdim,G),set),convert(map(linalg:-rowdim,G),set);\nif nops(rdi
m) <> 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):\nva
rs:=sort([op(indets(f))]):\nnvars:=nops(vars):\nif nvars>size then \n \+
  error `number of indeterminates cannot exceed the size of group G ma
trices` \nend if:\n###################################################
#########\nif nops(varslist) <> size then \n                          \+
error `size of the matrix A must equal the length of the variable list
` \nend if:\n#########################################################
###\nif not `subset`(indets(f),convert(varslist,set)) then \n   error \+
`variables in polynomial f must be a subset of the set of the ring ind
eterminates`\nend if;\n###############################################
#############\nflag:=true:\nfor A in G while flag do\n      flag:=eval
b(f=SP:-MatrixAction(A,f,varslist)):\nend do:\nreturn flag;\nend proc:
\n\n############################################################\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 551 "Reynolds:=proc(f::polynom, varslis
t::list(\{symbol,indexed\})) local G,m:\noptions `Copyright (c) 2007-2
008 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`,re
member;\n#############################################################
###################\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;\nm:=nops(G):\nadd(1/m*SP:-
MatrixAction(G[i],f,varslist),i=1..m);\nend proc:\n\n#################
###########################################\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1249 "isContained:=proc(fp::polynom,ff::list(polynom)) lo
cal n,m,varsF,vars,xi,yi,k,T,i,F,GB,r,flag,_f,_y;\noptions `Copyright \+
(c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights res
erved.`,remember;\n###################################################
#############################\nm:=nops(ff):\nif not m>=1 then error `l
ist ff must have at least one polynomial` end if:\nvarsF:=`union`(seq(
indets(ff[i]),i=1..m));\nn:=nops(varsF):\nvars:=indets(fp):\n#if not `
subset`(vars,varsF) then error `indeterminates in fp must be a subset \+
of indeterminates of ff` end if;\nvars:=sort(convert(vars,list)):\n###
#####################################################################
\n#### if n<=2 we will use Maple but if n>=3 then we will use FGb pack
age\n#################################################################
#######\nxi:=op(varsF):\nyi:=seq(_y[k],k=1..m);\nT:=plex(xi,yi);\nif n
<=1 then \n   F:=[seq(ff[i] - _y[i],i=1..m)];\n   GB:=Groebner:-Basis(
F,T);\n   else\n   GB:=FGb:-fgb_gbasis([seq(ff[i]-_y[i],i=1..m)],0,[xi
,yi],[]);\nend if:\nr:=Groebner:-NormalForm(fp,GB,T);\nflag:=`subset`(
indets(r),\{yi\});\nif not flag or nargs=2 then return flag end if;\ns
ubs(\{seq(_y[i]=_f[i],i=1..m)\},r);\nend proc:\n\n####################
########################################\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 459 "SymmetricGroup:=proc(n::posint) local II,m,i,L,G;\no
ptions `Copyright (c) 2007-2008 by Rafal Ablamowicz and Bertfried Faus
er. All rights reserved.`,remember;\n#################################
###############################################\nII:=linalg:-diag(1$n)
:\nL:=combinat:-permute(n);\nG:=[]:\nfor m in L do\nG:=[op(G),linalg:-
augment(seq(linalg:-col(II,m[i]),i=1..n))];\nend do;\nreturn G;\nend p
roc:\n\n############################################################\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 853 "permsign:=proc(L::list) local \+
newbas,ss,a,n12,s12,L1,L2,N,f,dummy_set,K,x;\noptions `Copyright (c) 2
007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved
.`;\n#################################################################
######\nL1:=L:\nN:=nops(L1):\nif N=1 then return 1 end if:\n##########
######## new\nn12,s12:=selectremove(member,L1,\{1,2,3,4,5,6,7,8,9\});
\n#s12:=remove(member,L1,\{1,2,3,4,5,6,7,8,9\});\nL2:=[op(sort(n12)),o
p(sort(s12))];\n################## new\nf:=proc() end proc:\nfor ss fr
om 1 to N do\n  f(L2[ss]):=L1[ss];\nend do;\ndummy_set:=convert(L1,set
);\nK:=0:\nwhile dummy_set <> \{\} do\n  a:=dummy_set[1]:\n  dummy_set
:=dummy_set[2..-1];\n  x:=a:\n  while f(x)<>a do\n      x:=f(x);\n    \+
  dummy_set:=dummy_set minus \{x\};\n      K:=K+1;\n  end do:\nend do;
\nreturn (-1)^K;\nend proc:\n\n#######################################
#####################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 572 "Alterna
tingGroup:=proc(n::posint) local II,m,i,L,Leven,G;\noptions `Copyright
 (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights re
served.`, remember;\n#################################################
#######################################\nII:=linalg:-diag(1$n):\nL:=co
mbinat:-permute(n);\nLeven:=[]:\nfor 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:\n\n########################################
####################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1473 "generat
eGinvariants:=proc(vars::list(monomial)) local G,ord,nvars,T,i,j,INV,m
,n,s,alpha,mons,Rmons,rdim,cdim,size;\noptions `Copyright (c) 2007-200
8 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`, rem
ember:\n##############################################################
#########################\nG:=op(procname);\nif not type(G,list(\{name
,symbol,matrix,algebraic\})) then \n   error `index G must be of type \+
list(\{name,symbol,matrix,algebraic\})` end if;\nrdim,cdim:=convert(ma
p(linalg:-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):
\nnvars:=nops(vars):\nif nvars<>size then error `number of indetermina
tes must equal the size of group G matrices` end if:\n################
############################################\nord:=nops(G):\nT:=combin
at:-cartprod([seq([seq(i,i=0..ord)],j=1..nvars)]):\nINV:=[]:\nwhile no
t T[finished] 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:=[se
q(mul(vars[i]^alpha[i],i=1..nvars),alpha=INV)];\nRmons:=map(SP:-Reynol
ds[G],mons,vars);\nRmons:=convert(convert(Rmons,set),list);\nRmons:=re
move(type,Rmons,numeric);\n#return Rmons;\nRmons:=[seq(mm/content(mm),
mm=Rmons)]:\nreturn Rmons; \nend proc:\n\n############################
################################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
2551 "reduceGinvariants:=proc(L::list(polynom),varslist::list(\{symbol
,indexed\})) local i,m,mm,temp,Rmons,flag,stopflag,G,rdim,cdim,size,va
rs,nvars:\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz and Be
rtfried Fauser. All rights reserved.`, remember:\n####################
####################################################################\n
Rmons:=convert(convert(L,set),list);\nRmons:=remove(type,Rmons,numeric
):\n###########################################################\nG:=op
(procname);\n#########################################################
##\nif not type(G,list(\{name,symbol,matrix,algebraic\})) then \n   er
ror `index G must be of type list(\{name,symbol,matrix,algebraic\})` e
nd if;\nrdim,cdim:=convert(map(linalg:-rowdim,G),set),convert(map(lina
lg:-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#################################################
###########\nvars:=`union`(op(map(indets,L))):\nif not `subset`(vars,c
onvert(varslist,set)) then \n   error `variables in polynomial f must \+
be a subset of the set of the ring indeterminates`\nend if;\n#########
###################################################\nsize:=op(rdim):\n
nvars:=nops(vars):\nif nvars>size then \n   error `number of indetermi
nates cannot exceed the size of group G matrices` \nend if:\n#########
###################################################\nif nops(varslist)
 <> size then \n                          error `size of group matrice
s must equal the length of the variable list` \nend if:\n#############
###############################################\nif not \{op(map(SP:-i
sGinvariant[G],Rmons,varslist))\}=\{true\} then\n   error `at least on
e of the polynomials in the list is not a G invariant`\nend if;\n#####
#######################################################\nif nops(Rmons
)=1 then return Rmons end if:\n#######################################
#####################\nRmons:=sort(Rmons, (m1,m2)->Groebner:-TestOrder
(m1,m2,'tdeg'(op(varslist))));\nRmons:=[seq(Rmons[nops(Rmons)-i+1],i=1
..nops(Rmons))];\n####################################################
########\nstopflag:=false:\ni:=1:\nwhile not stopflag do\n i;\n Rmons;
\n m:=Rmons[i]:\n temp:=subsop(i=NULL,Rmons):\n flag:=SP:-isContained(
m,temp);\n if flag then Rmons:=temp else\n    if nops(Rmons)>i then i:
=i+1 else stopflag:=true end if;\n end if;\nif i>nops(Rmons) then stop
flag:=true end if:\nend do:\n#return Rmons;\nRmons:=[seq(mm/content(mm
),mm=Rmons)]:\nreturn Rmons;\nend proc:\n\n###########################
#################################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
1010 "SyzygyIdeal:=proc(ff::list(polynom)) local m,i,varsF,n,xi,yi,T,F
,_y,GB,IdealF;\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz a
nd Bertfried Fauser. All rights reserved.`;\n#########################
##############################################\nm:=nops(ff):\nif not m
>=1 then error `list ff must have at least one polynomial` end if:\n##
###################################\nvarsF:=sort(convert(`union`(seq(i
ndets(ff[i]),i=1..m)),list));\nn:=nops(varsF):\nxi:=op(varsF):\nyi:=se
q(_y[k],k=1..m);\nT:=plex(xi,yi);\nif n<=1 then \n   F:=[seq(ff[i] - _
y[i],i=1..m)];\n   GB:=Groebner:-Basis(F,T);\n   else\n   GB:=FGb:-fgb
_gbasis([seq(ff[i]-_y[i],i=1..m)],0,[xi,yi],[]);\nend if:\n###########
##########################\nif GB=[0] then return GB end if:\n########
#############################\nIdealF:=[]:\nfor m in GB do\n    if `su
bset`(indets(m),\{yi\}) then IdealF:=[op(IdealF),m] end if;\nend do;\n
if not IdealF=[] then return IdealF else return [0] end if;\nend proc:
\n\n############################################################\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 422 "Dpolynom:=proc() local i,j,n;\nopt
ions `Copyright (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser
. All rights reserved.`;\n############################################
###########################\nn:=nargs:\nif not n>=2 then error `number
 of indexed arguments needs to be at least two` end if:\n`*`(seq(seq(a
rgs[i]-args[j],j=(i+1)..n),i=1..(n-1)));\nend proc:\n\n###############
#############################################\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 651 "ispartition:=proc(L::list(nonnegint)) local i,flag,n
:\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz and Bertfried \+
Fauser. All rights reserved.`;\n######################################
#################################\nif nargs=2 and not type(args[2],non
negint) then error `second argument, if used, must of type nonnegint` \+
end if:\nn:=nops(L):\nif nargs>2 then return args \n   elif nargs=2 th
en flag:=evalb(add(i,i=L)=args[2]) else flag:=true \nend if;\nif not f
lag then return flag end if;\nfor i from 1 to n-1 while flag do\n    f
lag:=evalb(L[i]>=L[i+1])\nend do:\nreturn flag;\nend proc:\n\n########
####################################################\n " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 543 "create_partitions:=proc(d::nonnegint,n::nonne
gint) local i,j,INV,m,s,T;\noptions `Copyright (c) 2007-2008 by Rafal \+
Ablamowicz and Bertfried Fauser. All rights reserved.`;\n#############
##########################################################\nT:=combina
t:-cartprod([seq([seq(i,i=0..d)],j=1..n)]):\nINV:=[]:\nwhile not T[fin
ished] do\n      m:=T[nextvalue]():\n      s:=add(i,i=m):\n      if s=
d and s>0 and SP:-ispartition(m) then INV:=[op(INV),m] end if: \nend d
o;\nreturn INV;\nend proc:\n\n########################################
####################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 617 "maxminde
gree:=proc(f::polynom,T::\{MonomialOrder,ShortMonomialOrder\}) local d
egrees,c,m,L,ff;\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz
 and Bertfried Fauser. All rights reserved.`;\n#######################
################################################\nff:=expand(f):\ndegr
ees:=[]:\nwhile ff<>0 do\nc,m:=Groebner:-LeadingTerm(ff,T);\nif nops(i
ndets(m))>1 then\n   L:=[op(m)];\n   degrees:=[op(degrees),`+`(op(map(
degree,L)))];\nelse\n   degrees:=[op(degrees),degree(m)];\nend if;\nff
:=ff-c*m;\nend do;\nreturn [max(op(degrees)),min(op(degrees))];\nend p
roc:\n\n############################################################\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 829 "Molien_series:=proc(indet::\{s
ymbol\}) local G,n,m,In,Gord,S,N;\noptions `Copyright (c) 2007-2008 by
 Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\n######
#################################################################\nG:=
op(1,procname);\nGord:=nops(G);\nn:=convert(map(linalg:-coldim,G),set)
;\nm:=convert(map(linalg:-rowdim,G),set);\nif nargs=2 and not type(arg
s[2],posint) then \n   error `second argument, when used, must be a po
sitive integer`\nend if:\nif nargs=1 then N:=Order else N:=args[2] end
 if:\nif nops(n)<>1 or nops(m)<>1 or not evalb(n=m) then \n   error `m
atrices must be square` \nend if;\nn:=op(n):\nIn:=linalg:-diag(1$n);\n
S:=0:\nfor m in G do\nS:=S+linalg:-det(In-indet*evalm(m))^(-1);\nend d
o;\nS:=series(normal(S/Gord),indet,N):\nreturn S;\nend proc:\n\n######
######################################################\n" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1077 "Hilbert_series:=proc(R::list(homogeneouspol
ynom),indet::symbol) local F,numerz,degF,N,m,vars,d,i;\noptions `Copyr
ight (c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All right
s reserved.`;\n#######################################################
################\nF:=op(1,procname):\nif not type(F,\{symbol,name,homo
geneouspolynom\}) then \n   error `index F when used must of type homo
geneouspolynom` \nend if;\nif evalb(F='F') then numerz:=1 else\n   deg
F:=degree(F,convert(indets(F),list));\n   numerz:=indet^degF;\nend if:
\nif not evalb(SP:-SyzygyIdeal(R)=[0]) then \n   error `polynomials in
 list R must be algebraically independent, but they are not: Their syz
ygy ideal is not [0].`\nend if;;\nif nargs=3 and not type(args[3],posi
nt) then \n   error `third argument, when used, must be a positive int
eger`\nend if:\nif nargs=2 then N:=Order else N:=args[3] end if:\nm:=n
ops(R):\nvars:=op(sort(convert(`union`(op(map(indets,R))),list)));\nd:
=map(degree,R);\nseries(numerz*`mul`((1-indet^d[i])^(-1),i=1..m),indet
,N);\nend proc:\n\n###################################################
#########\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3089 "FiniteGroups:=proc
(G::symbol) local k,A,A1,A2,A12,A21,_C2,_C3,_C4,_C6,_C8,_D6,_D8,_V4,_G
8,n,T,K:\noptions `Copyright (c) 2007-2008 by Rafal Ablamowicz and Ber
tfried Fauser. All rights reserved.`, remember;\n#####################
###################################################################\n#
if nargs=0 then \n#   K:=op(sort(map(op,[indices(op(4,eval(SP:-FiniteG
roups)))]))):\n#   print(K);\n#   return Null; \n#end if;\n###########
######################################\nif G='C2' then \n   A:=matrix(
2,2,[-1,0,0,-1]);\n   _C2:=map(evalm,[linalg:-diag(1$2),A]);\n   retur
n _C2;\n##################################################\nelif G='C3
' then\n   A:=matrix(2,2,[0,-1,1,-1]);\n   _C3:=[linalg:-diag(1$2),seq
(evalm(A^k),k=1..2)]:\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='C6' then\n   A:=matrix(2,2,
[0,1,-1,1]);\n   _C6:=[linalg:-diag(1$2),seq(evalm(A^k),k=1..5)]:\n###
###############################################\nelif G='C8' then\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]:=mat
rix(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]:=matrix(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:=ma
trix(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 &* A2, A12
^6,A12 &* A1,A2,\n                                       A12^4,A12^2,A
1,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##################################################\neli
f 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     re
turn _V4;\n##################################################\nelif G=
'G8' then\n     n:=3: ##group will have order 2^n\n     T:=combinat:-c
artprod([seq([1,-1],i=1..n)]):\n     _G8:=[]:\n     while not T[finish
ed] do _G8:=[op(_G8),linalg:-diag(op(T[nextvalue]()))] end do:\n     r
eturn _G8;\n##################################################\nelse \+
\nprintf(\"Group %s is not known to FiniteGroups. You can make it know
n by assigning it to FiniteGroups('%s').\\n\",G,G);\nend if;\nend proc
:\n\n############################################################\n" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 646 "load_remember_table:=proc(NAME::
\{string,symbol\}) local REM_TABLE,IND,ENTRIES,i;\noptions `Copyright \+
(c) 2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights res
erved.`;\n############################################################
###########\nREM_TABLE:=readlib(cat('_',NAME,'_rem_table'));\nif evalb
(indices(REM_TABLE)=NULL)=true then return NULL end if;\nIND:=[indices
(REM_TABLE)];\nENTRIES:=[entries(REM_TABLE)]:\nfor i from 1 to nops(IN
D) do\n    NAME(op(IND[i])):=op(ENTRIES[i]);\nend do;\nprintf(\"Rememb
er table of %s has been read and assigned\\n\",NAME);\nreturn NULL;\ne
nd proc:\n\n##########################################################
##\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1149 "ModuleLoad:=proc() local \+
i,n,K;global `type/homogeneouspolynom`;\noptions `Copyright (c) 2007-2
008 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\n
######################################################################
#\nSP:-load_remember_table('SymmetricGroup');\nSP:-load_remember_table
('AlternatingGroup');\nSP:-load_remember_table('Reynolds');\nSP:-load_
remember_table('FiniteGroups');\nSP:-load_remember_table('generateGinv
ariants');\n##############################################\n#n:=nargs:
\n#for i from 1 to n do\n#    sigma[i]:=SP:-Sigma[i](args);\n#end do:
\n###############################################\n`type/homogeneouspo
lynom`:=proc(f::polynom) local vars,maxmin,T;\noptions `Copyright (c) \+
2007-2008 by Rafal Ablamowicz and Bertfried Fauser. All rights reserve
d.`;\n################################################################
#######\n     vars:=op(sort(convert(indets(f),list))):\n     T:=plex(v
ars):\n     maxmin:=SP:-maxmindegree(f,T):\n     evalb(maxmin[1]=maxmi
n[2]):\nend proc:\n###############################################\nun
protect(D);\nreturn NULL;\nend proc:\n\n##############################
###############################\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
1495 "ModuleUnload:=proc() global _SymmetricGroup_rem_table,_Alternati
ngGroup_rem_table,\n_Reynolds_rem_table,_generateGinvariants_rem_table
,_FiniteGroups_rem_table;\noptions `Copyright (c) 2007-2008 by Rafal A
blamowicz and Bertfried Fauser. All rights reserved.`;\n##############
#########################################################\n###########
############################################\n_SymmetricGroup_rem_tabl
e:=op(4,eval(SP:-SymmetricGroup)):\nif not _SymmetricGroup_rem_table=N
ULL then savelib(_SymmetricGroup_rem_table) end if;\n#################
######################################\n_AlternatingGroup_rem_table:=o
p(4,eval(SP:-AlternatingGroup)):\nif not _AlternatingGroup_rem_table=N
ULL then savelib(_AlternatingGroup_rem_table) end if;\n###############
########################################\n_Reynolds_rem_table:=op(4,ev
al(SP:-Reynolds)):\nif not _Reynolds_rem_table=NULL then savelib(_Reyn
olds_rem_table) end if;\n#############################################
##########\n_FiniteGroups_rem_table:=op(4,eval(FiniteGroups)):\nif not
 _FiniteGroups_rem_table=NULL then savelib(_FiniteGroups_rem_table) en
d if;\n#######################################################\n_gener
ateGinvariants_rem_table:=op(4,eval(SP:-generateGinvariants)):\nif not
 _generateGinvariants_rem_table=NULL then savelib(_generateGinvariants
_rem_table) end if;\n#################################################
######\n\n#n:=nargs:\n#for i from 1 to n do\n#    sigma[i]:=SP:-Sigma[
i](args);\n#end do:\nreturn NULL;\nend proc:\n\nend module:" }}{PARA 
6 "" 1 "" {TEXT -1 59 "Remember table of SymmetricGroup has been read \+
and assigned" }}{PARA 6 "" 1 "" {TEXT -1 61 "Remember table of Alterna
tingGroup has been read and assigned" }}{PARA 6 "" 1 "" {TEXT -1 53 "R
emember table of Reynolds has been read and assigned" }}{PARA 6 "" 1 "
" {TEXT -1 57 "Remember table of FiniteGroups has been read and assign
ed" }}{PARA 6 "" 1 "" {TEXT -1 64 "Remember table of generateGinvarian
ts has been read and assigned" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "savelibname;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7C:\\Maple11/Cl
iffordlib6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Note: If you hav
e not created yet your library archive with march command before, unre
mark this next command and execute it.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "with(LibraryTools);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#72%1ActivationModuleG%1AddFromDirectoryG%'AuthorG%'BrowseG%3BuildFro
mDirectoryG%/ConvertVersionG%'CreateG%'DeleteG%,FindLibraryG%,PrefixMa
tchG%)PriorityG%%SaveG%-ShowContentsG%*TimestampG%4UpdateFromDirectory
G%*WriteModeG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ShowConten
ts(libname[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#797&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$7&Q/code_suppo
rt.mF&7(F(F)F*F+\"#X\"#J\"'K5f\"$w%7&Q)Define.mF&7(F(F)F*F+\"#9\"#d\"'
Q.a\"$$[7&Q&GTP.mF&7(F(F)F*F+\"#`\"#@\"'%Q8'\"$\"G7&Q&GfG.mF&7(F(F)F*F
+\"#^\"\"*\"'&e,'\"$:(7&Q>_AlternatingGroup_rem_table.mF&7(F(F)F*F2\"#
I\"#C\"'$Q;*FQ7&Q6_Reynolds_rem_table.mF&7(F(F)F*F2Ffn\"#D\"'km\"*\"$L
%7&Q<_SymmetricGroup_rem_table.mF&Fen\"'*49*\"%%G#7&QA_generateGinvari
ants_rem_table.mF&F[o\"'&G<*\"$#>7&Q+matcompL.mF&7(F(F)F*F+\"#8FG\"'D#
[$\"&=X(7&Q+matcompR.mF&Fio\"'VFU\"&^M(7&Q+matquatL.mF&Fio\"'%>'\\\"&9
@#7&Q+matquatR.mF&Fio\"'3$=&\"&I?#7&Q+matrealL.mF&Fio\"'na6\"'9k67&Q+O
ctonion.mF&7(F(F)F*F+FH\"\"%\"'J7i\"$P%7&Q,RJgrobner.mF&7(F(F)F*F+\"#f
\"#]\"'y#G'\"%]57&Q%SP.mF&7(F(F)F*F2\"#HF@\"'!p$*)\"%867&Q+SchurFkt.mF
&7(F(F)F*F2\"#6\"#F\"'$p;(\"%DQ7&Q:_FiniteGroups_rem_table.mF&F[o\"'(4
<*\"$)=7&Q<_implicitbezierpolynomial.mF&7(F(F)F*F2\"\"!FO\"'dkk\"&@7\"
7&Q+matrealR.mF&Fio\"'\")=B\"'Wj6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
177 "\nOnce you have executed the above command, either you get a mess
age that archive already exists, or, no error messag is returned. In e
ither case, remark back the above command.\n" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 85 "Do not execute the next command unless \"savelibname\" \+
above returns \".../Cliffordlib\"\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "savelibname;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q7C:
\\Maple11/Cliffordlib6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"savelib('SP');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ShowCont
ents(libname[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#797&Q*Bigebra.m6
\"7(\"%2?\"#7\"#?\"#:\"#<F*\"'/da\"%Z77&Q+Clifford.mF&7(F(F)F*\"#;F*\"
#S\"'3dx\"%JC7&Q*Cliplus.mF&7(F(F)F*F+\"#>\"#\\\"'Ywc\"$x$7&Q/code_sup
port.mF&7(F(F)F*F+\"#X\"#J\"'K5f\"$w%7&Q)Define.mF&7(F(F)F*F+\"#9\"#d
\"'Q.a\"$$[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
\"#I\"#C\"'$Q;*FQ7&Q6_Reynolds_rem_table.mF&7(F(F)F*F2Ffn\"#D\"'km\"*
\"$L%7&Q<_SymmetricGroup_rem_table.mF&Fen\"'*49*\"%%G#7&QA_generateGin
variants_rem_table.mF&F[o\"'&G<*\"$#>7&Q+matcompL.mF&7(F(F)F*F+\"#8FG
\"'D#[$\"&=X(7&Q+matcompR.mF&Fio\"'VFU\"&^M(7&Q+matquatL.mF&Fio\"'%>'
\\\"&9@#7&Q+matquatR.mF&Fio\"'3$=&\"&I?#7&Q+matrealL.mF&Fio\"'na6\"'9k
67&Q+Octonion.mF&7(F(F)F*F+FH\"\"%\"'J7i\"$P%7&Q,RJgrobner.mF&7(F(F)F*
F+\"#f\"#]\"'y#G'\"%]57&Q%SP.mF&7(F(F)F*F2Ffn\"#G\"'xu\"*\"%867&Q+Schu
rFkt.mF&7(F(F)F*F2\"#6\"#F\"'$p;(\"%DQ7&Q:_FiniteGroups_rem_table.mF&F
[o\"'(4<*\"$)=7&Q<_implicitbezierpolynomial.mF&7(F(F)F*F2\"\"!FO\"'dkk
\"&@7\"7&Q+matrealR.mF&Fio\"'\")=B\"'Wj6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#Delete
('SP',libname[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "#Show
Contents(libname[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#D
elete('_FiniteGroups_rem_table',libname[1]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "ShowContents(libname[1]);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#797&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>_Alternati
ngGroup_rem_table.mF&7(F(F)F*F2\"#I\"#C\"'$Q;*FQ7&Q6_Reynolds_rem_tabl
e.mF&7(F(F)F*F2Ffn\"#D\"'km\"*\"$L%7&Q<_SymmetricGroup_rem_table.mF&Fe
n\"'*49*\"%%G#7&QA_generateGinvariants_rem_table.mF&F[o\"'&G<*\"$#>7&Q
+matcompL.mF&7(F(F)F*F+\"#8FG\"'D#[$\"&=X(7&Q+matcompR.mF&Fio\"'VFU\"&
^M(7&Q+matquatL.mF&Fio\"'%>'\\\"&9@#7&Q+matquatR.mF&Fio\"'3$=&\"&I?#7&
Q+matrealL.mF&Fio\"'na6\"'9k67&Q+Octonion.mF&7(F(F)F*F+FH\"\"%\"'J7i\"
$P%7&Q,RJgrobner.mF&7(F(F)F*F+\"#f\"#]\"'y#G'\"%]57&Q%SP.mF&7(F(F)F*F2
Ffn\"#G\"'xu\"*\"%867&Q+SchurFkt.mF&7(F(F)F*F2\"#6\"#F\"'$p;(\"%DQ7&Q:
_FiniteGroups_rem_table.mF&F[o\"'(4<*\"$)=7&Q<_implicitbezierpolynomia
l.mF&7(F(F)F*F2\"\"!FO\"'dkk\"&@7\"7&Q+matrealR.mF&Fio\"'\")=B\"'Wj6" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 80 "##################################################
##############################" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The name of t
he package is SP for Symmetric Polynomials package. You can load it vi
a the with(SP); command:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "with(SP);" }}{PARA 6 "" 1 "" {TEXT -1 59 "Remember table of Symmetr
icGroup has been read and assigned" }}{PARA 6 "" 1 "" {TEXT -1 61 "Rem
ember table of AlternatingGroup has been read and assigned" }}{PARA 6 
"" 1 "" {TEXT -1 53 "Remember table of Reynolds has been read and assi
gned" }}{PARA 6 "" 1 "" {TEXT -1 57 "Remember table of FiniteGroups ha
s been read and assigned" }}{PARA 6 "" 1 "" {TEXT -1 64 "Remember tabl
e of generateGinvariants has been read and assigned" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7?%1AlternatingGroupG%)DpolynomG%-FiniteGroupsG%/Hilb
ert_seriesG%-MatrixActionG%+ModuleLoadG%-ModuleUnloadG%.Molien_seriesG
%)ReynoldsG%*SPversionG%.Schur_polynomG%&SigmaG%/SymmetricGroupG%,Syzy
gyIdealG%*a_polynomG%2create_partitionsG%4generateGinvariantsG%)gpolyn
omG%)hpolynomG%,isContainedG%-isGinvariantG%,isSymmetricG%,ispartition
G%4load_remember_tableG%-maxmindegreeG%)permsignG%)powersumG%2reduceGi
nvariantsG%2sigma_to_powersumG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 54 "SymmetricGroup(2),SymmetricGroup(3),SymmetricGroup(4);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6%7$K%'matrixG6#7$7$\"\"\"\"\"!7$F*F)Q(p
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