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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: with(group)
: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected
 names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "We will use the notational conven
tions on p.10 in the paper \"Chern approximations for generalised grou
p cohomology\" by Neil Strickland for the character table of S_3" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "e:=[]: a:=[[1,2]]: b:=[[1,2,
3]]: CCR:=[e,a,b]: # ordered list of conj cl reps" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "S3:=permgroup(3,\{a,b\}):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "elements(S3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<(7#7$\"\"\"\"\"#7#7%F&F'\"\"$7#7$F'F*7#7$F&F*7#7%F&F*F
'7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "nops(elements(S3));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 363 "mulp2:=proc(perm1,perm2)\ndescription \"Multiply 2
 permutations from right to left (which is conventional) instead of le
ft to right (as mulperms does).\";\n#  Takes a list of permuations and
 reverses their order before feeding it to mulperms.\";\n# mulperms(se
q(args[nargs-i],i=0..(nargs-1)));\n# argh! mulperms only takes two inp
uts!!!!\nmulperms(perm2,perm1);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "mulp2(a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$
\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "mymulp:=proc
()\ndescription \"multiply any number of permutations from right to le
ft\";\nlocal accumulated,m;\n    accumulated := mulp2(args[1],args[2])
; \n    for m from 3 to nargs do\n        accumulated := mulp2(accumul
ated,args[m]);\n    end do;\naccumulated;\nend proc:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 116 "conj1by2:=proc(elt1,elt2)\ndescription \+
\"Computes (elt2)(elt1)(elt2)^(-1)\";\nmymulp(elt2,elt1,invperm(elt2))
; end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "conjugacycl
ass:=proc(elt,gp) local Lgp:\nLgp:=[op(elements(gp))]:\n\{seq(conj1by2
(elt,Lgp[i]),i=1..nops(Lgp))\};\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "conjugacyclass(e,S3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "conjugacycla
ss(a,S3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%7#7$\"\"\"\"\"#7#7$F'\"
\"$7#7$F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "conjugacycla
ss(b,S3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$7#7%\"\"\"\"\"#\"\"$7#7
%F&F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "ordconjcl:=proc
(ccreps,gp) local Lgp;\ndescription \"Takes a list of conjugacy class \+
representatives and a group and produces a list of conjugacy classes i
n the group in the same order as the list of representatives\";\nLgp:=
[op(elements(gp))]: # list of group elts\n#\{seq(conjugacyclass(Lgp[i]
,gp),i=1..nops(Lgp))\}; \n[ seq( [ op(conjugacyclass(ccreps[i],S3)) ],
i=1..nops(ccreps)) ];\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "ordconjcl(CCR,S3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#7%7#7\"7%7#7$\"\"\"\"\"#7#7$F*\"\"$7#7$F)F-7$7#7%F)F*F-7#7%F)F-F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "ordconjcl(CCR,S3)[2]; # th
is allows me to index both the list of representatives and their conju
gacy classes by the same number" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7
#7$\"\"\"\"\"#7#7$F'\"\"$7#7$F&F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "nops(ordconjcl(CCR,S3)[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 "Let's de
fine the character matrix of S_3 with the irreducible representations \+
1, epsilon, sigma indexing the columns from left to right and the conj
ugacy class representatives e=[], a=[1,2], b=[1,2,3] indexing the rows
 from top to bottom." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Cha
rTableS3:=matrix([[1,1,2],[1,-1,0],[1,1,-1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%,CharTableS3G-%'matrixG6#7%7%\"\"\"F*\"\"#7%F*!\"\"\"
\"!7%F*F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "subvector(Ch
arTableS3, 1..3, 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%
\"\"#\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "definec
haracter:=proc(CharName, CC, gp, MtxCol)\ndescription \"CharName is th
e name of the character, and its values on conjugacy class representat
ives are listed in order in the matrix column of the character table c
orrespoding to the elements in a list of conjugacy class representativ
es\";\nlocal i,j;\nfor i from 1 to nops(CC) \n   do \n      #for j fro
m 1 to nops(ordconjcl(CC,gp)[i]) do print(ordconjcl(CC,gp)[i,j],MtxCol
[i]);\n      for j from 1 to nops(ordconjcl(CC,gp)[i]) do `CharName`(o
rdconjcl(CC,gp)[i,j]):=MtxCol[i];\n      od:\n   od:\nreturn: end proc
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "definecharacter(ONE,CC
R,S3,subvector(CharTableS3, 1..3, 1));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "# ONE(e); ONE(a);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 188 "defchars:=proc(CharNames, CCRep, gp, Mtx)\nlocal i;
\n for i from 1 to nops(CharNames) do definecharacter(CharNames[i], CC
Rep, gp, subvector(Mtx, 1..nops(CharNames), i)) od: return:\nend proc:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "defchars( [ONE,epsilon,
sigma], CCR, S3, CharTableS3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 120 "ONE(e); ONE(a); ONE(b); epsilon(e); epsilon(a); epsilon(b); s
igma(e); sigma(a);  sigma(b); # check to make sure it works" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 
"psi:=proc(k,character,conjclassrep)\nlocal i,j; \nseq(`character`( my
mulp( conjclassrep[i]$k) ),i=1..nops(conjclassrep));\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "psi(2,sigma,CCR);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"\"#F#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "sigma([]); sigma([]); sigma([[1,3,2]]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "IChar:=
[1,epsilon,sigma];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ICharG7%\"\"
\"%(epsilonG%&sigmaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "ps
ik2lincombi:=proc(k,character,CC,IrrChar,Mtx)\nlocal CoeffVect;\nCoeff
Vect:=subvector( gaussjord(augment(Mtx,[ psi(k,`character`,CC) ])), 1.
.nops(CC), nops(CC)+1);\n#print(CoeffVect);\nsum(CoeffVect[i]*IrrChar[
i],i=1..nops(IrrChar));\n#seq(IrrChar[i],i=1..nops(IrrChar));\n#add(Co
effMtx[i,nops(CC)+1]*CC[i],i=1..nops(CC));\nend proc:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "psik2lincombi(2,sigma,CCR,IChar,Cha
rTableS3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$%(epsilonG!\"
\"%&sigmaGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "listAdamsO
ps:=proc(i,j,CC,IrrChar,Mtx)\n#seq(seq( lprint(k,IrrChar[l],psik2linco
mbi(k,IrrChar[l],CC,IrrChar,Mtx)), l=1..nops(L)), k=i..j);\nseq(seq( p
rint([psi^k,IrrChar[l],`=`,psik2lincombi(k,IrrChar[l],CC,IrrChar,Mtx)]
), l=1..nops(IrrChar)), k=i..j);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 40 "listAdamsOps(2,6,CCR,IChar,CharTableS3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"F(%\"=GF(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%(epsilonG%\"=GF(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%&sigmaG%\"=G,(F
(F(%(epsilonG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG
\"\"$\"\"\"F(%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"
\"$\"\"\"%(epsilonG%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$p
siG\"\"$\"\"\"%&sigmaG%\"=G,&F(F(%(epsilonGF(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"F(%\"=GF(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"%(epsilonG%\"=GF(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"%&sigmaG%\"=G,(F(F(%(epsilon
G!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"F(
%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%(eps
ilonG%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"
%&sigmaG%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"
\"\"F(%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"\"
\"%(epsilonG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"
'\"\"\"%&sigmaG%\"=G\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Let'
s check a few of these to make sure we've done things correctly" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sigma(mymulp(e,e,e)); sigma(
mymulp(a,a,a)); sigma(mymulp(b,b,b));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gau
ssjord(augment(CharTableS3,[2,0,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'matrixG6#7%7&\"\"\"\"\"!F)F(7&F)F(F)F(7&F)F)F(F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "mymulp(b,b,b,b); # should be b" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 71 "sigma(mymulp(e,e,e,e)); sigma(mymulp(a,a,
a,a)); sigma(mymulp(b,b,b,b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "aug
ment(CharTableS3,[2,2,-1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr
ixG6#7%7&\"\"\"F(\"\"#F)7&F(!\"\"\"\"!F)7&F(F(F+F+" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "gaussjord(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7%7&\"\"\"\"\"!F)F(7&F)F(F)!\"\"7&F)F)F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 30 "We have done things correctly." }}}}{MARK "1 0 0" 60 }
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