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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart: with(linalg
): with(group):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected
 names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "See p. 42-43 in \"Linear Represen
tations of Finite Groups\" by J.P. Serre, and p. 170, 179-180, in \"Re
presentations and Characters of Groups (first edition)\" by James and \+
Liebeck." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "x:=[[1,2],[3,4]
]:  y:=[[1,3],[2,4]]:  z:=[[1,4],[2,3]]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Representativ
es of the five conjugacy classes in S_4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "a:=[[1,2]]: b:=[[1,2],[3,4]]: c:=[[1,2,3]]: d:=[[1,2,
3,4]]: e:=[]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "S4:=permg
roup(4, \{a,b,c,d,e\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
elements(S4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<:7\"7$7$\"\"\"\"\"#7
$\"\"$\"\"%7$7$F'F*7$F(F+7$7$F'F+7$F(F*7#F&7#7%F'F(F*7#7&F'F(F*F+7#7&F
'F+F*F(7#7&F'F+F(F*7#7&F'F*F+F(7#7&F'F*F(F+7#7&F'F(F+F*7#7%F'F*F(7#7%F
'F*F+7#F.7#7%F(F+F*7#F07#7%F(F*F+7#F17#F-7#7%F'F+F*7#7%F'F+F(7#F)7#7%F
'F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grouporder(S4);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Let's make s
ome procedures to multiply two or three permutations together using th
e right to left convention that is standard" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 61 "# mulperms(b,x); # this multiplication is left to \+
right!!!!!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mulp2:=proc
(perm1,perm2)\ndescription \"Multiply 2 permutations from right to lef
t (which is conventional) instead of left to right (as mulperms does).
\";\n#  Takes a list of permuations and reverses their order before fe
eding it to mulperms.\";\n# mulperms(seq(args[nargs-i],i=0..(nargs-1))
);\n# argh! mulperms only takes two inputs!!!!\nmulperms(perm2,perm1);
\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "reverselist
:=proc()\n#mulperms(seq(args[nargs-i],i=0..(nargs-1)));\nseq(args[narg
s-i],i=0..(nargs-1));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "reverselist(b,c,d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6%7#7&\"\"\"\"\"#\"\"$\"\"%7#7%F%F&F'7$7$F%F&7$F'F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 264 "mymulp:=proc()\ndescription \"multiply a
ny number of permutations from right to left\";\nlocal accumulated,m;
\n    accumulated := mulp2(args[1],args[2]); \n    for m from 3 to nar
gs do\n        accumulated := mulp2(accumulated,args[m]);\n    end do;
\naccumulated;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "mymulp(b,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7%\"\"#\"\"%\"\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mymulp(b,c,d);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "mymulp(b,x,invperm(b));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "mulp2(b,mulp2(c,d));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7#7$\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "#p
rint(mulp2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "mulp2(b,c);
 # this is right to left multiplication, which agrees with Serre and c
onvention" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7%\"\"#\"\"%\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "mulp2(b,mulp2(x,invperm(b)))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "mulp3:=proc()\ndescription \+
\"Multiply 3 permutations together from right to left.\";\nmulp2(args[
1],mulp2(args[2],args[3]));\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 33 "Let's make some coset procedures:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 140 "lcoset:=proc(g,subgp) local Lsubgp; # list su
bgp elts \nLsubgp:=[op(elements(subgp))];\n\{seq(mulp2(g,Lsubgp[i]),i=
1..nops(Lsubgp))\};\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "lcosets:=proc(g
p,subgp) local Lgp;\nLgp:=[op(elements(gp))]:\n\{seq(lcoset(Lgp[i],sub
gp), i=1..nops(Lgp))\};\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "#lcosets(A4,K);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 121 "rcoset:=proc(subgp,g) local Lsubgp;\nLsubgp:=[op(ele
ments(subgp))];\n\{seq(mulp2(Lsubgp[i],g),i=1..nops(Lsubgp))\};\nend p
roc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#rcoset(K,[[1,2],[3
,4]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "rcosets:=proc(su
bgp,gp) local Lgp;\nLgp:=[op(elements(gp))]:\n\{seq(rcoset(subgp,Lgp[i
]), i=1..nops(Lgp))\};\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "#rcosets(K,A4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
33 "Let's get some conjugacy classes:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "conj1by2:=proc(elt1,elt2)\ndescription \"Computes (e
lt2)(elt1)(elt2)^(-1)\";\nmulp3(elt2,elt1,invperm(elt2)); end proc:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "conj1by2(x,b);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "#centralizer:=proc(elt,gp) local L
gp:\n#Lgp:=[op(elements(gp))]:\n#\{[]\} union \{seq(conj1by2(elt,Lgp[i
]),i=1..nops(Lgp))\};\n#end proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 118 "conjugacyclass:=proc(elt,gp) local Lgp:\nLgp:=[op(el
ements(gp))]:\n\{seq(conj1by2(elt,Lgp[i]),i=1..nops(Lgp))\};\nend proc
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "conjugacyclass([],S4);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#7\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "conjugacyclass(x,S4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<%7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F&F)7$F'F*7$7$F&F*7$F'F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "conjugacyclass(b,S4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<%7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F&F)7$F'F*7$7$F&F*7$F'F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "conjugacyclass(mulp2(b,b),S4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "conjugacyclasses:=proc(gp)
 local Lgp;\nLgp:=[op(elements(gp))]:\n\{seq(conjugacyclass(Lgp[i],gp)
,i=1..nops(Lgp))\}; end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "conjugacyclasses(S4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<'<#7\"
<%7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F)F,7$F*F-7$7$F)F-7$F*F,<(7#F(7#F07#F
27#F37#F/7#F+<*7#7%F)F*F,7#7%F)F,F*7#7%F)F,F-7#7%F*F-F,7#7%F*F,F-7#7%F
)F-F,7#7%F)F-F*7#7%F)F*F-<(7#7&F)F*F,F-7#7&F)F-F,F*7#7&F)F-F*F,7#7&F)F
,F-F*7#7&F)F,F*F-7#7&F)F*F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "`&*`:=(PERM1,PERM2)->mulp2(PERM1,PERM2):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "`&-`:=(PERM1,PERM2)->mulp2(PERM1,invperm(PERM2
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a &* b;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "a &- b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$\"\"$\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A4:=permgroup(4, \{c
,x,y,z\} ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "elements(A4)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.7\"7$7$\"\"\"\"\"#7$\"\"$\"\"%7
$7$F'F*7$F(F+7$7$F'F+7$F(F*7#7%F'F(F*7#7%F'F*F(7#7%F'F*F+7#7%F(F+F*7#7
%F(F*F+7#7%F'F+F*7#7%F'F+F(7#7%F'F(F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "ListGroup:=proc(gp)\ndescription \"Make a veritable \+
list of group elements.\";\n[op(elements(gp))]; end proc:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ListGroup(A4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7.7\"7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F'F*7$F(F+7$7$F'F+7
$F(F*7#7%F'F(F*7#7%F'F*F(7#7%F'F*F+7#7%F(F+F*7#7%F(F*F+7#7%F'F+F*7#7%F
'F+F(7#7%F'F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(ele
ments(A4)); #grouporder(A4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "isabelian(A4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 60 "S_4 is the semidirect product of L and the normal subgroup H" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "H:=permgroup(4,\{x,y,z\}):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "elements(H);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<&7\"7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F'F*7$F(
F+7$7$F'F+7$F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "groupor
der(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 15 "isnormal(S4,H);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "lcoset([[1
,2],[3,4]], H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&7\"7$7$\"\"\"\"\"
#7$\"\"$\"\"%7$7$F'F*7$F(F+7$7$F'F+7$F(F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "rcoset(H, [[1,2],[3,4]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<&7\"7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F'F*7$F(F+7$7$F'F+7
$F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "L:=permgroup(4,\{a
,c\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "elements(L);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<(7\"7#7$\"\"\"\"\"#7#7%F'F(\"\"$7#7%F
'F+F(7#7$F(F+7#7$F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gr
ouporder(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "isnormal(S4,L);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "lcoset([[1,2]], L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(7\"7#7$\"
\"\"\"\"#7#7%F'F(\"\"$7#7%F'F+F(7#7$F(F+7#7$F'F+" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "lcosets(L,S4);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#<#<:7\"7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F(F+7$F)F,7$7$F(F,7$F)F+7#
F'7#7%F(F)F+7#7&F(F)F+F,7#7&F(F,F+F)7#7&F(F,F)F+7#7&F(F+F,F)7#7&F(F+F)
F,7#7&F(F)F,F+7#7%F(F+F)7#7%F(F+F,7#F/7#7%F)F,F+7#F17#7%F)F+F,7#F27#F.
7#7%F(F,F+7#7%F(F,F)7#F*7#7%F(F)F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "rcosets(L,S4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<&<
(7$7$\"\"\"\"\"#7$\"\"$\"\"%7#7&F'F(F*F+7#7&F'F*F+F(7#7%F'F*F+7#7%F(F*
F+7#F)<(7$7$F'F*7$F(F+7#7&F'F*F(F+7#7&F'F(F+F*7#F87#7%F(F+F*7#7%F'F(F+
<(7$7$F'F+7$F(F*7#7&F'F+F*F(7#7&F'F+F(F*7#FD7#7%F'F+F*7#7%F'F+F(<(7\"7
#F&7#7%F'F(F*7#7%F'F*F(7#FE7#F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "inter(H,L); # co
mputes the intersection of H and K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%*permgroupG6$\"\"%<\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b &* x;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
b &* x &- y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$\"\"\"\"\"$7$\"\"#
\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mulp3(b,x,invperm(
y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$\"\"\"\"\"$7$\"\"#\"\"%" }
}}{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 "" {TEXT -1 316 "Let's list off the conjugacy clas
ses of each of the conjugacy class representatives of S4.  Note: we na
me the conjugacy classes by the letter corresponding to their generato
r and their size (because \"D\" is a protected variable).  Also, we co
nvert them from sets to lists so that the elements remain in a fixed o
rder." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A6:=[op(conjugacyc
lass(a,S4))]; nops(A6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A6G7(7#7
$\"\"\"\"\"#7#7$F)\"\"%7#7$F(F,7#7$F)\"\"$7#7$F(F17#7$F1F," }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 41 "B3:=[op(conjugacyclass(b,S4))]; nops(B3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B3G7%7$7$\"\"\"\"\"#7$\"\"$\"\"%7$7$F(F+7$F)F,7$7$F(
F,7$F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 41 "C8:=[op(conjugacyclass(c,S4))]; nops(C8);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C8G7*7#7%\"\"\"\"\"#\"\"$7#7%F(F*F
)7#7%F(F*\"\"%7#7%F)F/F*7#7%F)F*F/7#7%F(F/F*7#7%F(F/F)7#7%F(F)F/" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "D6:=[op(conjugacyclass(d,S4))]; nops(D6);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#D6G7(7#7&\"\"\"\"\"#\"\"$\"\"%7#7&F(F+F*F
)7#7&F(F+F)F*7#7&F(F*F+F)7#7&F(F*F)F+7#7&F(F)F+F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E1:
=[op(conjugacyclass(e,S4))]; nops(E1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#E1G7#7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 182 "Let us define a procedure for defining characters for S_
4 according to the naming conventions in \"Chern approximations for ge
neralised group cohomology\" by Neil Strickland on page 21" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 490 "definecharacter:=proc(val0, val1, \+
val2, val3, val4, val5)\ndescription \"val0 is the name of the charact
er, and val1 through val5 are the values that this character takes on \+
e,a,b,c,d (respectively) and extends this to conjugacy classes.\";\nlo
cal i;\n`val0`([]):=val1:\nfor i from 1 to nops(A6) do `val0`(A6[i]):=
val2 od:\nfor i from 1 to nops(B3) do `val0`(B3[i]):=val3 od:\nfor i f
rom 1 to nops(C8) do `val0`(C8[i]):=val4 od:\nfor i from 1 to nops(D6)
 do `val0`(D6[i]):=val5 od:\nreturn: end proc:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Define ONE, the triv
ial character of degree n_1=1." }{TEXT 256 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "definecharacter(ONE,1,1,1,1,1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "ONE([]); ONE([[1,2]]); ONE([[1,2],[
3,4]]); ONE([[1,2,3]]); ONE([[1,2,3,4]]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{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 "" {TEXT -1 46 "Define e
psilon, a character of degree n_2 = 1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "definecharacter(epsilon,1,-1,1,1,-1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "epsilon([]); epsilon([[1,2]]); epsi
lon([[1,2],[3,4]]); epsilon([[1,2,3]]); epsilon([[1,2,3,4]]);" }}
{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 "" {TEXT -1 44 "Define sigma, a character of deg
ree n_3 = 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "definechara
cter(sigma,2,0,2,-1,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "
sigma([]); sigma([[1,2]]); sigma([[1,2],[3,4]]); sigma([[1,2,3]]); sig
ma([[1,2,3,4]]);" }}{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 "" {TEXT -1 42 "Define rh
o, a character of degree n_4 = 3." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "definecharacter(rho,3,1,-1,0,-1);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "rho(e); rho(a); rho(b); rho(c); rho(d);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Define alpha = epsilon * rho, a ch
aracter of degree n_5 = 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "definecharacter(alpha,3,-1,-1,0,1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "alpha(e); alpha(a); alpha(b); alpha(c); alpha(d);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{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 94 "epsilon(e)*rho(e); epsilon(a
)*rho(a); epsilon(b)*rho(b); epsilon(c)*rho(c); epsilon(d)*rho(d);" }}
{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 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 58 "Let's verify the relations among the characters in R(S_4)
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "CCR:=[e,a,b,c,d]: # a \+
list of conjugacy class representatives" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 243 "for i from 1 to nops(CCR) do lprint(epsilon(CCR[i])*
epsilon(CCR[i]), ONE(CCR[i])) od;\n# on each line, print the value of \+
epsilon^2 and ONE on a conjugacy class representative\n# if the values
 on each line agree, then the characters are the same" }}{PARA 6 "" 1 
"" {TEXT -1 4 "1, 1" }}{PARA 6 "" 1 "" {TEXT -1 4 "1, 1" }}{PARA 6 "" 
1 "" {TEXT -1 4 "1, 1" }}{PARA 6 "" 1 "" {TEXT -1 4 "1, 1" }}{PARA 6 "
" 1 "" {TEXT -1 4 "1, 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "
for i from 1 to nops(CCR) do lprint(epsilon(CCR[i])*sigma(CCR[i]), sig
ma(CCR[i])) od;" }}{PARA 6 "" 1 "" {TEXT -1 4 "2, 2" }}{PARA 6 "" 1 "
" {TEXT -1 4 "0, 0" }}{PARA 6 "" 1 "" {TEXT -1 4 "2, 2" }}{PARA 6 "" 
1 "" {TEXT -1 6 "-1, -1" }}{PARA 6 "" 1 "" {TEXT -1 4 "0, 0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "for i from 1 to nops(CCR) d
o lprint(sigma(CCR[i])^2, ONE(CCR[i]) + epsilon(CCR[i]) + sigma(CCR[i]
)) od;" }}{PARA 6 "" 1 "" {TEXT -1 4 "4, 4" }}{PARA 6 "" 1 "" {TEXT 
-1 4 "0, 0" }}{PARA 6 "" 1 "" {TEXT -1 4 "4, 4" }}{PARA 6 "" 1 "" 
{TEXT -1 4 "1, 1" }}{PARA 6 "" 1 "" {TEXT -1 4 "0, 0" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 95 "for i from 1 to nops(CCR) do lprint(sigma
(CCR[i])*rho(CCR[i]), rho(CCR[i]) + alpha(CCR[i])) od;" }}{PARA 6 "" 
1 "" {TEXT -1 4 "6, 6" }}{PARA 6 "" 1 "" {TEXT -1 4 "0, 0" }}{PARA 6 "
" 1 "" {TEXT -1 6 "-2, -2" }}{PARA 6 "" 1 "" {TEXT -1 4 "0, 0" }}
{PARA 6 "" 1 "" {TEXT -1 4 "0, 0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "for i from 1 to nops(CCR) do lprint(rho(CCR[i])^2, O
NE(CCR[i]) + sigma(CCR[i]) + rho(CCR[i]) + alpha(CCR[i]) ) od;" }}
{PARA 6 "" 1 "" {TEXT -1 4 "9, 9" }}{PARA 6 "" 1 "" {TEXT -1 4 "1, 1" 
}}{PARA 6 "" 1 "" {TEXT -1 4 "1, 1" }}{PARA 6 "" 1 "" {TEXT -1 4 "0, 0
" }}{PARA 6 "" 1 "" {TEXT -1 4 "1, 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 679 "It is much easier to define and work with Adams operations on \+
the character ring rather than the representation ring.  Lemma 3.61 on
 p. 69 of Adams \"Lectures on Lie Groups\" and p.7 of Strickland's \"C
hern approximations...\" paper contain a very useful formula that tell
s that the value of the character chi associated to the representation
 psi^k (V) on an element g in the group is the value of the character \+
chi associated to V on the element g^k.  That is, chi_\{psi^k(V)\} (g)
 = chi_V (g^k).  In terms of our notation (using the same name for bot
h a representation and its character), we might rephrase this as psi^k
(sigma)(g) = sigma (g^k) where sigma = V is the representation." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "evalchar:=proc(character,co
njclassrep) local i;\nseq(`character`(conjclassrep[i]),i=1..nops(conjc
lassrep));\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 75 "Let's make the Adams operation psi^2 that takes
 a character to a character." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 136 "psi2:=proc(character,conjclassrep)\nlocal i; \nseq(`character`(
mulp2(conjclassrep[i],conjclassrep[i])),i=1..nops(conjclassrep));\nend
 proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "psi^2(ONE) = ONE" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi2(ONE,[e,a,b,c,d]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"F#F#F#F#" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 20 "psi^2(epsilon) = ONE" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "psi2(epsilon,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"\"F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "
psi^2(sigma) = ONE - epsilon + sigma" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "psi2(sigma,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"#F#F#!\"\"F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 40 "evalchar(ONE-epsilon+sigma,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"#F#F#!\"\"F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 "psi^2(rho) = ONE + sigma + rho - alpha" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "psi2(rho,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"$F#F#\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 102 "evalchar(ONE + sigma + rho - epsilon*rho,[e,a,b,c,d]
);\n# I'm baffled as to why epsilon*rho works here!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6'\"\"$F#F#\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "evalchar(ONE + sigma + rho - alpha, [e,a,b,c,d]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F#\"\"!!\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 63 "psi^2(epsilon * rho) = psi^2(alpha) = ONE + sig
ma + rho - alpha" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "psi2(al
pha,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F#\"\"!!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# see previous calcul
ation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "psi2(epsilon*rho,[
e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F#\"\"!!\"\"" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Let's
 make the Adams operation psi^3 that takes a character to a character.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "psi3:=proc(character,c
onjclassrep)\nlocal i; \nseq(`character`(mulp3(conjclassrep[i],conjcla
ssrep[i],conjclassrep[i])),i=1..nops(conjclassrep));\nend proc:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "psi^3(ONE) = ONE" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "psi3(ONE,[e,a,b,c,d]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6'\"\"\"F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
24 "psi^3(epsilon) = epsilon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "psi3(epsilon,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"
\"\"!\"\"F#F#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalchar
(epsilon,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"!\"\"F
#F#F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "psi^3(sigma) = ONE + eps
ilon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "psi3(sigma,[e,a,b,c
,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"!F#F#F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalchar(ONE + epsilon,[e,a,b,c,d])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"!F#F#F$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "psi^3(rho) = ONE + epsilon - sigma + rho
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi3(rho,[e,a,b,c,d]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"\"!\"\"F#F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "evalchar(ONE + epsilon - sigma + rh
o,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"\"!\"\"F#F
%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "psi^3(epsilon * rho) = psi^3
(alpha) = ONE + epsilon - sigma + rho" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "psi3(alpha,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"$!\"\"F$F#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "# see previous calculation" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 30 "psi3(epsilon*rho,[e,a,b,c,d]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6'\"\"$!\"\"F$F#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Let's make the Adams operation \+
psi^4 that takes a character to a character." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 171 "psi4:=proc(character,conjclassrep)\nlocal i; \n
seq(`character`( conjclassrep[i] &* conjclassrep[i] &* conjclassrep[i]
 &* conjclassrep[i]),i=1..nops(conjclassrep));\nend proc:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "psi^4(ONE) = ONE" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "psi4(ONE,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"\"F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "
psi^4(epsilon) = epsilon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"psi4(epsilon,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"F
#F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "psi^4(sigma) = ONE - epsilon + sigma" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "psi4(sigma,[e,a,b,c,d]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#F#F#!\"\"F#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 44 "evalchar(ONE - epsilon + sigma,[e,a,b,c,d]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#F#F#!\"\"F#" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 36 "psi^4(rho) = 2*ONE - epsilon + sigma" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "psi4(rho,[e,a,b,c,d]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F#\"\"!F#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 169 "CharacterTable:=matrix([[1,1,2,3,3],[1,-1,0,
1,-1],[1,1,2,-1,-1],[1,1,-1,0,0],[1,-1,0,-1,1]]);\n# the ordered basis
 here is \{ONE, epsilon, sigma, rho, epsilon*rho = alpha\}" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%/CharacterTableG-%'matrixG6#7'7'\"\"\"F*\"
\"#\"\"$F,7'F*!\"\"\"\"!F*F.7'F*F*F+F.F.7'F*F*F.F/F/7'F*F.F/F.F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "augment(CharacterTable,[3,3
,3,0,3]);\n# we augment the matrix above by attaching the rightmost co
lumn which contains the solution set we want: psi^4(rho)( [e,a,b,c,d] \+
) = 3,3,3,0,3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7(\"\"
\"F(\"\"#\"\"$F*F*7(F(!\"\"\"\"!F(F,F*7(F(F(F)F,F,F*7(F(F(F,F-F-F-7(F(
F,F-F,F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "gaussjord(au
gment(CharacterTable,[3,3,3,0,3]));\n# this says that we need 2*ONE - \+
epsilon + sigma to get the linear combination of characters that equal
s the character psi^4(rho)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matri
xG6#7'7(\"\"\"\"\"!F)F)F)\"\"#7(F)F(F)F)F)!\"\"7(F)F)F(F)F)F(7(F)F)F)F
(F)F)7(F)F)F)F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eval
char(2*ONE - epsilon + sigma,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"$F#F#\"\"!F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
61 "psi^4(epsilon * rho) = psi^3(alpha) = 2*ONE - epsilon + sigma" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "psi4(alpha,[e,a,b,c,d]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F#\"\"!F#" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "# see previous calculation" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "psi4(epsilon*rho,[e,a,b,c,d]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6'\"\"$F#F#\"\"!F#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 86 "Let's make the Adams operation psi^k in general that takes a ch
aracter to a character." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "
# mymulp( a$5 ); a &* a &* a &* a &* a; mymulp( c$5 ); c &* c &* c &* \+
c &* c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "psi:=proc(k,cha
racter,conjclassrep)\nlocal i,j; \nseq(`character`( mymulp( conjclassr
ep[i]$k) ),i=1..nops(conjclassrep));\nend proc:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 16 "psi^5(ONE) = ONE" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "psi(5,ONE,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6'\"\"\"F#F#F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "psi^5(eps
ilon) = epsilon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "psi(5,ep
silon,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"!\"\"F#F#
F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalchar(epsilon,CCR)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"!\"\"F#F#F$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "psi^5(sigma) = sigma" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "psi(5,sigma,[e,a,b,c,d]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6'\"\"#\"\"!F#!\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "gaussjord(augment(CharacterTable,[ psi(5,sigma,[e,a,b
,c,d]) ]));\n# this says we want 1*sigma" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7'7(\"\"\"\"\"!F)F)F)F)7(F)F(F)F)F)F)7(F)F)F(F)F)F(7
(F)F)F)F(F)F)7(F)F)F)F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "evalchar(sigma,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"
\"#\"\"!F#!\"\"F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "psi^5(rho) =
 rho" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "psi(5,rho,[e,a,b,c,
d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"\"!\"\"\"\"!F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "#CharacterTable:=matrix([[1,
1,2,3,3],[1,-1,0,1,-1],[1,1,2,-1,-1],[1,1,-1,0,0],[1,-1,0,-1,1]]);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "#augment(CharacterTable,[ p
si(5,rho,[e,a,b,c,d]) ]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "gaussjord(augment(CharacterTable,[ psi(5,rho,[e,a,b,c,d]) ]));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7(\"\"\"\"\"!F)F)F)F)7(F
)F(F)F)F)F)7(F)F)F(F)F)F)7(F)F)F)F(F)F(7(F)F)F)F)F(F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalchar(rho,[e,a,b,c,d]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"\"!\"\"\"\"!F%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 22 "psi^5(epsilon * rho) =" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "psi(5,alpha,[e,a,b,c,d]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6'\"\"$!\"\"F$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "# see previous calculation" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "psi(5,epsilon*rho,[e,a,b,c,d]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6'\"\"$!\"\"F$\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 70 "gaussjord(augment(CharacterTable,[ psi(5,epsilon
*rho,[e,a,b,c,d]) ]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#
7'7(\"\"\"\"\"!F)F)F)F)7(F)F(F)F)F)F)7(F)F)F(F)F)F)7(F)F)F)F(F)F)7(F)F
)F)F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalchar(epsil
on*rho,[e,a,b,c,d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$!\"\"F$\"
\"!\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 514 "So what have we lea
rned here?  We've learned that thinking of the character table as a ma
trix A, if we are given a list b of values that a character chi takes \+
on the conjugacy class representatives, we can solve the system Ax=b f
or x and thereby obtain the (coefficients in) linear combination of th
e characters ONE, epsilon, sigma, rho, epsilon*rho that equals chi.  T
hus, given any character chi, we can express it as a linear combinatio
n of ONE, epsilon, sigma, rho, epsilon*rho just by a little linear alg
ebra." }}}{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 "" {TEXT -1 195 "Now it's time to get serious.  Let's write a p
rocedure that takes input psi^k(character) and gives as output the lin
ear combination of ONE, epsilon, sigma, rho, and epsilon*rho which com
prise it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 426 "psik2lincombi
:=proc(k,character)\nlocal CoeffMtx;\n#print(psi(k,`character`,[e,a,b,
c,d]));\nCoeffMtx:=gaussjord(augment(CharacterTable,[ psi(k,`character
`,[e,a,b,c,d]) ]));\nCoeffMtx[1,6] + CoeffMtx[2,6]*epsilon + CoeffMtx[
3,6]*sigma + CoeffMtx[4,6]*rho + CoeffMtx[5,6]*epsilon*rho;\n#evalchar
(\n#CoeffMtx[1,6] + CoeffMtx[2,6]*epsilon + CoeffMtx[3,6]*sigma + Coef
fMtx[4,6]*rho + #CoeffMtx[5,6]*epsilon*rho\n#, [e,a,b,c,d]);\nend proc
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "psik2lincombi(5,rho);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$rhoG" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "psik2lincombi(6,rho);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,*\"\"#\"\"\"%(epsilonGF%%$rhoGF%*&F&F%F'F%!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "listAdamsOps:=proc(i,j,L)\n#seq(se
q( lprint(k,L[l],psik2lincombi(k,L[l])), l=1..nops(L)), k=i..j);\nseq(
seq( print([psi^k,L[l],`=`,psik2lincombi(k,L[l])]), l=1..nops(L)), k=i
..j);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 464 "Note: the Ad
ams operations on R(G) for a finite group G are always periodic with p
eriod the order of G, and perhaps even of smaller period, as is the ca
se for S_4 in which the Adams operations are periodic with period 12 =
 |S_4|/2.  I had suspected that there was periodicity when I saw that \+
psi^k(epsilon)=epsilon^k.  The periodicity theorem is Corollary 3.2 on
 p. 274 in Atiyah and Tall's paper \"Group Representations, $\\lambda$
-rings, and the $J$-homomorphism\"." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "listAdamsOps(2,12,[ONE,epsilon,sigma,rho,epsilon*rho]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%$ONEG%\"=
GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%(epsilon
G%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%&si
gmaG%\"=G,(F(F(%(epsilonG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7&*$)%$psiG\"\"#\"\"\"%$rhoG%\"=G,*F(F(%&sigmaGF(F)F(*&%(epsilonGF(F)F
(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"*&%(ep
silonGF(%$rhoGF(%\"=G,*F(F(%&sigmaGF(F+F(F)!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%$ONEG%\"=GF(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%(epsilonG%\"=GF)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%&sigmaG%\"=G,&F(F(%(epsil
onGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%$rhoG%
\"=G,*F(F(%(epsilonGF(%&sigmaG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&*$)%$psiG\"\"$\"\"\"*&%(epsilonGF(%$rhoGF(%\"=G,*F(F(F*F(%&sig
maG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"
%$ONEG%\"=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\"\"%\"\"\"%$rhoG%\"=G,(\"\"#F(%(epsilonG!\"
\"%&sigmaGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"
*&%(epsilonGF(%$rhoGF(%\"=G,(\"\"#F(F*!\"\"%&sigmaGF(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%$ONEG%\"=GF(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%(epsilonG%\"=GF)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%&sigmaG%\"=GF)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%$rhoG%\"=GF
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"*&%(epsilon
GF(%$rhoGF(%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'
\"\"\"%$ONEG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"
'\"\"\"%(epsilonG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psi
G\"\"'\"\"\"%&sigmaG%\"=G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$
)%$psiG\"\"'\"\"\"%$rhoG%\"=G,*\"\"#F(%(epsilonGF(F)F(*&F-F(F)F(!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"\"\"*&%(epsilonG
F(%$rhoGF(%\"=G,*\"\"#F(F*F(F+F(F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&*$)%$psiG\"\"(\"\"\"%$ONEG%\"=GF(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%(epsilonG%\"=GF)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%&sigmaG%\"=GF)" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%$rhoG%\"=GF)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"*&%(epsilonGF(%$rhoGF(
%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%$ONE
G%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%(ep
silonG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"
\"%&sigmaG%\"=G,(F(F(%(epsilonG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&*$)%$psiG\"\")\"\"\"%$rhoG%\"=G,(\"\"#F(%(epsilonG!\"\"%&sigma
GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"*&%(epsil
onGF(%$rhoGF(%\"=G,(\"\"#F(F*!\"\"%&sigmaGF(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"*\"\"\"%$ONEG%\"=GF(" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7&*$)%$psiG\"\"*\"\"\"%(epsilonG%\"=GF)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"*\"\"\"%&sigmaG%\"=G,&F(F(%(epsil
onGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"*\"\"\"%$rhoG%
\"=G,*F(F(%(epsilonGF(%&sigmaG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&*$)%$psiG\"\"*\"\"\"*&%(epsilonGF(%$rhoGF(%\"=G,*F(F(F*F(%&sig
maG!\"\"F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#5\"\"\"%
$ONEG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#5\"\"\"%
(epsilonG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#5\"
\"\"%&sigmaG%\"=G,(F(F(%(epsilonG!\"\"F)F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"#5\"\"\"%$rhoG%\"=G,*F(F(%&sigmaGF(F)F(*&%
(epsilonGF(F)F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#
5\"\"\"*&%(epsilonGF(%$rhoGF(%\"=G,*F(F(%&sigmaGF(F+F(F)!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#6\"\"\"%$ONEG%\"=GF(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#6\"\"\"%(epsilonG%\"=GF)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#6\"\"\"%&sigmaG%\"=G
F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#6\"\"\"%$rhoG%\"=G
F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#6\"\"\"*&%(epsilon
GF(%$rhoGF(%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#7
\"\"\"%$ONEG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"#7
\"\"\"%(epsilonG%\"=GF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG
\"#7\"\"\"%&sigmaG%\"=G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%
$psiG\"#7\"\"\"%$rhoG%\"=G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*
$)%$psiG\"#7\"\"\"*&%(epsilonGF(%$rhoGF(%\"=G\"\"$" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "205 0 0" 0 }
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