{VERSION 5 0 "IBM INTEL LINUX" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg
):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm a
nd trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 293 "Let's look at the quaternion group of order 8.  See
 p.6 in \"Representations of Compact Lie Groups\" by Br\\\" oker and t
om Dieck for notation, and p.189 of Hopkins, Kuhn, and Ravenel's \"Mor
ava K-theories of classifying spaces and generalized characters for fi
nite groups\" for the character table." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 133 "We're going to define the elements of Q_8 to be matrices
, that way we can multiply them easily without having to define proced
ures..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "G:=[\n[[1,0],[0
,1]], # 1\n[[0,1],[-1,0]], # j\n[[-1,0],[0,-1]], #-1 \n[[0,-1],[1,0]],
# -j \n[[I,0],[0,-I]], # i \n[[0,-I],[-I,0]], # -k \n[[-I,0],[0,I]], #
 -i \n[[0,I],[I,0]] # k\n]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "#evalm( [[0,1],[-1,0]] &* [[0,1],[-1,0]] );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 61 "#evalm( [[0,1],[-1,0]] &* [[0,1],[-1,0]] &* [[
0,1],[-1,0]] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "#evalm( \+
[[I,0],[0,-I]] &* [[I,0],[0,-I]] &* [[I,0],[0,-I]] );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "#evalm( [[0,I],[I,0]] &* [[0,I],[I,
0]] &* [[0,I],[I,0]] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 
"g(G[1]):=e: g(G[2]):=j: g(G[3]):=-e: g(G[4]):=-j: g(G[5]):=i: g(G[6])
:=-k: g(G[7]):=-i: g(G[8]):=k:\n# assign the matrices to names that wo
n't be evaluated by maple (that way,\n# the names are immutable just l
ike basis elements!\n# we could do this with lists, too, e.g. g:=[seq(
f(G[i]),i=1..nops(G))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 
"mm:=(A,B)->[[A[1,1]*B[1,1]+A[1,2]*B[2,1], A[1,1]*B[1,2]+A[1,2]*B[2,2]
], [A[2,1]*B[1,1] +A[2,2]*B[2,1],A[2,1]*B[1,2]+A[2,2]*B[2,2]]]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g(mm(G[2],G[3]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$%\"jG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "g(G[2]) &* g(G[3]) := g( mm(G[2],G[3]) );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>-%#&*G6$%\"jG,$%\"eG!\"\",$F'F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "j &* (-e);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%\"jG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
156 "# tell maple how to multiply the names\nfor a from 1 to nops(G) d
o\n  for b from 1 to nops(G) do\n    g(G[a]) &* g(G[b]) := g(mm(G[a],G
[b]));\n  end do;\nend do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "(-j) &* (-k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"iG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k &* (-j);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"iG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "mymulp:=proc()\ndescriptio
n \"multiply any number of group elts\";\nlocal accumulated,m;\n    ac
cumulated := args[1] &* args[2]; \n    for m from 3 to nargs do\n     \+
   accumulated := accumulated &* args[m];\n    end do;\naccumulated;\n
end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mymulp(k,k,k);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"kG!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "Q8List:=[e,j,-e,-j,i,-k,-i,k]:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 94 "Let's compute the conjugacy classes in Q_
8. (skip) \nThe conjugacy class of e is obviously \{e\}." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 33 "The conjugacy class of -e is \{-e\}" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The conjugacy class of i is \{i, -
i\}" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The conjugacy class of j i
s \{j, -j\}" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "The conjugacy clas
s of k is \{k, -k\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Conj
ClList:=[ [e], [-e], [i,-i], [j,-j], [k,-k] ]:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "nops(ConjClList);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nop
s(ConjClList[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ConjClList[3,2];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$%\"iG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "CCR:=[e,-e,i,j,k]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Let's
 define the character table. (p 161 James and Liebeck)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "CharTableTransposed:=\n[[1,1,1,1,1]
,\n[1,1,1,-1,-1],\n[1,1,-1,1,-1],\n[1,1,-1,-1,1],\n[2,-2,0,0,0]]:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "In this character table the colum
ns are indexed by the irreducible characters chi1,chi2,... from left t
o right, and the rows are indexed by the conjugacy class representativ
es e, r2, r, f, rf   from top to bottom." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "CharTable:=transpose(CharTableTransposed);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%*CharTableG-%'matrixG6#7'7'\"\"\"F*F*F*\"
\"#7'F*F*F*F*!\"#7'F*F*!\"\"F/\"\"!7'F*F/F*F/F07'F*F/F/F*F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 437 "definecharacter:=proc(CharN
ame, CCL, MtxCol)\ndescription \"CharName is the name of the character
, and its values on conjugacy class representatives are listed in orde
r in the matrix column of the character table correspoding to the elem
ents in a list of conjugacy class representatives\";\nlocal a,b;\nfor \+
a from 1 to nops(CCL) \n   do \n      for b from 1 to nops(CCL[a]) do \+
`CharName`(CCL[a,b]):=MtxCol[a];\n      od:\n   od:\nreturn: end proc:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "definecharacter(chi1,Co
njClList,subvector(CharTable,1..nops(ConjClList), 1));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "chi1(e); chi1(-e); chi1(i); chi1(-i
); chi1( i &* i);" }}{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 0 176 "d
efchars:=proc(CharNames, CCL, Mtx)\nlocal a;\n for a from 1 to nops(Ch
arNames) do definecharacter(CharNames[a], CCL, subvector(Mtx, 1..nops(
CharNames), a)) od: return:\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "IChar:=[chi1,chi2,chi3,chi4,chi5]:\n#IChar:=[ONE,alph
a,beta,epsilon,chi5]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "de
fchars(IChar,ConjClList,CharTable);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "#beta(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
128 "psi:=proc(K,character,conjclassrep)\nlocal a,b; \nseq(`character`
( mymulp(conjclassrep[a]$K) ),a=1..nops(conjclassrep));\nend proc:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "psi(2,chi5,CCR);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6'\"\"#F#!\"#F$F$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "augment(CharTable,[psi(2,chi5,CCR)]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7(\"\"\"F(F(F(\"\"#F)7(F(F(F(F(!\"
#F)7(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 13 "gaussjord(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7'7(\"\"\"\"\"!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 0 302 "psik2lincombi:=proc(K,character,CC,IrrChar,Mtx)\nloc
al CoeffVect;\nCoeffVect:=subvector( gaussjord(augment(Mtx,[ psi(K,`ch
aracter`,CC) ])), 1..nops(CC), nops(CC)+1);\nCoeffVect[1]*IrrChar[1] +
 CoeffVect[2]*IrrChar[2] + CoeffVect[3]*IrrChar[3] + CoeffVect[4]*IrrC
har[4] + CoeffVect[5]*IrrChar[5];\nend proc:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "psik2lincombi(2,chi5,CCR,IChar,CharTable);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%%chi1G!\"\"%%chi2G\"\"\"%%chi3GF'%%
chi4GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "psik2lincombi(2,
chi4,CCR,IChar,CharTable);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%chi1G
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "listAdamsOps:=proc(a,b
,CC,IrrChar,Mtx)\nseq(seq( print([psi^K,IrrChar[l],`=`,psik2lincombi(K
,IrrChar[l],CC,IrrChar,Mtx)]), K=a..b), l=1..nops(IrrChar));\nend proc
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "listAdamsOps(2,8,CCR,I
Char,CharTable);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"
\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$
\"\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"
\"%\"\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG
\"\"&\"\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$ps
iG\"\"'\"\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$
psiG\"\"(\"\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)
%$psiG\"\")\"\"\"%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*
$)%$psiG\"\"#\"\"\"%%chi2G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&*$)%$psiG\"\"$\"\"\"%%chi2G%\"=GF)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"%%chi2G%\"=G%%chi1G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%%chi2G%\"=GF)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"\"\"%%chi2G%\"=G%%chi1G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%%chi2G%\"=GF)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%%chi2G%\"=G
%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%%chi
3G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"
\"%%chi3G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"
\"\"%%chi3G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG
\"\"&\"\"\"%%chi3G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$ps
iG\"\"'\"\"\"%%chi3G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&
*$)%$psiG\"\"(\"\"\"%%chi3G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7&*$)%$psiG\"\")\"\"\"%%chi3G%\"=G%%chi1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%%chi4G%\"=G%%chi1G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%%chi4G%\"=GF)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"%%chi4G%\"=G%%chi1G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%%chi4G%\"=GF)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"\"\"%%chi4G%\"=G
%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%%chi
4G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%%c
hi4G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"
\"\"%%chi5G%\"=G,*%%chi1G!\"\"%%chi2GF(%%chi3GF(%%chi4GF(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%%chi5G%\"=GF)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"\"%%chi5G%\"=G,$*&\"\"#F
(%%chi1GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"
%%chi5G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"\"
\"%%chi5G%\"=G,*%%chi1G!\"\"%%chi2GF(%%chi3GF(%%chi4GF(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%%chi5G%\"=GF)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%%chi5G%\"=G,$*&\"\"#F
(%%chi1GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK 
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