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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 -1 190 "Let's look at the dihedral group of order 8.  See p
.31 in \"Representations and Characters of Groups (first edition)\" by
 James and Liebeck for notation, and p.160-161 for the character table
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "We're going to define the e
lements of D_8 to be matrices, that way we can multiply them easily wi
thout having to define procedures..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 134 "G:=[\n[[1,0],[0,1]],\n[[0,1],[-1,0]],\n[[-1,0],[0,-1
]],\n[[0,-1],[1,0]],\n[[1,0],[0,-1]],\n[[0,-1],[-1,0]],\n[[-1,0],[0,1]
],\n[[0,1],[1,0]]\n]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "g
(G[1]):=e: g(G[2]):=r: g(G[3]):=r2: g(G[4]):=r3: g(G[5]):=f: g(G[6]):=
rf: g(G[7]):=r2f: g(G[8]):=r3f:\n# assign the matrices to names that w
on't be evaluated by maple (that way,\n# the names are immutable just \+
like 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#%#r3G" }}}{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$%\"rG%#r2G%#r3G" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "r &* r2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%#r3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "# tell maple how \+
to multiply the names\nfor i from 1 to nops(G) do\n  for j from 1 to n
ops(G) do\n    eval(g(G[i])) &* eval(g(G[j])) := eval(g(mm(G[i],G[j]))
);\n  end do;\nend do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "r \+
&* r2f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$r3fG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 235 "mymulp:=proc()\ndescription \"multiply any \+
number of group elts\";\nlocal accumulated,m;\n    accumulated := args
[1] &* args[2]; \n    for m from 3 to nargs do\n        accumulated :=
 accumulated &* args[m];\n    end do;\naccumulated;\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mymulp(r,r,r);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%#r3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "D8List:=[e,r,r2,r3,f,rf,r2f,r3f]:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 94 "Let's compute the conjugacy classes in D_8. (skip) \nThe \+
conjugacy class of e is obviously \{e\}." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 33 "The conjugacy class of r2 is \{r2\}" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 36 "The conjugacy class of r is \{r3,  r\}" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "The conjugacy class of f is \{r2f, f\}" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The conjugacy class of rf is \{r
3f, rf\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ConjClList:=[ [
e], [r2], [r,r3], [f,r2f], [rf,r3f] ]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "nops(ConjClList);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nops(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#%#r3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "CCR:=[e,r2,r,f,r
f]:" }}}{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 columns are indexed b
y the irreducible characters chi1,chi2,... from left to right, and the
 rows are indexed by the conjugacy class representatives 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(CharName, CCL, MtxCol)
\ndescription \"CharName is the name of the character, and its values \+
on conjugacy class representatives are listed in order in the matrix c
olumn of the character table correspoding to the elements in a list of
 conjugacy class representatives\";\nlocal i,j;\nfor i from 1 to nops(
CCL) \n   do \n      for j from 1 to nops(CCL[i]) do `CharName`(CCL[i,
j]):=MtxCol[i];\n      od:\n   od:\nreturn: end proc:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "definecharacter(chi1,ConjClList,sub
vector(CharTable,1..nops(ConjClList), 1));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 49 "chi1(e); chi1(r2); chi1(r); chi1(r3); chi1(r&*r);" 
}}{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 "defchars:=proc(CharNames
, CCL, Mtx)\nlocal i;\n for i from 1 to nops(CharNames) do definechara
cter(CharNames[i], CCL, subvector(Mtx, 1..nops(CharNames), i)) od: ret
urn:\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "IChar:=[
chi1,chi2,chi3,chi4,chi5]:\n#IChar:=[ONE,alpha,beta,epsilon,chi5]:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "defchars(IChar,ConjClList,C
harTable);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "beta(f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%betaG6#%\"fG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 128 "psi:=proc(k,character,conjclassrep)\nlocal i,
j; \nseq(`character`( mymulp(conjclassrep[i]$k) ),i=1..nops(conjclassr
ep));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "psi(2,c
hi5,CCR);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#F#!\"#F#F#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "augment(CharTable,[psi(2,chi
5,CCR)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7(\"\"\"F(F
(F(\"\"#F)7(F(F(F(F(!\"#F)7(F(F(!\"\"F-\"\"!F+7(F(F-F(F-F.F)7(F(F-F-F(
F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gaussjord(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7(\"\"\"\"\"!F)F)F)F(7(F
)F(F)F)F)!\"\"7(F)F)F(F)F)F(7(F)F)F)F(F)F(7(F)F)F)F)F(F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 426 "psik2lincombi:=proc(k,character,CC
,IrrChar,Mtx)\nlocal CoeffVect;\nCoeffVect:=subvector( gaussjord(augme
nt(Mtx,[ psi(k,`character`,CC) ])), 1..nops(CC), nops(CC)+1);\n#print(
CoeffVect);\n#print(nops(CC));\nCoeffVect[1]*IrrChar[1] + CoeffVect[2]
*IrrChar[2] + CoeffVect[3]*IrrChar[3] + CoeffVect[4]*IrrChar[4] + Coef
fVect[5]*IrrChar[5];\n#seq(IrrChar[i],i=1..nops(IrrChar));\n#add(Coeff
Mtx[i,nops(CC)+1]*CC[i],i=1..nops(CC));\nend proc:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "psik2lincombi(2,chi5,CCR,IChar,CharTable);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%%chi1G\"\"\"%%chi2G!\"\"%%chi3G
F%%%chi4GF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "listAdamsOp
s:=proc(i,j,CC,IrrChar,Mtx)\n#seq(seq( lprint(k,IrrChar[l],psik2lincom
bi(k,IrrChar[l],CC,IrrChar,Mtx)), l=1..nops(L)), k=i..j);\nseq(seq( pr
int([psi^k,IrrChar[l],`=`,psik2lincombi(k,IrrChar[l],CC,IrrChar,Mtx)])
, k=i..j), l=1..nops(IrrChar));\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 121 "The correct way to interpret the output is, for example,
 psi^2(\\chi_1) = \\chi_1.  In other words, just ignore the commas." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "listAdamsOps(2,8,CCR,IChar
,CharTable);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"
%%chi1G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"
\"%%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&*$)%$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\"\"#\"\"\"%%chi2G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7&*$)%$psiG\"\"$\"\"\"%%chi2G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#7&*$)%$psiG\"\"%\"\"\"%%chi2G%\"=G%%chi1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"&\"\"\"%%chi2G%\"=GF)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7&*$)%$psiG\"\"'\"\"\"%%chi2G%\"=G%%chi1G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%%chi2G%\"=GF)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%%chi2G%\"=G%%ch
i1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"\"%%chi3G%
\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%
%chi3G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"%\"\"
\"%%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&*$
)%$psiG\"\"(\"\"\"%%chi3G%\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&
*$)%$psiG\"\")\"\"\"%%chi3G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&*$)%$psiG\"\"#\"\"\"%%chi4G%\"=G%%chi1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&*$)%$psiG\"\"$\"\"\"%%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%%ch
i1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"(\"\"\"%%chi4G%
\"=GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\")\"\"\"%%chi4
G%\"=G%%chi1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&*$)%$psiG\"\"#\"\"
\"%%chi5G%\"=G,*%%chi1GF(%%chi2G!\"\"%%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,*%%chi1GF(%%chi2G!\"\"%%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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1 }