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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "                          \+
           " }{TEXT 256 7 "       " }{TEXT 257 7 "Crystal" }{TEXT 258 
18 " test worksheet #1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 29 "8-27-99,crytest1.mws, wdj+rem" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 343 "  This maple worksheet w
ill explain how to use crystal to compute \"3 tensor 3 tensor 3\" - th
e triple tensor product of the identity representation of SU(3) with i
tself. \"3 tensor 3\" breaks up into an irreducible 6 dimensional repr
esentation plus the \"3-bar\" (the contragredient of 3). 3 tensor 3-ba
r is an irreducible 8 dimensional plus a 1.  " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "with(plots):\nwith(share);\nwith(coxeter);\nwith(weyl);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#7@%%baseG%*char_polyG%*class_repG%+class_sizeG%+cox_matrixG%+cox_numb
erG%(degreesG%+descent_gfG%(diagramG%*exponentsG%-highest_rootG%,inter
ior_ptG%&iprodG%*length_gfG%,longest_eltG%)multpermG%(name_ofG%)num_re
flG%&orbitG%+orbit_sizeG%*perm_charG%)perm_repG%*pos_rootsG%-presentat
ionG%%rankG%'reduceG%(reflectG%,root_coordsG%%sizeG%'vec2fcG" }}{PARA 
6 "" 1 "" {TEXT -1 20 "Share Library:  weyl" }}{PARA 6 "" 1 "" {TEXT 
-1 25 "Author: Stembridge, John." }}{PARA 6 "" 1 "" {TEXT -1 217 "Desc
ription:  The weyl package is a supplement to the coxeter package that
 contains 7 procedures for manipulating weight vectors and computing m
ultiplicities for irreducible representations of semisimple Lie algebr
as." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)%$rhoG%&storeG%.weight_coords
G%-weight_multsG%+weight_sysG%(weightsG%)weyl_dimG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "read(`e:/maplestuff/crystal/crystal26.mpl`)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 " One must always initialize \+
crystal as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "init
_crystal():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Crystal~initialized.G
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "The identity representation (
i.e., \"3\") is represented by the list L1 of its weights as follows:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "weyl[weights](A2);\nL1:
=crystal[weight_system](-(1/3*e2-2/3*e1+1/3*e3),A2);\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7$,(%#e2G#\"\"\"\"\"$%#e1G#!\"#F(%#e3GF&,(F%#!\"\"
F(F)F.F,#\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G7%7$\"\"\"\"
\"!7$!\"\"F'7$F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 " We use L2 \+
to denote another copy of the same representation as L1." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "weyl[weyl_dim](1/3*e2-2/3*e1+1/3*e3
,A2);\nL2:=crystal[weight_system](-(1/3*e2-2/3*e1+1/3*e3),A2);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#L2G7%7$\"\"\"\"\"!7$!\"\"F'7$F(F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "To see the crystal graph of 3, simply type the commands:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "crystal[graphrep](L1,A2
,G1);\ncrystal[graphrep](L2,A2,G2);\ncrystal[showgraph](G1,1,3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~formed.G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.Graph~formed.G" }}{PARA 13 "" 1 "" {GLPLOT2D 385 385 
385 {PLOTDATA 2 "6+-%'POINTSG6$7$\"\"\"\"\"!-%'SYMBOLG6#%'CIRCLEG-F$6$
7$\"\"#F(F)-F$6$7$\"\"$F(F)-%'CURVESG6#7$F&F/-F66#7$F/F3-%%TEXTG6%7$#F
4F0F(Q\"16\"%+ALIGNABOVEG-F=6%7$#\"\"&F0F(Q\"2FBFC-%(SCALINGG6#%,CONST
RAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 0 2 9 1 1 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The crystal graph \+
product of the two is denoted G3:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "crystal[graphprodrep](G1,G2,G3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.Graph~formed.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
37 "To see this, simply type the command:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "crystal[showgraphprod](G3,1,3,1,3);" }}{PARA 13 "" 
1 "" {GLPLOT2D 385 385 385 {PLOTDATA 2 "6=-%'POINTSG6$7$\"\"\"F'-%'SYM
BOLG6#%'CIRCLEG-F$6$7$\"\"#F'F(-F$6$7$\"\"$F'F(-F$6$7$F'F/F(-F$6$7$F/F
/F(-F$6$7$F3F/F(-F$6$7$F'F3F(-F$6$7$F/F3F(-F$6$7$F3F3F(-%'CURVESG6#7$F
&F6-FG6#7$F6F?-FG6#7$F6F9-FG6#7$F?FB-FG6#7$F9FB-FG6#7$F.F2-FG6#7$FBFE-
FG6#7$F2F<-%%TEXTG6%7$F'#F3F/Q\"16\"%*ALIGNLEFTG-Fjn6%7$F'#\"\"&F/Q\"2
F_oF`o-Fjn6%7$F]oF/F^o%+ALIGNABOVEG-Fjn6%7$F]oF3F^oFjo-Fjn6%7$F/FdoFfo
F`o-Fjn6%7$FdoF'FfoFjo-Fjn6%7$FdoF3FfoFjo-Fjn6%7$F3F]oF^oF`o-%(SCALING
G6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 0 2 9 1 1 1 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "We pick
 out the component of this graph which contains the highest weight vec
tor 2*e1 and call it G4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 
"crystal[linsubgraphrep](2*e1,A2,G3,G4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%.Graph~formed.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "In oth
er words, we denote by G4 the component of G3 which contains the highe
st weight vertex v1X1 (the so-called \"Cartan product\"):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "crystal[linsubgraphrep](v1X1,A2,G3,
G4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~formed.G" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 68 "By the way, to see that v1X1 is the highe
st weight, simply type the " }{TEXT 259 7 "vweight" }{TEXT -1 46 " com
mand to see it's associated weight vector:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 53 "networks[vweight](G3);\nprint(networks[vertices](G3
));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6$%'sparseG7+/%%v3X3G7
$\"\"!!\"#/%%v1X3G7$\"\"\"!\"\"/%%v3X1GF//%%v1X2G7$F+F0/%%v2X2G7$F,\"
\"#/%%v2X3G7$F1F+/%%v2X1GF6/%%v1X1G7$F:F+/%%v3X2GF=" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#<+%%v2X1G%%v2X2G%%v2X3G%%v1X2G%%v3X2G%%v3X1G%%v3X3G%%
v1X1G%%v1X3G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Though we don't n
eed this, to see the (root) labels of the edges, type:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "networks[eweight](G3);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7*/%#e2G\"\"#/%#e3G\"\"\"/%#e8GF,/%#
e7GF)/%#e5GF)/%#e6GF)/%#e4GF,/%#e1GF," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 55 "The graph G4 can be viewed as a linear graph using the " 
}{TEXT 260 9 "showgraph" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "crystal[showgraph](G4,1,6);" }}{PARA 13 "" 1 "
" {GLPLOT2D 385 385 385 {PLOTDATA 2 "66-%'POINTSG6$7$\"\"\"\"\"!-%'SYM
BOLG6#%'CIRCLEG-F$6$7$\"\"#F(F)-F$6$7$\"\"$F(F)-F$6$7$\"\"%F(F)-F$6$7$
\"\"&F(F)-F$6$7$\"\"'F(F)-%'CURVESG6#7$F&F/-FB6#7$F/F3-FB6#7&F/7$F4#F'
F0FKF7-FB6#7&F37$F8FLFPF;-FB6#7$F7F;-FB6#7$F;F?-%%TEXTG6%7$#F4F0F(Q\"1
6\"%+ALIGNABOVEG-FX6%7$#F<F0F(FfnFhn-FX6%FKQ\"2FgnFhn-FX6%FPF_oFhn-FX6
%7$#\"\"*F0F(FfnFhn-FX6%7$#\"#6F0F(F_oFhn-%(SCALINGG6#%,CONSTRAINEDG-%
*AXESSTYLEG6#%%NONEG" 1 2 0 1 0 2 9 1 1 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "What is 3 tensor 6
? If we denote it by G8," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 
"crystal[graphprodrep](G1,G4,G8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
.Graph~formed.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "then to see th
is, simply type the command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "crystal[showgraphprod](G8,1,3,1,6);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 384 384 384 {PLOTDATA 2 "6hn-%'POINTSG6$7$\"\"\"F'-%'SYMBOLG
6#%'CIRCLEG-F$6$7$\"\"#F'F(-F$6$7$\"\"$F'F(-F$6$7$\"\"%F'F(-F$6$7$\"\"
&F'F(-F$6$7$\"\"'F'F(-F$6$7$F'F/F(-F$6$7$F/F/F(-F$6$7$F3F/F(-F$6$7$F7F
/F(-F$6$7$F;F/F(-F$6$7$F?F/F(-F$6$7$F'F3F(-F$6$7$F/F3F(-F$6$7$F3F3F(-F
$6$7$F7F3F(-F$6$7$F;F3F(-F$6$7$F?F3F(-%'CURVESG6#7$F&FB-F_o6#7$FBFT-F_
o6#7$FBFE-F_o6#7$FTFW-F_o6#7$FEFW-F_o6#7$F.F2-F_o6#7$FEFH-F_o6#7$FWFZ-
F_o6#7&F.7$F3#F3F/FjpF6-F_o6#7&FW7$F3#\"\"(F/F_qFgn-F_o6#7$FHFZ-F_o6#7
&F27$F7F[qFhqF:-F_o6#7&FZ7$F7F`qF\\rFjn-F_o6#7$F6FK-F_o6#7$FKFN-F_o6#7
$FgnFjn-F_o6#7$F:F>-F_o6#7$FNFQ-F_o6#7$FjnF]o-F_o6#7$F>FQ-%%TEXTG6%7$F
'F[qQ\"16\"%*ALIGNLEFTG-Fcs6%7$F'#F;F/Q\"2FgsFhs-Fcs6%7$F[qF/Ffs%+ALIG
NABOVEG-Fcs6%7$F[qF3FfsFat-Fcs6%7$F/F\\tF]tFhs-Fcs6%7$F\\tF'FfsFat-Fcs
6%7$F\\tF/FfsFat-Fcs6%7$F\\tF3FfsFat-Fcs6%FjpF]tFat-Fcs6%F_qF]tFat-Fcs
6%7$F3F\\tF]tFhs-Fcs6%FhqF]tFat-Fcs6%F\\rF]tFat-Fcs6%FhqFfsFhs-Fcs6%7$
#\"\"*F/F/FfsFat-Fcs6%7$FavF3FfsFat-Fcs6%7$#\"#6F/F'F]tFat-Fcs6%7$FivF
/F]tFat-Fcs6%7$FivF3F]tFat-Fcs6%7$F?F[qFfsFhs-%(SCALINGG6#%,CONSTRAINE
DG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 0 2 9 1 1 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 253 "We shall not need this, but the restrict
ion of this representation to the subalgebra su(1) associated to the r
oot labeled \"1\" is easy to determine. It is obtained by omitting tho
se edges with label 2. Denote this restriction by G5 and view it using
 the " }{TEXT 261 9 "showgraph" }{TEXT -1 9 " command:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "crystal[branch]([2],G4,G5);\ncrysta
l[showgraph](G5,1,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Branch~grap
h~formed.G" }}{PARA 13 "" 1 "" {GLPLOT2D 385 385 385 {PLOTDATA 2 "60-%
'POINTSG6$7$\"\"\"\"\"!-%'SYMBOLG6#%'CIRCLEG-F$6$7$\"\"#F(F)-F$6$7$\"
\"$F(F)-F$6$7$\"\"%F(F)-F$6$7$\"\"&F(F)-F$6$7$\"\"'F(F)-%'CURVESG6#7$F
&F/-FB6#7$F/F3-FB6#7$F7F;-%%TEXTG6%7$#F4F0F(Q\"16\"%+ALIGNABOVEG-FL6%7
$#F<F0F(FPFR-FL6%7$#\"\"*F0F(FPFR-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTY
LEG6#%%NONEG" 1 2 0 1 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Returning to 3 tensor 3 tensor 3, \+
let G6 denote the smaller component of 3 tensor 3:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "crystal[linsubgraphrep](v1X2,A2,G3,G6);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~formed.G" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 24 "It is viewed as follows:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "crystal[showgraph](G6,1,3);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 384 384 384 {PLOTDATA 2 "6+-%'POINTSG6$7$\"\"\"\"\"!-%'SYMBO
LG6#%'CIRCLEG-F$6$7$\"\"#F(F)-F$6$7$\"\"$F(F)-%'CURVESG6#7$F&F/-F66#7$
F/F3-%%TEXTG6%7$#F4F0F(Q\"26\"%+ALIGNABOVEG-F=6%7$#\"\"&F0F(Q\"1FBFC-%
(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 0 2 9 1 1 1 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 
"This is obviously 3-bar. Let G7 denote the crystal graph product of G
1 and G6:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "crystal[graphp
rodrep](G1,G6,G7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~formed.G
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "It is an 8 plus 1:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "crystal[showgraphprod](G7,1,
3,1,3);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6=-%'PO
INTSG6$7$\"\"\"F'-%'SYMBOLG6#%'CIRCLEG-F$6$7$\"\"#F'F(-F$6$7$\"\"$F'F(
-F$6$7$F'F/F(-F$6$7$F/F/F(-F$6$7$F3F/F(-F$6$7$F'F3F(-F$6$7$F/F3F(-F$6$
7$F3F3F(-%'CURVESG6#7$F&F6-FG6#7$F6F?-FG6#7$F&F.-FG6#7$F?FB-FG6#7$F.F9
-FG6#7$F9F<-FG6#7$FBFE-FG6#7$F<FE-%%TEXTG6%7$F'#F3F/Q\"16\"%*ALIGNLEFT
G-Fjn6%7$F'#\"\"&F/Q\"2F_oF`o-Fjn6%7$F]oF'Ffo%+ALIGNABOVEG-Fjn6%7$F]oF
3FfoFjo-Fjn6%7$F/F]oF^oF`o-Fjn6%7$FdoF/F^oFjo-Fjn6%7$FdoF3F^oFjo-Fjn6%
7$F3FdoFfoF`o-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 
1 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "                                \+
                          " }{TEXT 263 11 " References" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "B. Simon, " }{TEXT 
262 44 "Representations of finite and compact groups" }{TEXT -1 34 ", \+
Grad Studies in Math., AMS, 1996" }}}}{MARK "0 2 0" 29 }{VIEWOPTS 1 1 
0 1 1 1803 }