{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "System" 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Syst em" 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 32 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 74 0 16 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 " \+ " }{TEXT 256 7 " " }{TEXT 257 7 "Crystal" }{TEXT 258 18 " test worksheet #3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "4-5-97,crytest25a.mws, wdj" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 343 " This maple worksheet will ex plain how to use crystal to compute \"3 tensor 3 tensor 3\" - the trip le tensor product of the identity representation of SU(3) with itself. \"3 tensor 3\" breaks up into an irreducible 6 dimensional representa tion plus the \"3-bar\" (the contragredient of 3). 3 tensor 3-bar is a n irreducible 8 dimensional plus a 1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "with(plots):\nwith(share);\nreadshare(coxeter,algebra);" }}{PARA 6 " " 1 "" {TEXT -1 70 "See ?share and ?share,contents for information abo ut the share library" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(coxeterG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "read(`c:/maplev4/share/algebra/crystal/crystal25.mpl` );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 " One must always initialize crystal as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ini t_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 "" {INLPLOT "6+-%'POI NTSG6$7$\"\"\"\"\"!-%'SYMBOLG6#%'CIRCLEG-F$6$7$\"\"#F(F)-F$6$7$\"\"$F( F)-%'CURVESG6#7$F&F/-F66#7$F/F3-%%TEXTG6%7$#F4F0F(%\"1G%+ALIGNABOVEG-F =6%7$#\"\"&F0F(%\"2GFB-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONE G" 2 385 385 385 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 4 18 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The crystal graph product of the two is d enoted G3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "crystal[graph prodrep](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[showgr aphprod](G3,1,3,1,3);" }}{PARA 13 "" 1 "" {INLPLOT "6=-%'POINTSG6$7$\" \"\"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$F6F9-FG6#7$F?FB-FG6#7$F9FB-FG6#7$F.F2 -FG6#7$FBFE-FG6#7$F2F<-%%TEXTG6%7$F'#F3F/%\"1G%*ALIGNLEFTG-Fjn6%7$F'# \"\"&F/%\"2GF_o-Fjn6%7$F]oF/F^o%+ALIGNABOVEG-Fjn6%7$F]oF3F^oFio-Fjn6%7 $F/FcoFeoF_o-Fjn6%7$FcoF'FeoFio-Fjn6%7$FcoF3FeoFio-Fjn6%7$F3F]oF^oF_o- %(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 2 385 385 385 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "We pick out the component of this graph which contains the highes t weight vector 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 other words, we denote by G4 the component of G3 whic h contains the highest weight vertex v1X1 (the so-called \"Cartan prod uct\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "crystal[linsubgr aphrep](v1X1,A2,G3,G4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~for med.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "By the way, to see that \+ v1X1 is the highest weight, simply type the " }{TEXT 259 7 "vweight" } {TEXT -1 46 " command to see it's associated weight vector:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "networks[vweight](G3);\nprin t(networks[vertices](G3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLE G6$%'sparseG7+/%%v1X1G7$\"\"#\"\"!/%%v1X3G7$\"\"\"!\"\"/%%v3X1GF//%%v2 X1G7$F,F0/%%v2X3G7$F1F,/%%v3X2GF9/%%v1X2GF6/%%v2X2G7$!\"#F+/%%v3X3G7$F ,FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+%%v1X1G%%v2X1G%%v1X2G%%v2X2G% %v2X3G%%v3X2G%%v3X1G%%v1X3G%%v3X3G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Though we don't need this, to see the (root) labels of the edge s, type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "networks[eweigh t](G3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7*/%#e4G\"\"\"/% #e5G\"\"#/%#e2GF,/%#e8GF)/%#e6GF,/%#e1GF)/%#e3GF)/%#e7GF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The graph G4 can be viewed as a linear gr aph 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 "" {INLPLOT "66-%'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&F/7$F4#F'F0FKF7 -FB6#7&F37$F8FLFPF;-FB6#7$F7F;-FB6#7$F;F?-%%TEXTG6%7$#F4F0F(%\"1G%+ALI GNABOVEG-FX6%7$#F " 0 "" {MPLTEXT 1 0 32 "crysta l[graphprodrep](G1,G4,G8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~ formed.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "then to see this, sim ply type the command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cr ystal[showgraphprod](G8,1,3,1,6);" }}{PARA 13 "" 1 "" {INLPLOT "6hn-%' POINTSG6$7$\"\"\"F'-%'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(-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(-%'C URVESG6#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$F jnF]o-F_o6#7$F>FQ-%%TEXTG6%7$F'F[q%\"1G%*ALIGNLEFTG-Fcs6%7$F'#F;F/%\"2 GFgs-Fcs6%7$F[qF/Ffs%+ALIGNABOVEG-Fcs6%7$F[qF3FfsF`t-Fcs6%7$F/F[tF\\tF gs-Fcs6%7$F[tF'FfsF`t-Fcs6%7$F[tF/FfsF`t-Fcs6%7$F[tF3FfsF`t-Fcs6%FjpF \\tF`t-Fcs6%F_qF\\tF`t-Fcs6%7$F3F[tF\\tFgs-Fcs6%FhqF\\tF`t-Fcs6%F\\rF \\tF`t-Fcs6%FhqFfsFgs-Fcs6%7$#\"\"*F/F/FfsF`t-Fcs6%7$F`vF3FfsF`t-Fcs6% 7$#\"#6F/F'F\\tF`t-Fcs6%7$FhvF/F\\tF`t-Fcs6%7$FhvF3F\\tF`t-Fcs6%7$F?F[ qFfsFgs-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 2 384 384 384 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 0 2 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 253 "We shall not need this, but the restriction of this representation to the subalgeb ra su(1) associated to the root labeled \"1\" is easy to determine. It is obtained by omitting those edges with label 2. Denote this restric tion by G5 and view it using the " }{TEXT 261 9 "showgraph" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "crystal[bra nch]([2],G4,G5);\ncrystal[showgraph](G5,1,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Branch~graph~formed.G" }}{PARA 13 "" 1 "" {INLPLOT "6 0-%'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(%\"1G%+ALIGNABOVEG-FL6%7 $#F " 0 "" {MPLTEXT 1 0 39 "crystal[li nsubgraphrep](v1X2,A2,G3,G6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Gra ph~formed.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "It is viewed as fo llows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "crystal[showgraph ](G6,1,3);" }}{PARA 13 "" 1 "" {INLPLOT "6+-%'POINTSG6$7$\"\"\"\"\"!-% 'SYMBOLG6#%'CIRCLEG-F$6$7$\"\"#F(F)-F$6$7$\"\"$F(F)-%'CURVESG6#7$F&F/- F66#7$F/F3-%%TEXTG6%7$#F4F0F(%\"2G%+ALIGNABOVEG-F=6%7$#\"\"&F0F(%\"1GF B-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 2 384 384 384 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 4 18 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "This is obviously 3-bar. Let G7 denote the crystal graph product o f G1 and G6:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "crystal[gra phprodrep](G1,G6,G7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Graph~forme d.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 "" {INLPLOT "6=-%'POINTSG6$7$\"\"\"F'-%'SYMBO LG6#%'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$F&F 6-FG6#7$F6F?-FG6#7$F&F.-FG6#7$F?FB-FG6#7$F.F9-FG6#7$F9F<-FG6#7$FBFE-FG 6#7$F " 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 "Representation s of finite and compact groups" }{TEXT -1 34 ", Grad Studies in Math., AMS, 1996" }}}}{MARK "33 0 0" 51 }{VIEWOPTS 1 1 0 1 1 1803 }