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1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 257 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 66 "Maple Programs for Binary Tensor Invariants and Outerplanar Graphs" }}{PARA 19 "" 0 "" {TEXT -1 14 "Alec Mihailovs" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 80 "Invaria nts and decompositions of tensor powers of the standard representation of" }{TEXT 256 3 " SL" }{TEXT -1 4 "(2)." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The dimension of the space of " }{TEXT 257 6 "SL(2)-" } {TEXT -1 14 "invariants in " }{TEXT 258 1 "n" }{TEXT -1 69 "-fold tens or product of the standard 2-dimensional representation of " }{TEXT 259 2 "SL" }{TEXT -1 46 "(2) can be determined using following command :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "DimInvSL2 := n->`if`(n mod 2 = 0, \n binomial(n,n/2)/(n/2+1), \n 0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "DimInvSL2(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" =cWxBOlghvd;Ey>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Here is the li st of the dimensions for " }{TEXT 260 1 "n" }{TEXT -1 14 " from 0 to 2 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for n from 0 to 20 \n do A[n] := DimInvSL2(n) end do: \n(op(op((A))));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#77/\"\"!\"\"\"/F&F%/\"\"#F&/\"\"$F%/\"\"%F)/\"\"&F%/ \"\"'F//\"\"(F%/\"\")\"#9/\"\"*F%/\"#5\"#U/\"#6F%/\"#7\"$K\"/\"#8F%/F6 \"$H%/\"#:F%/\"#;\"%I9/\"#F%/\"#?\"&'z;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "We can also put the formula in a spreadsh eet, so that it can be used without typing the command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(Spread):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "i2 := CreateSpreadsheet():\nfor n to 10 \n \+ do SetCellFormula(i2,n,2,evaln(DimInvSL2(cat(~A,n)))) od; \nSetSelect ion(i2,1$4):" }}{PARA 0 "" 1 "" {SPREADSHEET {NAME "SpreadSheet009" } {ROWHEIGHTS } {COLWIDTHS 2 124 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }487 291 291 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "22" 20 "6#\"#A" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "DimInvSL2(~A1)" 20 "6#\"&'ye" }0 }{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "24" 20 "6#\"#C" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "DimInvSL2(~A2)" 20 "6#\"'7!3#" }0 } {CELL 3 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "26" 20 "6#\"#E" }0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "DimInvSL2(~A3)" 20 "6#\"'+Hu" }0 }{CELL 4 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2 8" 20 "6#\"#G" }0 }{CELL 4 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ " DimInvSL2(~A4)" 20 "6#\"(SWn#" }0 }{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "30" 20 "6#\"#I" }0 }{CELL 5 2 {CELLOPTS 0 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will produce the dimensions of the invariant spaces in column " }{TEXT 262 1 "B" }{TEXT -1 206 " after the evaluating of \+ the worksheet. For example, try to enter some numbers. Reentering othe r numbers and reevaluating the worksheet will correct the dimensions. \+ Another way of doing it is using Maplets." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "with(Maplets[Elements]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "m_i2 := Maplet( \nWindow( 'title'=\"Dimensions of Invariants\", [\n [\"Enter the tensor power: \", \n TextFiel d['TF1'](26, onchange=Evaluate('TB1' = evaln(DimInvSL2(TF1))))],\n \+ [\"Here is the dimension: \", \n TextBox['TB1']('editable' = ' false',3..40 )],\n [Button(\"Evaluate the dimension\", Evaluate('TB 1' = evaln(DimInvSL2(TF1)))),\n Button(\"Close\", Shutdown(['TF1 ', 'TB1'])),\n Button(\"Clear\", SetOption('TF1' = \"\"))]\n] ) \+ ):\nMaplets[Display]( m_i2 ):" }}{PARA 6 "" 1 "" {TEXT -1 38 "Initiali zing Java runtime environment." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The maplet looks as follows:" }}}{EXCHG {PARA 256 "" 0 "" {BITMAP 442 183 183 1 "?TMGVgA:;JUJNJWA:i?=JFJ:::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::yF^YvU;uQYER:::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::<:;jIsmYHjyF^xSByB::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::jq:;jIsmZY<::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::jq:;jIsmZYLJySB=tI=C:: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::jI=sAyCY=>:y^I=xC;_Z<:::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYvavGJ:vMxFra> N:y^I=xC;_Z<:::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYvavGJ:vMx Fra>N:y^I=xC;_Z<:::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYvavGJ :vMxFra>N:y^I=xC;_Z<:::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYv avGJ:vMxFra>N:y^I=xC;_Z<:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: vMYvavGJ:vMxFra>NNNTJr::::::::::::::::::::::: ::::::::;J:NZZLJ;:::::::::::::::BnZZ<::::::::::::::::::::::::::: ::JrZ:>NN::::::::::::::::::::::::::::::ZZLJ;:: :::::::::::::ynAyCY=>:y^I=xC;_Z<::::::::::::::Jb;;?::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Zj;;Bb> N::::::::::::::::HB:;_Z<::::::::::: ::::Jb;;?:::::::::::::::::::::::::::::Zj;;Bb>N:::::::::::::::jIwkYlq:; 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::::::::::::::::::::Y@vM:::::::jq::y>:::::::vGJ::Y@vM:::::::jq:;:vMxCY =>:y^I=xC;Z:jyCZ;FZ;Gb;?Y_ZDb[yyJrkysqxDjy?Zy>dyyM;_Zlyyqyre>::::::::::::::::::::::ZJJB:;J::::::::::::::::::::::::<;>:::: :::::::::::::::::::ZJJB:;J:::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::ja;y>:::::::vG:jI;:::::: :Y=>:ja;y>:::::::vGJ::y^Ylq:;jIsmZYL:B:vyy?dixi:>j:@jZDbB:yrE[LbN::: ::::vG:jI;:::::::Y=>:ja;y>:::::::vGJ::y^Ylq:;jIsmZYL:B:vYyueymQCZZ:vSxF::::::::B:<:jasm::::::::Z:B::y^YlqB:> :y^ImIsa>:<:vOxD:::::::::;B::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::vMxCY=>:y^I=xC;Z::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::jIsavGJ:vMxFra:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vG J:vMxFra:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::vGJ:vMxFY@:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::vgZ;;jIsa::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::Z:J:3:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Similarly, we can produce the procedure, the spreadsheet, and the maplet for the decomposition \+ of the tensor powers of the standard representation of " }{TEXT 263 2 "SL" }{TEXT -1 4 "(2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "M ultSL2 := (n,k)->`if`(n-k mod 2 = 0, \n binomial(n,(n-k)/2)*(k+1)/(n -(n-k)/2+1),\n 0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For examp le, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "MultSL2(6,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Here is the corresponding spreadsheet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "m2 := CreateSpreadsheet():\nfor n to 10 \+ \n do SetCellFormula(m2,n,3,evaln(MultSL2(cat(~A,n),cat(~B,n)))) od; \nSetSelection(m2,1$4):" }}{PARA 0 "" 1 "" {SPREADSHEET {NAME "Spread Sheet002" } {ROWHEIGHTS } {COLWIDTHS 3 150 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }487 291 291 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2" 20 "6#\"\"#" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "0" 20 "6#\"\"!" }0 }{CELL 1 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A1,~B1)" 20 "6#\"\"\"" }0 }{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "4" 20 "6#\"\"%" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2" 20 "6#\"\"#" }0 }{CELL 2 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A2,~B2)" 20 "6# \"\"$" }0 }{CELL 3 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "6" 20 "6 #\"\"'" }0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "4" 20 " 6#\"\"%" }0 }{CELL 3 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL 2(~A3,~B3)" 20 "6#\"\"&" }0 }{CELL 4 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "9" 20 "6#\"\"*" }0 }{CELL 4 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "1" 20 "6#\"\"\"" }0 }{CELL 4 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A4,~B4)" 20 "6#\"#U" }0 }{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "12" 20 "6#\"#7" }0 }{CELL 5 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "12" 20 "6#\"#7" }0 }{CELL 5 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A5,~B5)" 20 "6#\" \"\"" }0 }{CELL 6 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "15" 20 "6 #\"#:" }0 }{CELL 6 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "15" 20 " 6#\"#:" }0 }{CELL 6 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2 (~A6,~B6)" 20 "6#\"\"\"" }0 }{CELL 7 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "20" 20 "6#\"#?" }0 }{CELL 7 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "10" 20 "6#\"#5" }0 }{CELL 7 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A7,~B7)" 20 "6#\"&f1\"" }0 }{CELL 8 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "25" 20 "6#\"#D" }0 }{CELL 8 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "19" 20 "6#\"#>" }0 }{CELL 8 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A8,~B8)" 20 "6# \"%+?" }0 }{CELL 9 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "30" 20 " 6#\"#I" }0 }{CELL 9 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "26" 20 "6#\"#E" }0 }{CELL 9 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL 2(~A9,~B9)" 20 "6#\"$0%" }0 }{CELL 10 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "37" 20 "6#\"#P" }0 }{CELL 10 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "17" 20 "6#\"#<" }0 }{CELL 10 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A10,~B10)" 20 "6#\"*;l#RA" }0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "Entering the tensor power in the A colum n and the index of the irreducible representation in the B column, we \+ obtain the multiplicity of the representation in the tensor power of t he standard representation of " }{TEXT 264 2 "SL" }{TEXT -1 25 "(2). H ere is the maplet: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 730 "m_m 2 := Maplet( \nWindow( 'title'=\"Multiplicities of Representations\", \+ [\n [\"Enter the tensor power: \", \n TextField['TF1'](26,onc hange=Evaluate('TB1' = evaln(MultSL2(TF1,TF2)))), \n Button(\"Cl ear\", SetOption('TF1' = \"\"))], \n [[\"Enter the index\", \"of th e representation:\"], \n TextField['TF2'](26,onchange=Evaluate(' TB1' = evaln(MultSL2(TF1,TF2)))), \n Button(\"Clear\", SetOption ('TF2' = \"\"))],\n [\"Here is the multiplicity: \", \n Tex tBox['TB1']('editable' = 'false',3..50 )],\n [Button(\"Evaluate the multiplicity\", Evaluate('TB1' = evaln(MultSL2(TF1,TF2)))),\n B utton(\"Close\", Shutdown(['TF1', 'TF2', 'TB1'])),\n Button(\"Cl ear\", SetOption('TB1' = \"\"))]\n] ) ):\nMaplets[Display]( m_m2 ):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The maplet looks as follows:" }} }{EXCHG {PARA 256 "" 0 "" {BITMAP 507 236 236 1 "?TMGSYE:;JUJNjwA:SA=J FJ:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::jI=sAyPjxoq<@::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::>:<:y^Iys:ymJta[v]:::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::rUZ:jIsmZY<::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::x@B:vMxFra>nY@lZxmJ<::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZY@lIxMx@B :vMxFra>NNNNN:::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::ZZxM;?:::::::::::::::::::::::::Z:B::::::::::::::::::::::: ::::::::::;_Z<:::::::::::::::::::::::::J:>:::::::::::::::::::::::::::: ::::ZZLJ;::::::::::::::::::::raxrIsUZ:jIsmZYLJCb::::::::::::::::::::>t JysjwgZ?hi?F:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::>:;Bb>N:::::::::::::::::::::::::::;Bb>N:::::::::::::::::::::xcIxMx@B:vMxFra>NZZLJ;:::::::::::::::::::: :::::BnZZ<:::::::::::::::::::::::::::::JrZ:>NN::::::::::::::::::::::::::::::ZZLJ;::::::::::::::::::::raxrI sUZ:jIsmZYLJCb::::::::::::::::::::>TJJ;::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::BVJ:TJJ;:::::::::::::::::::::::::::::BVJ:NB:;?NB:;?NB:;?NB:;?NB:;?NB:;?NB:;?N B:;?NB:;?NB:;?NB:;?NB:;?NB:;?NB:;? 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::ravMx@B:vMxFra>:<:::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :ravMx@B:vMxFra::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::Zi;<:y^I=xC:::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::rUZ:jIsmvS:::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::xPjZ:jIsa:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>: <:1:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Comparing \+ with the calculation done by the previous maplet shows that the result s are the same. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "The followin g formula, the spreadsheet, and the maplet give all the multiplicities at once as the coefficients of a polynomial. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "MultPolSL2 := (n,x)->sum(evaln(MultSL2(n,'k')) *x^'k','k'=0..n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For example, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MultPolSL2(10,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"#U\"\"\"*&\"#!*F%)%\"xG\"\"#F%F%*& \"#vF%)F)\"\"%F%F%*&\"#NF%)F)\"\"'F%F%*&\"\"*F%)F)\"\")F%F%*$)F)\"#5F% F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "means that that the 10-fold tesor power of the standard representation of " }{TEXT 265 2 "SL" } {TEXT -1 69 "(2) equals the direct sum of the 42-dimensional space of \+ invariants, " }}{PARA 0 "" 0 "" {TEXT -1 331 "90 irreducible represent ations of dimension 3, 75 irreducible representations of dimension 5, \+ 35 irreducible representation of dimension 7, 9 irreducible representa tions of dimension 9, and 1 irreducible representation of dimension 11 . The following formula gives the dimension of the representation pres ented by such a polynomial:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dimSL2 := p->subs(x=1,diff(x*p,x)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "We can use this command to check the dimension of the rep resentation above, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dimS L2(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%C5" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 34 "The spreadsheet can be defined as " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "p2 := CreateSpreadsheet():\nfor n \+ to 10 do SetCellFormula(p2,n,2,evaln(MultPolSL2(cat(~A,n),x))) od; \nS etSelection(p2,1$4):" }}{PARA 0 "" 1 "" {SPREADSHEET {NAME "SpreadShee t010" } {ROWHEIGHTS 2 34 3 34 4 34 5 34 6 34 7 34 8 34 9 34 10 34 } {COLWIDTHS 2 271 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }487 291 291 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "1" 20 "6#\" \"\"" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL 2(~A1,x)" 20 "6#%\"xG" }0 }{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "2" 20 "6#\"\"#" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A2,x)" 20 "6#,&\"\"\"F$*$)%\"xG\"\"#F$F$" }0 }{CELL 3 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "3" 20 "6#\"\"$" } 0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A3,x )" 20 "6#,&%\"xG\"\"#*$)F$\"\"$\"\"\"F)" }0 }{CELL 4 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "4" 20 "6#\"\"%" }0 }{CELL 4 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A4,x)" 20 "6#,(\"\"#\"\"\"*& \"\"$F%)%\"xGF$F%F%*$)F)\"\"%F%F%" }0 }{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "5" 20 "6#\"\"&" }0 }{CELL 5 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A5,x)" 20 "6#,(%\"xG\"\"&*&\"\"%\" \"\")F$\"\"$F(F(*$)F$F%F(F(" }0 }{CELL 6 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "6" 20 "6#\"\"'" }0 }{CELL 6 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A6,x)" 20 "6#,*\"\"&\"\"\"*&\"\"*F%)%\"xG \"\"#F%F%*&F$F%)F)\"\"%F%F%*$)F)\"\"'F%F%" }0 }{CELL 7 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "7" 20 "6#\"\"(" }0 }{CELL 7 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A7,x)" 20 "6#,*%\"xG\"#9*&F %\"\"\")F$\"\"$F'F'*&\"\"'F')F$\"\"&F'F'*$)F$\"\"(F'F'" }0 }{CELL 8 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "8" 20 "6#\"\")" }0 }{CELL 8 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A8,x)" 20 "6#,, \"#9\"\"\"*&\"#GF%)%\"xG\"\"#F%F%*&\"#?F%)F)\"\"%F%F%*&\"\"(F%)F)\"\"' F%F%*$)F)\"\")F%F%" }0 }{CELL 9 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "9" 20 "6#\"\"*" }0 }{CELL 9 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A9,x)" 20 "6#,,%\"xG\"#U*&\"#[\"\"\")F$\"\"$F (F(*&\"#FF()F$\"\"&F(F(*&\"\")F()F$\"\"(F(F(*$)F$\"\"*F(F(" }0 }{CELL 10 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "10" 20 "6#\"#5" }0 } {CELL 10 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultPolSL2(~A10,x) " 20 "6#,.\"#U\"\"\"*&\"#!*F%)%\"xG\"\"#F%F%*&\"#vF%)F)\"\"%F%F%*&\"#N F%)F)\"\"'F%F%*&\"\"*F%)F)\"\")F%F%*$)F)\"#5F%F%" }0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Here is the maplet:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 490 "m_p2 := Maplet( \nWindow( 'title'=\"Multiplicities of Representations\", [\n [\"Enter the tensor power: \", \n \+ TextField['TF1'](26), \n Button(\"Clear\", SetOption('TF1' = \" \"))], \n [\"Here are the multiplicities:\", \n TextBox['TB1' ]('editable' = 'false',10..50 )],\n [Button(\"Evaluate the polynomi al\", Evaluate('TB1' = evaln(MultPolSL2(TF1,x)))),\n Button(\"Cl ose\", Shutdown(['TF1', 'TB1'])),\n Button(\"Clear\", SetOption( 'TB1' = \"\"))]\n] ) ):\nMaplets[Display]( m_p2 ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "However, for large tensor powers, the results ne ed to be sorted. 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::vMxCY=>:y^I=xC;Z::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::jIsa vGJ:vMxFra:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::vGJ:vMxFra:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::vGJ:vMxFY@:::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::vgZ;;jIsa::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::Z:J:3:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "C omparing the pictures of the Maplets with the Maple source, one can ea sily change them accordingly her/his computing needs. " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outerplanar Graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The formula for the dimension of the space of " } {TEXT 267 1 "n" }{TEXT -1 97 "-fold tensor invariants gives us the for mula for the number of outerplanar 1-regular graphs with " }{TEXT 268 1 "n" }{TEXT -1 81 " vertices. For instance, the number of the outerpl anar graphs with 6 vertices is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "DimInvSL2(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Writing " }{TEXT 278 1 "x" }{TEXT -1 37 " at the left vertix of every arc and " }{TEXT 279 1 "y" }{TEXT -1 69 " at the right vertex of every arc, we obtain a word in the alph abet \{" }{TEXT 280 4 "x, y" }{TEXT -1 81 "\}. They are so-called bala nced lattice words. \"Balanced\" mean that the number of " }{TEXT 283 1 "x" }{TEXT -1 29 "'s is equal to the number of " }{TEXT 284 1 "y" } {TEXT -1 53 "'s. The following recursive procedure determines the " } {TEXT 281 1 "k" }{TEXT -1 36 "-th balanced lattice word of length " } {TEXT 282 1 "n" }{TEXT -1 39 " in the lexicographical order, so that \+ " }{TEXT 285 13 "xx...xyy...y " }{TEXT -1 23 " is the first word and \+ " }{TEXT 286 9 "xyxy...xy" }{TEXT -1 57 " is the last one. The procedu re outputs a list of length " }{TEXT 287 1 "n" }{TEXT -1 21 " with 1's instead of " }{TEXT 288 1 "x" }{TEXT -1 23 "'s and -1's instead of " }{TEXT 289 1 "y" }{TEXT -1 97 "'s. A \"lattice\" word in this context \+ means that the sum of 1's and -1's from 1st position to the " }{TEXT 291 1 "m" }{TEXT -1 37 "-th position is non-negative for all " }{TEXT 292 1 "m" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "BLW := proc(n,k) \noption remember; \nlocal i,j, numset; \nnumset := [0$n/2+1]; \nfor i from 2 to n/2+1 \ndo numset[i] := numset[i-1]+D imInvSL2(n-2*(i-1))*DimInvSL2(2*(i-2)) od; \nj:=1; \nwhile (k>numset[j ]) \ndo j:=j+1 od; \n[1, \n op(BLW(n-2*(j-1), \n iquo(k-numset[j- 1]-1, \n DimInvSL2(2*(j-2)))+1)), \n -1, \n op(BLW(2*(j-2), \n \+ irem(k-numset[j-1]-1, \n DimInvSL2(2*(j-2)))+1))] \nend proc: \nBLW (0,1) := []:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "For example, here is the list of the balanced lattice words for " }{TEXT 290 1 "n" } {TEXT -1 5 " = 6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for k \+ from 1 to DimInvSL2(6) \n do BLW(6,k) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"F$F$!\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7(\"\"\"F$!\"\"F$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"F$! \"\"F%F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"!\"\"F$F$F%F%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"!\"\"F$F%F$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "To draw the outerplanar graphs, it is bet ter to have lists of the non-intersecting arcs instead: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "OPG := proc(n,k) \noption remember ; \nlocal i,j, numset; \nnumset:=array([0$n/2+1]); \nfor i from 2 to n /2+1 \n do numset[i] := numset[i-1]+DimInvSL2(n-2*(i-1))*DimInvSL2( 2*(i-2)) od;\nj:=1; \nwhile (k>numset[j]) \n do j:=j+1 od; \n[[1,n-2 *(j-2)], \n op(map(x->[x[1]+1, x[2]+1], \n OPG(n-2*(j-1), iquo( k-numset[j-1]-1,DimInvSL2(2*(j-2)))+1))), \n op(map(x->[x[1]+n-2*(j- 2), x[2]+n-2*(j-2)], \n OPG(2*(j-2), irem(k-numset[j-1]-1,DimInvS L2(2*(j-2)))+1)))] \nend proc: \nOPG(0,1):=[]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "For example, here is the list of the lists of the no n-intersecting arcs for " }{TEXT 275 1 "n" }{TEXT -1 5 " = 6." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for k from 1 to DimInvSL2(6) \n do OPG(6,k) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$\"\"\"\"\" '7$\"\"#\"\"&7$\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$\"\" \"\"\"'7$\"\"#\"\"$7$\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7 $\"\"\"\"\"%7$\"\"#\"\"$7$\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7%7$\"\"\"\"\"#7$\"\"$\"\"'7$\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$\"\"\"\"\"#7$\"\"$\"\"%7$\"\"&\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The following procedure draws " }{TEXT 269 2 "cr" }{TEXT -1 25 " outerplanar graphs with " }{TEXT 271 1 "n" } {TEXT -1 26 " vertices, starting from " }{TEXT 273 1 "k" }{TEXT -1 1 "," }{TEXT 274 1 " " }{TEXT -1 11 "displaying " }{TEXT 272 2 "c " } {TEXT -1 23 "graphs in each row, in " }{TEXT 270 1 "r" }{TEXT -1 36 " \+ rows, numbering vertices with step " }{TEXT 293 1 "s" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 943 "GridOPG := proc (n,k,c,r ,s) \nlocal nr,nlr,cr,i1,opg,lin,i,ir,ic,lab,w,t; \nw:=interface(warnl evel); \ninterface(warnlevel=0);\nwith(plots); \nwith(plottools); \nin terface(warnlevel=w);\nnr := 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