MVR5 I'hidden=6"%&falseGE\[lZ%-genhallmonomGR6#'%'WeightG-%%listG6#%(integerG6-%%Gen sG%$AlgG%$SetG%"iG%"jG%"kG%"UG%"VG%"AG%"BG%$AddG6$%'systemG%)rememberGF$C*@'2"" #9#-%&ERRORG6$%3too~many~argumentsG9"/FCFBC%>8$%(ALGEBRAG>8%&FH6#FB@$0FPFM-%$us eG6#FO-%'RETURNG6#-F'6$9$FM>8-7#-%$seqG6$""!/8'Ffn@'-%%&!>=G6$FfnFhn>8.%%trueG- %%&!<=GFco>FeoF%C$-FU6#FL-FE6%%'weightGFfn%Mhas~to~be~either~positive,~negative ~or~zero.G>8&7"?(F$"""FepF$0FhnFfnC$?(F_oFepFep-%%nopsG6#FhnFfo@%/&Fhn6#F_o&Ffn F_q>F^qF]oC$@%Feo>F^q,&F^qFepFepFep>F^q,&F^qFep!""Fep%&breakG@$FfpC%>8*-F'6$,&F fnFepFhnFhqFO>8+-F'6$FhnFO?(8(FepFep-Fjp6#F]rFfo?(8)FepFep-Fjp6#FbrFfo@$-%#&Fbp7$& FbpF$-%&evalnG6#-%#&*GF`s@$-%'memberG6%Ffn7#-F[o6$-%#wtG6$F_oFO/F_o%+GENERATORS G.Fjr>Fbp7$Fct&FduFds@$3FI0FMFLF[p-FX6#Fbp6#%RGenerates~Hall~Monomials~of~the~s pecified~degree.G6$FduFMF$%&AnmatGR6#'%"nGF/F$FF]r-F_t6$""%-%%evalG6#%'simpleG>Fbp&F]r6$Fgx%% typeG-%*unprotectG6#-Fet6#%,MATRIX_TYPEG>F[z&FbpFQ-%(protectGFhy@%0FgxFMC&>FbrF 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cl>FO-Fht6$FOF[cl-FXFV6#%PConverts~a~list~to~a~lieexpression~(with~&*-s).GF$F$% /epsilon_type_DGR6%'F9F,'F:F,'%"MG%'matrixG6%F4F5F__lF$F$C%>FbpF]o?(FLFepFepFe` lFfo?(FOFepFep-FjpFezFfo@$2FOFL>Fbp,&FbpFep-%$modG6$*(F[clFep&FgxFVFep&9&6$FLFO FepFBFep-FX6#-F`el6$FbpFB6#%!GF$F$%6triangularsimplifyoldGR6$'%"EGFhwFiw6,F4F__ lFbx%%TempG%"PG%"SG%"NG%(formsumG%$LhsG%$RhsGFFbrR6&'FhflFhw'%%Rhs1GFhw'% %Rhs2GFhwFiw6$FcflF__lFFL-%-mainsimplifyG6$-Fht6$FfnFgx9'@%-Fey6$FL%"*G>F O*&-F_t6$FepFLFep-Fggl6$-Fht6$-F_t6$FBFLFeelF[hlFep>FO-Fggl6$-Fht6$FLFeelF[hl>F L-Fggl6$-Fht6$FfnFeelF[hl@&F]hl>FO,&FOFep*&FbhlFep-Fggl6$-Fht6$FgxFhhlF[hlFepFe p0FLF]o>FO,&FOFep-Fggl6$-Fht6$FgxFLF[hlFepFjclF$F$F$>Fbp-F_t6$F]y-F_y6#%+triang ularG>F_o7#&FbpFez>Ffr&F_oF][l>Fjr&F_oFQ>F]r&F_o6#""$@%5F]al-Fey6$FfnFi\lF]\lC$ >8,F`\l>FhnFc\l@+3-Fhs6$7#-%(extractG6$Fj[mFgxFfr-Fhs6$7#-Fb\m6$FhnFgxFfr>FO-Fg gl6$FfnFfr3-Fhs6$F`\mF]r-Fhs6$Ff\mF]r>FO-Fggl6$FfnF]r3-FhsFc\m-FhsFh\m>FO-FcwFj 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bm6'F]rFOF_oFfrFjrFigmF[flF$F$%,double_listGR6#'%$SumGFhw6(Fedm%"TGFeflF4F2F1F$ F$C(@'FAFDFI@%0FMFPC%>FjrFM>FfrFP-FUFbsC$>FjrFM>FfrFMC$>FjrFM>FfrFM>FLFcp>FOFcp @$0FfnF]oC$@%-Fey6$FfnFbal>Fbp7#-F_tFj[l>FbpF_`l?(F_oFepFep-FjpF]vFfo@%-Fey6$&F bpF_qF_hlC$>FO7$Fd]l-F_t6$FepFhjm>FL7$Fb]l-F_t6$FBFhjmC$>FO7$&FOF$Fep>FL7$Fb]lF hjm@$3FI0FjrFM-FUFds-FX6#7$FLFO6#%_vConverts~a~sum~into~a~pair~of~lists~-~first ~list~is~a~list~of~monomials,~the~second~is~the~list~of~coefficients.|+~~~~~~~~ ~~~~~Returns:~[[list~of~monomials],~[list~of~coefficients]]G6#FMF$%&DnmatGRFcv6 #FcdlFFL-F_w6$Ffn,&FfnFepFhqFep>&FL6$,&FfnFep!"#FepFfnFhq>&F L6$FfnF_]nFhq>&FLF`wFB-FX6#-F_yF\p6#%7generates~an~Dn~matrixGF$F$%&fieldGRF$F$F $F$@'2FepFC-FE6$%Yfield()~expects~zero~or~one~(integer)~argument,~receivedGFH/F CF]oC'@$/%)MODVALUEG.Ff^n-FE6#%Nthere~is~no~MODVALUE~defined,~can't~set~flag.G- Fgy6#-Fet6#%(MODFLAGG>F__n4F__n-F_zF\_n@%F__n-FX6#/-Fet6#%/characteristicGFf^n- FX6#%3characteristic~offGC(@$4-Fey6$&FHF][lF/-FE6$%Nfield()~expects~an~integer~ argument,~receivedGFb`n-Fgy6$-Fet6#Ff^nF]_n>Ff^nFb`n>F__nFfo-F_zFg`nFd_n6#%Sthr ough~field()~it~is~possible~to~set~the~MODVALUEG6$F__nFf^nF$%&EnmatGRFcvFd\nFFLFi\n>&FL6$,&FfnFep!"$FepFfnFhq>&FL6$FfnFianFhq>Fe]nFBFf]n6#%7 generates~an~En~matrixGF$F$%:simplefunction_ADE_matrixGR6$'FaflFealFbdl6&%#LIG% #RIGFhflFiflF$F$C'>FbpF`\l>F_oFc\l>FL7#-F_tF]v>FO7#-F_tF_q@-553-%(hastypeG6$FL% )negativeG-Ffcn6$FL%)positiveG3-Ffcn6$FOFhcn-Ffcn6$FOF[dn/FbpF_oFe_m3/-F_t6$F]o &FduFQ-F_t6$F]oFbp/Fddn-F_t6$F]oF_oFe_mFcdn@%3/-%%scpdG6%FOFOFgxFB32F]o&FLF$42- FjpFVFden-FX6#*&-%$addG6$*&&FO6#F5Fep&Fgx6$FdenF5Fep/F5;FepFgenFep&-F_t6$F]o&Fd uF][l6#Fe[nFepFe_mFidn@%3/-F`en6%FLFLFgxFB32F]oFe[n42-FjpF\pFe[n-FX6#,$*&-F\fn6 $*&&FLF`fnFep&Fgx6$Fe[nF5Fep/F5;FepFcgnFep&Fffn6#FdenFepFhqFe_m3F\gnF^en@'/-F`e n6%FLFOFgxF`]n-FX6#*&)Fhq-(%)epsilon_GF[zFfhnFep-F\fn6$*&F[hnFep&FddnF`fnFepF^h 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'%*MonomialBGFhw6(F4%&LimitG%&CompAG%&CompBG%%VarAG%%VarBGF$F$C'>Fbp7#-Fb\mFen> F_o7#-Fb\m6$FeelFM@%2Fdjm-FjpF_q>FOFdjm>FOF`_u?(FLFepFepFOFfoC)>FfrF]o>FjrF]o-F [u6%&FbpF\pFdu.Ffr@$3/FfrF]o-Fey6$Fi_uFi\l-F[u6%-F_t6$F]oFi_u-Fgbm6$RF[cpF$F$F$ @%Ff[m-F_t6$F]oFfnFfnF$F$F$FduFj_u-F[u6%&F_oF\pFduFeu@$3/FjrF]o-Fey6$F\auFi\l-F [u6%-F_t6$F]oF\au-Fgbm6$RF[cpF$F$F$@%Ff[mFh`uFfnF$F$F$FduFeu@$0FfrFjr@'3/Fgx%"> G2FjrFfr-FX6#Ffo3/Fgx%"FLF^w>&FL6$F_]nF[]nF`]nFf]n6#%7generates~an ~Fn~matrixGF$F$FghtR6$'%#E1GFhw'%#E2GFhw6+F4F9F:F__l%$RemGF1F2%#L1G%#L2GF$F$C+@ 'FhasFDFiasC%>FjrFM>F]rF]bs@$0FMF]r-FUFhrC$>FjrFM>F]rFM>Fbr7#-Fb\m6$-FhalFc]mF] r>Fj[m7#-Fb\m6$-Fhal6$FgxF]rF]r@%2Fj]tF[sC$>FO&Fbr6#;,(F[sFepFj]tFhqFepFepF[s>F fr&Fbr6#;Fep,&F[sFepFj]tFhqC$>FOFbr>FfrFcp>FbpFj[m>F_oFcp?(FLFepFepFgenFfo@$/&F O6#;FLFgen&Fbp6#;Fep,(FgenFepFepFepFLFhq>F_o7$&F_oF$7%7$Feep&&FOF]]sF$F_gu&Fbp6 #;,(FgenFepFBFepFLFhqFdjm@$Fj[nF[\n-FXF_q6#%droverlap()~picks~out~the~overlaps~ of~the~end~of~A~(1st~arguments)~and~the~|+~~~~~~~~~~~~~~~beginning~of~B~(second ~argument)GF$F$%%dumpGRF$F$F$F$@'Fb^n-FX6#/FM-%-anglebracketG6#&FduF$-Fey6$Fb`n F_]l@%0-F_t6$F]y-F_y6#Fb`nFh`l-FX6#FciuFf`l-FX6#Fd[l6#%indebugging~tool,~dumps~ a~procedures~hash~table~(remember~table)GF$F$%/sorted_leadingGRFdhmF$F$F$@%F]jm -FXFfhrF]\l6#%dpReturns~the~leading~term~of~the~sorted~sum|+~~~~~~~~~~~~~~~~wit h~the~leading~coefficient~1GF$F$FhalRFdhm6%%(MinTermGF2F1F$F$C&@'FAFDFI@%F\imC% >FbpFM>FOFPFTC$>FbpFM>FOFMC$>FbpFM>FOFM@%F]jm>FL&-F^dp6$F`jmF_sF][l>FLFfn@$3FI0 FbpFM-FUF]vFhjr6#%esPicks~out~the~leading~term~from~the~sum~(the~term~with~the~ maximum~value),|+~~~~~~~~~~~~~~~~and~also~checks~for~the~terms~to~be~homogenous .GFa\nF$%/epsilon_type_EGRF_dl6&F4F5F__lFevF$F$C&>FbpF]o>F_o-Fjp6#-%(convertG6$ -&%'linalgG6#%$rowG6$FeelFepF-?(FLFepFepFe`lFfo?(FOFepFepFjdlFfo@$5553/FL,&FOFe pFepFep1FL,&F_oFepFjanFep3/FLFi]v/FO,&F_oFepF`]nFep3/FLF]^v/FO,&F_oFepFhqFep3F[ ^v/FOF_o>FbpF^elFgel6#%@epsilon~function~for~En~algebraGF$F$%$&!FLF^w>FbdqFjanFf]n6#%7generates~an~Gn~matrixGF$F$F_aqRF [du6(F4F1F2F__lF9F:F$F$C(@'FhasFDFiasC%>FOFM>FbpF]bs@$0FMFbpF[\vC$>FOFM>FbpFM>F fr7#-Fb\m6$-Fhal6$FfnFbpFbp>Fjr7#-Fb\m6$-Fhal6$FgxFbpFbp@%/FfrFjr>F_o7%FcpFfnFc pC$>F_oFcp?(FLFepFep,(Fi[rFepFacoFhqFepFepFfo@$/Ffr&Fjr6#;FL,(FLFepFacoFepFhqFe pC$>F_o7%&FjrF]]sFfn&Fjr6#;,&FLFepFacoFepFi[rFiq@$0FOFMFTFbhu6#%`qsubwd()~picks ~out~the~subword~E1~in~E2~and~returns~a~and~b~in~a~list,|+~~~~~~~~~~~~~~~~where ~E2~=~aE1bGF$F$F]_vR6%'%#W1GF,'%#OPGF]^u'%#W2GF,6$F4%%FlagGF$F$C&@&0Fe`l-FjpFec p-FE6'%,the~weightsGFfn%$andGFeel%Gare~of~non-equal~length,~can't~compareG4-F[u 6$Fgx<'%"=G%#>=GF_bu%#<=GFebu-FE6%%*operationGFgx%1is~not~supportedG>FOFfo?(FLF epFepFe`lFfo@$5553Fdbu42F[cl&FeelF\p3F^bu42FbevF[cl3/FgxFddvFeev3/FgxFcdvFaevC$ >FOF%FiqFjcl6#%bschecks~for~W1~to~be~less~than~or~equal~to~W2~(this~relationshi p|+~~~~~~~~~~~~~~~has~to~hold~for~all~corresponding~elements~in~the~lists)GF$F$ %$&!>GRFi^vF$F$F$-FX6#-F]_v6%FfnF_buFgx6#%dqlist~&!>~list~defines~the~operator~ "&!>",~which~is~interface~to~the~|+~~~~~~~~~~~~~~~~~function~comparewtsGF$F$%/e psilon_type_AGRF_dlF`\vF$F$C&>FbpF]o>F_oFd\v?(FLFepFepFe`lFfo?(FOFepFepFjdlFfo@ $53Ff]v1,$FOFBF_o3/FL,&FOFepFhqFep2F_o,$FLFB>FbpF^elFgel6#%@epsilon~function~fo r~An~algebraGF$F$%(rankingGR6#'%)MonomialGFhw6)F4F5Fedm%&valueGFffl%"fG%)positi onG6#F>F$C(>FbpFh^u>F_oF]o>Ffr,&-Fjp6#FduFepFepFep>Fjr*$FfrFhq?(FLFepFepFdjmFfo C%-F[u6%Fi_uFdu.F]r>F_o,&F_oFep*&F]rFepFjrFepFep>Fjr*&FjrFepFfrFhqFbhu6#%epgive s~a~numerical~rank~to~a~monomial;~to~be~used|+~~~~~~~~~~~~~for~comparison~of~mo nomials;GF$F$%*sortscaleGRFdhm6'F2F1FcflFaxF6F$F$C%@'FAFDFI@%F\imC%>FOFM>FLFPF[ pC$>FOFM>FLFMC$>FOFM>FLFM@'F]jmC&>FbpFe[v>F_oFep@$-Fey6$F]gmF_hl>F_o-F_t6$FepF] gm-FX6#*&-F\fn6$Ffr/FfrFbpFepF_oFhqF^cm-FX6#*&FfnFepF`\lFhqF]\l@$3FIFhbvFT6#%eq Sorts~the~Sum~(the~leading~term~goes~first)~and|+~~~~~~~~~~~~~~~~rescales~to~ma ke~the~leading~coefficient~1GFa\nF$%%grabGR6#Fiw6#FbxF$F$C$>FL-Fehu6#%+generato rsG@%30FLFh`l-F[u6$F_`l7#-%(indicesGF\p-FX6#7#&FLFj[lFiiu6#%Qreturns~the~genera tors~for~the~requested~algebraGF$F$%8simplefunction_F_matrixGR6$'F^jpFealFbdl6+ FfbnFgbnF4FgjnFa_p%#EIGFhflFifl%#EMGF$F$C'>F]rF`\l>FbrFc\l>FLFh\p>FOFg_o@)3/Fdd nFc\pFa`oFe_mFd^w@%55/-%)F4LookupGFVFd[lF\aoFfenFe_mFhenFa`o@%55/-Fj^wF\pFd[lF\ boFbgnFe_mFdgnC*@$0FLF[co@$/-Fj^w6#F`jnFd[lFe_m>Fj[mF__w@%/Fj[mFd[lFe_m>Fjr-F\f n6$Fbp/Fbp-Fgbm6%RFjbmF$FjinF$F]jnF$F$F$Fj[mFffn>Fj[mFi^w@%Fi_wFe_m>Fjr-Fht6$Fj r-F\fn6$Fbp/Fbp-Fgbm6%RFjbmF$FjinF$F]jnF$F$F$Fj[mFffn>Fj[m-F`an6#""'@'-Fey6$Fjr Fbal>Fjr-Fgbm6%RFjbmF$FjinF$@%F^cmF\foF^foF$F$F$FjrFj[m-Fey6$FjrF_hl>Fjr*&-F_t6 $FepFjrFep-Fabn6$-F_t6$FBFjrFj[mFep>Fjr-Fabn6$FjrFj[m@%F_auFe_mC%@%Faaw>Fj[mF\b w>Fj[mFjr@%-Fey6$Fj[mF_hlC$>F_oF[`m>Fj[mF]`m>F_oFep@%/Fe`oFddn-FX6#-F]ho6$<%/&F ddn6#""&F]o/&FddnF^aw&FddnFb`p/F^dwF]oFjrC%>FLFj_o>FO7&,&&FLF][lFep&FLFjcwFep,& &FLFQFep&FLFb`pFep&FLFb[m&FLF^aw-FX6#*&F_oFep&Fe`oFifnFep6#%]qthis~is~the~funct ion~for~a~simple~lie~algebra,~that~does~the~simplification~of~algebras~of~type~ F4GF$F$Fg[lRFew6(%&tableGF4F\xF]x%%tempG%(entriesGFFL-F_t6$F]y-F_y6#%&sto reG@%3-Fey6$FfnFeal2FepFe`lC$>FbpF`\l>F_oFc\l-FE6$%Jnot~a~valid~expression~to~f ind~a~rule~forGFfn@%FcasFiiu>FjrF[gm?(FOFepFepFi[rFfoC$>Ffr-F_t6$Fep&FjrFV@)3-F ey6$FbpFi\l-Fey6$F_oFi\l@&33333/Fbp-F_t6$FepFfr/F_o-F_t6$FBFfr-Fey6$F\cn-F-6#Fi ap/F\cn7#-F_t6#F^hw-Fey6$F_cnFehw/F_cn7#-F_t6#Fahw-FX6#-F_t6$FBF`gw333/Fgdn-F_t 6$F]o&-F_tFbsF][l/Fjdn-F_t6$F]o&F\jwFQ/-Fjp6#F\cn-Fjp6#-%'removeG6%Fey7#-F_t6#F [jwF//-Fjp6#F_cn-Fjp6#-Fgjw6%Fey7#-F_t6#F`jwF/@%/FihwF_iw-FX6#-%(unapplyG6$Fciw Fihw-FX6#-F[\x6%FciwFihwF_iwFcgw@&FjgwFaiw33Fhiw/F_oF`jwFajw@%/FihwF_o-FX6#-F[\ x6$FciwF_o-FX6#-F[\x6%FciwFihwF_oFegw@&33F\hwF[iwF]iwFaiw33/FbpF[jwF]jwF\[x@%/F bpF_iw-FX6#-F[\x6$FciwFbp-FX6#-F[\x6%FciwFbpF_iw@$3Fd]xFd\xFaiwFiiu6#%\]lfuncti onize(expression),~if~possible,~finds~the~rule~in~the~lookup~table~that~|+~~~~~ ~~~~~~~~~~suits~the~given~expression,~and~returns~that~rule~converted~into~a~ma ple~function;|+~~~~~~~~~~~~~~~when~no~corresponding~rule~is~found~or~the~lookup ~table~is~empty~FAIL~is~|+~~~~~~~~~~~~~~~returnedGF$F$%$&!=GRFi^vF$F$F$-FX6#-%& evalbG6#/FfnFgx6#%dqlist~&!=~list~defines~the~operator~"&!=",~which~is~interfac e~to~the~|+~~~~~~~~~~~~~~~~~function~comparewtsGF$F$FhtR6$'F9Ffar'F:FfarF$F$F$@ -5-Fey6$FfnFiap-Fey6$FgxFiap-FX6#*&FfnFepFgxFep35-Fey6$FfnFjwF`fw5-Fey6$FgxFeal -Fey6$FgxFjw-FX6#-Fet6#FiglF^cm-FX6#*&F`\lFep-FhtFccmFepF[ir-FX6#*&FcirFep-Fht6 $FfnFeirFep5F]jmFjgr-FX6#-FbarFd`x-FE6%FfnFgx%Aare~not~defined~in~this~context. GF$F$F$FboRFi^vF$F$F$-FX6#-F]_v6%FfnFcdvFgx6#%fqlist~&!>=~list~defines~the~oper ator~"&!>=",~which~is~interface~to~the~|+~~~~~~~~~~~~~~~~~function~comparewtsGF $F$%/factorsimplifyGRF\jp6-FcjpF4F5FbxFdjpFcflF2FejpFgcmF1F_pFchvF$C,>F_oFjjp>F ]r&7#&F_oFezF][l>Fbr&FebxFQ>Feo-FauFc]m>FL-F^]q6$FeoFgx@$/FcgnF]oFe_m>Ffr&&F_o6 $FbrFg]qFQ@%-Fbo6$Feo7#-F[o6$F]o/FOFeo>Ffr-Fgbm6&RFc^qF$F$F$@$Fe^qFfnF$F$F$FfrF eoF]r>Ffr-Fgbm6&RFc^qF$F$F$@$F__qFfnF$F$F$FfrFeoF]r@$-F[u6$FfnFLF]\l?(FbpFepFep FacoFfoC$>Fjr-F_aq6%FdcoFfnF]r@$0FjrFcpC'>Fj[m,&FfnFep-Fggl6$-Fh^p6$FjesF]rF]rF hq@$/Fj[mF]oFe_m@$F[cw-FX6#*&F[`mFep-F^bx6$F]`mFgxFep@%Fads-FX6#-Fgbm6$RF[cpF$F $F$@%F^cm*&F`\lFep-F^bxF\emFep-F^bxFaemF$F$6$F`jpFgxFj[m-FX6#-F^bxFc\mFiqF$F$F$ %,independentGR6#'F]_lF-6-F4F5F6%*SimplListG%%VarsGF1F2%(VectorsG%.LinIndVector sGF__l%+addelementGF$F$C1>FeoR6$'%(ElementGFhw'F3%$setGF$F$F$@'F^cm-FX6#-%&unio nG6$Fgx<#Fc\l4Fc_x-FX6#-Fbhx6$Fgx<#Ffn-FXFezF$F$F$@$FAFD@%FI@%F\imC%>FjrFM>F]rF PF[euC$>FjrFM>F]rFM>F]rFM-%%withG6$F[]v%'vectorG-Fgix6$F[]v%&basisG>Ffr<">F_o-% &arrayG6#F^\o?(FLFepFepFe`lFfoC$>F\au-Fh`p6$F[clF]r@%-Fey6$F\auFbal?(FOFepFep-F jp6#F\auFfo>Ffr-Feo6$-F_t6$FOF\auFfr>Ffr-Feo6$F\auFfr>Fbr7#-F[o6$-Fiix6$FacoF]o /FL;Fep-Fjp6#7#-F_]wF_q?(FLFepFepF^\yFfo?(FOFepFepFacoFfo@(Fijx?(FbpFepFepF\[yF fo@&-Fey6$-F_t6$FbpF\auF_hl@$/-F_t6$FBFi\y&FfrFV>&&FbrF\pFV,&Fa]yFep-F_t6$FepFi \yFep/Fi\yF_]y>Fa]y,&Fa]yFepFepFep3-Fey6$F\auF_hl/-F_t6$FBF\auF_]y>Fa]y,&Fa]yFe p-F_t6$FepF\auFep/F\auF_]y>Fa]yFh]y>Fj[m-F\jxF\s>FhnFcp?(FLFepFepFj]tFfo@$-F[u6 %&Fj[mF\pFbr.Fbp>Fhn7$F_`rF[ds@$Fi[nF[\n-FXF[q6#%jqindependent([list~of~express ions])~picks~out~the~linearly~independent~expressions~|+~~~~~~~~~~~~~~~from~the ~listGF$F$FhoRFi^vF$F$F$-FX6#-F]_v6%FfnFddvFgx6#%fqlist~&!<=~list~defines~the~o perator~"&!<=",~which~is~interface~to~the~|+~~~~~~~~~~~~~~~~~function~comparewt sGF$F$%*checktabsGRF$6%FierFedm%$indGF$F$C$>FO7#-F_]w6#-F_t6$F]y-F_y6#Fggl?(FLF epFepFgenFfoC&>FbpFg\o-Fjft6#-F_t6$FepFbp@&-Fey6$F]ayFbal>&Fd`yF]v-Fet6#&Fd`y6# F]cn4-Fho6$-FauF\ay7%FepFepFep>Fcay-Fet6#Fcay-Fjft6#-Fjp6#Fa`yF[flF$F$%8simplef unction_G_matrixGRFh]w6'FfbnFgbnFhflFifl%)TrueGensGF$F$C(>FbpF`\l>F_oFc\l>FLF\c n>FOF_cn>Ffr&F[y6$%#G2GF_]w@-FbcnFe_mFbdnFe_mFcdn@%3-F[u6$FOFfrFbenFhenFe_mFidn @%3-F[u6$FLFfrF_gnFdgnFe_m3FhcyFdcyC%>FL-%)G2LookupG6#-Fet6#-Fht6$FbpF_o@$/FLFd [l>FL,$-F^dy6#-Fet6#-Fht6$F_oFbpFhqFhjrFe_m6#%]qthis~is~the~function~for~a~simp le~lie~algebra,~that~does~the~simplification~of~algebras~of~type~G2GF$F$Fj^wRF$ F$F$F$C$F>@%Fb^nCR>-Fj^w6#7&FepF]oF]oF]o7$7(FepF]oF]oF]oF]oF]o7(F]oF]oF]oF]oFep F]o>-Fj^w6#7&F]oFepF]oF]o7$7(F]oFepF]oF]oF]oF]o7(F]oF]oF]oFepF]oF]o>-Fj^w6#7&F] oF]oFepF]o7#7(F]oF]oFepF]oF]oF]o>-Fj^w6#7&F]oF]oF]oFep7#7(F]oF]oF]oF]oF]oFep>-F j^w6#7&FepFepF]oF]o7$7(FepFepF]oF]oF]oF]o7(F]oF]oF]oFepFepF]o>-Fj^w6#7&F]oFepFe pF]o7$7(F]oFepFepF]oF]oF]o7(F]oF]oFepFepF]oF]o>-Fj^w6#7&F]oF]oFepFep7#7(F]oF]oF epF]oF]oFep>-Fj^w6#7&FepFepFepF]o7$7(FepFepFepF]oF]oF]o7(F]oF]oFepFepFepF]o>-Fj ^w6#7&F]oFepFepFep7$7(F]oFepFepF]oF]oFep7(F]oF]oFepFepF]oFep>-Fj^w6#7&FepFepFep Fep7$7(FepFepFepF]oF]oFep7(F]oF]oFepFepFepFep>-Fj^w6#7&F]oFBFepF]o7#7(F]oFepFep FepF]oF]o>-Fj^w6#7&F]oFBFepFep7#7(F]oFepFepFepF]oFep>-Fj^w6#7&FepFBFepF]o7$7(Fe pFepFepFepF]oF]o7(F]oFepFepFepFepF]o>-Fj^w6#7&F]oFBFBFep7#7(F]oFepFBFepF]oFep>- Fj^w6#7&FepFBFepFep7$7(FepFepFepFepF]oFep7(F]oFepFepFepFepFep>-Fj^w6#7&FBFBFepF ]o7#7(FepFepFepFepFepF]o>-Fj^w6#7&FepFBFBFep7$7(FepFepFBFepF]oFep7(F]oFepFBFepF epFep>-Fj^w6#7&FBFBFepFep7#7(FepFepFepFepFepFep>-Fj^w6#7&FepFc[mFBFep7$7(FepFBF BFepF]oFep7(F]oFepFBFBFepFep>-Fj^w6#7&FBFBFBFep7#7(FepFepFBFepFepFep>-Fj^w6#7&F BFc[mFBFep7$7(FepFBFBFepFepFep7(FepFepFBFBFepFep>-Fj^w6#7&FBF]yFBFep7#7(FepFBFB FBFepFep>-Fj^w6#7&FBF]yFc[mFep7#7(FepFBFc[mFBFepFep>-Fj^w6#7&FBF]yFc[mFB7#7(Fep FBFc[mFBFepFB>-Fj^w6#7&FhqF]oF]oF]o7$7(FhqF]oF]oF]oF]oF]o7(F]oF]oF]oF]oFhqF]o>- 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#7&FhqFjanF`]nFhq7$7(FhqF`]nF`]nFhqF]oFhq7(F]oFhqF`]nF`]nFhqFhq>-Fj^w6#7&F`]nF` ]nF`]nFhq7#7(FhqFhqF`]nFhqFhqFhq>-Fj^w6#7&F`]nFjanF`]nFhq7$7(FhqF`]nF`]nFhqFhqF hq7(FhqFhqF`]nF`]nFhqFhq>-Fj^w6#7&F`]n!"%F`]nFhq7#7(FhqF`]nF`]nF`]nFhqFhq>-Fj^w 6#7&F`]nF^hzFjanFhq7#7(FhqF`]nFjanF`]nFhqFhq>-Fj^w6#7&F`]nF^hzFjanF`]n7#7(FhqF` ]nFjanF`]nFhqF`]nFiiu6#%fnthis~is~for~looking~up~what~indices~in~F4~unfold~into ~in~E6GF$F$%)symbasisGR6#'F+Fehw6)%&isposG%.GenerateBasisGFejpFbx%*IdealGensG%( AlgGensGF2FFOR6''F+F-'FfizF-'FgizF-Fiw'%$NegG%(booleanG6&F4%"WGFcflF__lF< F$C&>F_oF^jx?(FLFepFepFhcvFfoC$>FO-%$zipG6&RFjbmF$FjinF$,&FfnFepFgxFhqF$F$F$Ffn -Fau6$FbevF[hlF]o@&39(4F]dnC$>Fbp-Fj\s6'FOFgxFeelF[hlFb[[l>F_o-Fbhx6$F_o<#-F_t6 #-Fgbm6%RFjbmF$F$F$-Fet6#FhbrF$F$F$FbpFbev45Fb[[lF_dnC$>FbpFf[[l>F_o-Fbhx6$F_o< #-F_t6#-Fgbm6%RFjbmF$F$F$Fa\[lF$F$F$FbpFbev?(FLFepFepFjdlFfo@$/Ffn-Fau6$Fe\sF[h l>F_o-Fbhx6$F_o<#Fe\s-FX6#-F\gx6$F_cnF[hlF$F$6$FeizFO@'FAFDFI>FbpFP-FX6#-F_izFe n>F_o-F_t6$F]y-F_y6#%&idealG@%0F_oFh`l@%-F[u6$7#FbpF`\yC'>Ffr7#&F_oF]v>F]r&F_o6 $Fbp%(algebraG>F_o-F_t6$F]y-F_yFf\w@%Fj^[l>Fjr7#&F_oFhr-FE6#%Athere~are~no~gene rators~defined.G@'-Fbo6$Ffn7#-F[o6$F]o/F4Ffn@%/-Fgbm6%RFjbmF$F$F$@$-Fho6$-FauFj gl7#-F[o6$F]o/F4Faa[lFgbuF$F$F$FjrF]rFcp-FX6#-FO6'FfnFfrFjrF]rFfoF^^o-FhoFd`[l@ %/-Fgbm6%RFjbmF$F$F$@$-FboF`a[lFgbuF$F$F$FjrF]rFcp-FX6#-FO6'FfnFfrFjrF]rF%F^^o- FE6%%8received~a~mixed~weightGFfn%Jwhich~should~either~be~pos,~neg,~or~zero.G-F E6$%-not~an~idealGFbp-FE6#%=there~are~no~ideals~defined.G6#%fosymbasis([weight] )~generates~the~basis~of~an~algebra~for~a~specified~weightGF$F$%-leading_pairGR FdhmFdjuF$F$C&@'FAFDFI@%F\imC%>FbpFM>FOFPFTC$>FbpFM>FOFMC$>FbpFM>FOFM@%F]jm>FLF d[v>FLFfn@$Fi[vF[\v@%F]hl-FX6#7$FbhlFhhl-FX6#7$FepFL6#%`wPicks~out~the~leading~ term~from~the~sum~(the~smallest~term~with~respect~to~&<~order),|+~~~~~~~~~~~~~~ ~~and~also~checks~for~the~terms~to~be~homogenous.~~~~~~~~~~~~~Returns~a~list:~[ coefficient,~term]GFa\nF$FcgmRF_fl6.F4F__lFbxFcflFdflFeflFfflFhflFifl%%LgenG%%A lg1G%%Alg2GF$F$C/>FbpFdjl>F_oFjjl>FfrF][m>FjrF_[m>F]rFa[m@%Fe[mF]\lC$>FbrF`\l>F j[mFc\l>Fhn7#-Fb\m6$FbrFgx@)-Fhs6$FhnFfr>FeoFfr-Fhs6$FhnFjr>FeoFjr-Fhs6$FhnF]r> FeoF]rC&>F_o-FcgmF_f[l@$F]amFe_m@$Feam-FX6#*&FiamFep-Fcgm6$-Fht6$F]bmFj[mFgxFep @%-Fey6$F_oFbal-FX6#-Fgbm6$RF[cpF$F$F$@%F^cm*&F`\lFep-Fcgm6$-Fht6$Fc\lFj\sT&Fep -Fcgm6$-Fht6$FfnFj\sFdh[lF$F$6&FiflFj[mF2FgxF_o-FX6#-Fcgm6$-Fht6$F_oFj[mFgx>Fhn F`\m@)Faf[l>F^[oFfrFdf[l>F^[oFjrFgf[l>F^[oF]rC&>F_o-FcgmFc\m@$F]amFe_m@$Feam-FX 6#*&FiamFep-Fcgm6$-Fht6$FbrF]bmFgxFep@%Fgg[l-FX6#-Fgbm6$RF[cpF$F$F$@%F^cm*&F`\l Fep-Fcgm6$-Fht6$Fj\sFc\lFdh[lFep-Fcgm6$-Fht6$Fj\sFfnFdh[lF$F$6&FhflFbrF2FgxF_o- FX6#-Fcgm6$-Fht6$FbrF_oFgx@$/-F_yFceo-F_yFccpF]\l@$353-Fhs6$F]f[lFfrF]]m3-Fhs6$ F]f[lF]rF^\m45-Fbo6$,&-FauF_f[lFepF^_mFep7#-F[o6$F]o/FLFj\\l-FhoFh\\lFe_m@%-Fhs 6$FbrFeo@%-Fhs6$Fj[mF^[o-FX6#Fh]m-FX6#,&-Fcgm6$-Fht6$-Fht6$FbrF[`mF]`mFgxFep-Fc 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