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Specifically, we want to know what kind of " }}{PARA 201 "" 0 "" {TEXT 200 90 "cycle structure s appear when we view the Weyl group's action as a permutations of wei ghts." }}{PARA 201 "" 0 "" {TEXT 200 106 " In general, a representat ion's weights may split up into several orbits. This can happen even w hen the " }}{PARA 201 "" 0 "" {TEXT 200 113 "representation is irreduc ible. In fact, since the Weyl group action preserves inner products (a nd thus preserves " }}{PARA 201 "" 0 "" {TEXT 200 118 "lengths), we mu st have multiple orbits in some cases. For example, anytime the zero v ector is a weight, it is the sole" }{TEXT 200 33 "\noccupant of its or bit. However, " }{TEXT 200 76 "in the case of minuscule representation s, all weights lie in a single orbit." }}{PARA 201 "" 0 "" {TEXT 200 5 " A " }{TEXT 201 9 "minuscule" }{TEXT 200 107 " representation is \+ an irreducible representation in which all weights lie in a single Wey l group orbit. The" }}{PARA 201 "" 0 "" {TEXT 200 109 "highest weight \+ of such a representation is said to be a minuscule weight. The highest weight space is always " }}{PARA 201 "" 0 "" {TEXT 200 115 "one-dimen sional. Also, if two weights are in the same Weyl group orbit, then th eir weight spaces must have the same" }}{PARA 201 "" 0 "" {TEXT 200 94 "dimensions. Thus, all of the weight spaces of a minuscule represen tation are one-dimensional. " }{TEXT 200 118 "\n Thus minuscule repr esentations behave quite nicely with respect to the Weyl group action. In fact, in some cases, " }}{PARA 201 "" 0 "" {TEXT 200 118 "we can \+ \"see\" the irreducibility of the minuscule representation by inspecti ng the cycle structures of the permutations" }}{PARA 201 "" 0 "" {TEXT 200 114 "associated with the Weyl group's elements. An invariant subspace of a representation cannot share part of an orbit" }}{PARA 201 "" 0 "" {TEXT 200 120 "of weights, so if the cycle structures tell us that there is only one orbit, we must conclude that the representa tion is" }}{PARA 201 "" 0 "" {TEXT 200 116 "irreducible (this assumes \+ that the weight spaces are one-dimensional, which may not be true if t he highest weight is" }}{PARA 201 "" 0 "" {TEXT 200 16 "not minuscule) . " }}{PARA 201 "" 0 "" {TEXT 200 121 " In this worksheet, we will s ee that cycle structure determines irreducibility for minuscule repres entation associated " }}{PARA 201 "" 0 "" {TEXT 200 115 "with simple L ie algebras of types B2, B3, B5, B7, E6, and E7. The corresponding pap er covers the types An, Cn, and " }}{PARA 201 "" 0 "" {TEXT 200 114 "D n (all n). We will also see that cycle structure does not tell us enou gh in the case of B4 -- even though we are " }}{PARA 201 "" 0 "" {TEXT 200 16 "dealing with an " }{TEXT 201 11 "irreducible" }{TEXT 200 26 " minuscule representation." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 109 ": Please start with the \"A4: \+ Master Demo\" which demonstrates how to use each of the procedures def ined below." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 58 " I n this worksheet, we will work in the Euclidean space " }{TEXT 203 1 " C" }{TEXT 200 56 "^m (m-dimensional complex space). All weights (inclu ding" }}{PARA 201 "" 0 "" {TEXT 200 121 "simple roots, fundamental wei ghts, etc.) are written down in the standard basis: L_i = [0,...,0,1,0 ,...,0] (1 in the i-th" }}{PARA 201 "" 0 "" {TEXT 200 33 "position) fo r i=1,...,m. We give " }{TEXT 203 1 "C" }{TEXT 200 38 "^m the standard bilinear dot product: " }}{PARA 201 "" 0 "" {TEXT 200 30 " ( [ a_1,a_2,...,a_m] | " }{TEXT 200 20 "[b_1,b_2,...,b_m] ) " }{TEXT 200 36 "= a_1 b_1 + a_2 b_2 + ... + a_m b_m." }}{PARA 201 "" 0 "" {TEXT 200 118 " We represent cycle structures by partitions. For example, \+ suppose that some permutation sigma is the product of two" }{TEXT 200 13 "\n3-cycles, a " }{TEXT 200 111 "transposition, and a 1-cycle (all \+ disjoint). Therefore, sigma has cycle structure [3,3,2,1]. However, w e will " }{TEXT 200 17 "\nalways suppress " }{TEXT 200 69 "the 1's in \+ these partitions. Thus, sigma has cycle strucutre [3,3,2]." }{TEXT 200 111 "\n Throughout this worksheet, V(lambda) will denote the irr educible representation with highest weight lambda." }}{PARA 201 "" 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT 204 23 "Remember to Initialize!" } }{PARA 201 "" 0 "" {TEXT 200 113 "Place your mouse next to the \"resta rt\" command below and press enter three times (this will add the grou p package" }{TEXT 200 23 "\nand define all of the " }{TEXT 200 104 "ne cessary procedures in \"Simple Lie Algebra Background Information\" an d \"Weyl Group Action on Weights\")." }}}}{SECT 0 {PARA 200 "" 0 "" {TEXT -1 10 "Initialize" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(group):" }}}}{SECT 1 {PARA 200 "" 0 "" {TEXT -1 41 "Sim ple Lie Algebra Background Information" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 52 "SimpleLieTypeCheck -- checks for valid type and rank" } {TEXT 200 78 "\nCartanMatrix -- returns the Cartan matrix of the speci fied simple Lie algebra" }{TEXT 200 78 "\nDynkinDiagram -- plots the D ynkin diagram of the specified simple Lie algebra" }{TEXT 200 13 "\nSi mpleRoots " }{TEXT 200 70 "-- returns a base of simple roots for the s pecified simple Lie algebra" }{TEXT 200 91 "\n (from Fulton an d Harris: pages 324,332,333 & the C-type bases have been rescaled)" } {TEXT 200 20 "\nFundamentalWeights " }{TEXT 200 85 "-- returns the fun damental weights with respect to the base calculated in SimpleRoots" } }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 73 "# SimpleLieTypeCheck che cks if the combination of type and rank is valid." }{MPLTEXT 1 0 81 " \n# lie_type should be a capital letter A,B,...,G and lie_rank a posit ive integer." }{MPLTEXT 1 0 83 "\n# SimpleLieTypeCheck returns true fo r valid combinations. For invalid combinations" }{MPLTEXT 1 0 50 "\n# \+ false is returned along with a warning message." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 76 "\n# Note: Warnings are given for types C2 and D3 si nce they are more properly" }{MPLTEXT 1 0 51 "\n# referred to by the n ames B2 and A3 respectively." }{MPLTEXT 1 0 47 "\nSimpleLieTypeCheck : = proc(lie_type, lie_rank)" }{MPLTEXT 1 0 78 "\n if (lie_type = 'A' \+ and (not type(lie_rank,integer) or lie_rank <= 0)) then" }{MPLTEXT 1 0 95 "\n WARNING(cat(\"Invalid rank: \",lie_rank,\". Rank must be a positive integer for type A.\"));" }{MPLTEXT 1 0 13 "\n false; " }{MPLTEXT 1 0 80 "\n elif (lie_type = 'B' and (not type(lie_rank,i nteger) or lie_rank <= 1)) then" }{MPLTEXT 1 0 102 "\n WARNING(ca t(\"Invalid rank: \",lie_rank,\". Rank must be an integer greater than 1 for type B.\"));" }{MPLTEXT 1 0 13 "\n false;" }{MPLTEXT 1 0 80 "\n elif (lie_type = 'C' and (not type(lie_rank,integer) or lie_r ank <= 1)) then" }{MPLTEXT 1 0 102 "\n WARNING(cat(\"Invalid rank : \",lie_rank,\". Rank must be an integer greater than 1 for type C.\" ));" }{MPLTEXT 1 0 13 "\n false;" }{MPLTEXT 1 0 80 "\n elif (li e_type = 'D' and (not type(lie_rank,integer) or lie_rank <= 2)) then" }{MPLTEXT 1 0 102 "\n WARNING(cat(\"Invalid rank: \",lie_rank,\". Rank must be an integer greater than 2 for type D.\"));" }{MPLTEXT 1 0 13 "\n false;" }{MPLTEXT 1 0 84 "\n elif (lie_type = 'E' and \+ lie_rank <> 6 and lie_rank <> 7 and lie_rank <> 8) then" }{MPLTEXT 1 0 87 "\n WARNING(cat(\"Invalid rank: \",lie_rank,\". Rank must be 6, 7, or 8 for type E.\"));" }{MPLTEXT 1 0 13 "\n false;" } {MPLTEXT 1 0 48 "\n elif (lie_type = 'F' and lie_rank <> 4) then" } {MPLTEXT 1 0 78 "\n WARNING(cat(\"Invalid rank: \",lie_rank,\". R ank must be 4 for type F.\"));" }{MPLTEXT 1 0 13 "\n false;" } {MPLTEXT 1 0 48 "\n elif (lie_type = 'G' and lie_rank <> 2) then" } {MPLTEXT 1 0 78 "\n WARNING(cat(\"Invalid rank: \",lie_rank,\". R ank must be 2 for type G.\"));" }{MPLTEXT 1 0 13 "\n false;" } {MPLTEXT 1 0 63 "\n elif (not (lie_type in \{'A','B','C','D','E','F' ,'G'\})) then" }{MPLTEXT 1 0 70 "\n WARNING(cat(\"Invalid Type: \+ \",lie_type)); # Invalid letter type." }{MPLTEXT 1 0 13 "\n false ;" }{MPLTEXT 1 0 47 "\n elif (lie_type = 'C' and lie_rank = 2) then" }{MPLTEXT 1 0 65 "\n # Looking at Dynkin diagrams it is easy to \+ see that C2=B2." }{MPLTEXT 1 0 78 "\n WARNING(\"Type C2 is more c ommonly known as B2 (they are isomorphic).\");" }{MPLTEXT 1 0 12 "\n \+ true;" }{MPLTEXT 1 0 47 "\n elif (lie_type = 'D' and lie_rank = \+ 3) then" }{MPLTEXT 1 0 65 "\n # Looking at Dynkin diagrams it is \+ easy to see that D3=A3." }{MPLTEXT 1 0 78 "\n WARNING(\"Type D3 i s more commonly known as A3 (they are isomorphic).\");" }{MPLTEXT 1 0 12 "\n true;" }{MPLTEXT 1 0 8 "\n else" }{MPLTEXT 1 0 12 "\n \+ true;" }{MPLTEXT 1 0 11 "\n end if;" }{MPLTEXT 1 0 5 "\nend:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 74 "\n# CartanMatrix returns the Car tan matrix of the simple Lie algebra of the" }{MPLTEXT 1 0 27 "\n# spe cified type and rank." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 59 "\n# Not e: These matrices can be found in Humphreys, page 59." }{MPLTEXT 1 0 40 "\nCartanMatrix := proc(lie_type,lie_rank)" }{MPLTEXT 1 0 50 "\n \+ if(SimpleLieTypeCheck(lie_type,lie_rank)) then" }{MPLTEXT 1 0 37 "\n \+ if (lie_type = 'A') then " }{MPLTEXT 1 0 32 "\n # Typ e A Cartan Matrix" }{MPLTEXT 1 0 74 "\n linalg[matrix](lie_ran k, lie_rank, [seq(seq(`if`(abs(i-j)=1,-1, " }{MPLTEXT 1 0 74 "\n \+ `if`(i=j,2,0)), j=1..lie_rank), i=1..lie_rank)]);" } {MPLTEXT 1 0 33 "\n elif (lie_type = 'B') then" }{MPLTEXT 1 0 32 "\n # Type B Cartan Matrix" }{MPLTEXT 1 0 91 "\n linal g[matrix](lie_rank, lie_rank, [seq(seq(`if`(i=lie_rank-1 and j=lie_ran k, -2," }{MPLTEXT 1 0 96 "\n `if`(abs(i-j)=1, - 1, `if`(i=j,2,0))), j=1..lie_rank), i=1..lie_rank)]);" }{MPLTEXT 1 0 33 "\n elif (lie_type = 'C') then" }{MPLTEXT 1 0 32 "\n # Type C Cartan Matrix" }{MPLTEXT 1 0 92 "\n linalg[matrix](lie _rank, lie_rank, [seq(seq(`if`(i=lie_rank and j=lie_rank-1, -2, " } {MPLTEXT 1 0 96 "\n `if`(abs(i-j)=1, -1, `if`(i =j,2,0))), j=1..lie_rank), i=1..lie_rank)]);" }{MPLTEXT 1 0 33 "\n \+ elif (lie_type = 'D') then" }{MPLTEXT 1 0 32 "\n # Type D Ca rtan Matrix" }{MPLTEXT 1 0 91 "\n linalg[matrix](lie_rank, lie _rank, [seq(seq(`if`(i=lie_rank-2 and j=lie_rank, -1," }{MPLTEXT 1 0 99 "\n `if`(i=lie_rank-1 and j=lie_rank, 0, `if `(i=lie_rank and j=lie_rank-2, -1," }{MPLTEXT 1 0 82 "\n \+ `if`(i=lie_rank and j=lie_rank-1, 0, `if`(abs(i-j)=1, -1," } {MPLTEXT 1 0 78 "\n `if`(i=j,2,0)))))), j=1..li e_rank), i=1..lie_rank)]);" }{MPLTEXT 1 0 33 "\n elif (lie_type = 'E') then" }{MPLTEXT 1 0 32 "\n # Type E Cartan Matrix" } {MPLTEXT 1 0 96 "\n linalg[matrix](lie_rank, lie_rank, [seq(se q(`if`(i=1 and j=2, 0, `if`(i=1 and j=3, -1," }{MPLTEXT 1 0 97 "\n \+ `if`(i=2 and (j=1 or j=3), 0, `if`(i=2 and j=4, -1 , `if`(i=3 and j=2, 0," }{MPLTEXT 1 0 89 "\n `i f`(i=3 and j=1, -1, `if`(i=4 and j=2, -1, `if`(abs(i-j)=1, -1," } {MPLTEXT 1 0 81 "\n `if`(i=j,2,0))))))))), j=1. .lie_rank), i=1..lie_rank)]);" }{MPLTEXT 1 0 33 "\n elif (lie_typ e = 'F') then" }{MPLTEXT 1 0 32 "\n # Type F Cartan Matrix" } {MPLTEXT 1 0 87 "\n linalg[matrix](lie_rank, lie_rank, [2,-1,0 ,0,-1,2,-2,0,0,-1,2,-1,0,0,-1,2]); " }{MPLTEXT 1 0 33 "\n elif (l ie_type = 'G') then" }{MPLTEXT 1 0 32 "\n # Type G Cartan Matr ix" }{MPLTEXT 1 0 59 "\n linalg[matrix](lie_rank, lie_rank, [2 ,-1,-3,2]); " }{MPLTEXT 1 0 14 "\n end if;" }{MPLTEXT 1 0 11 "\n \+ end if;" }{MPLTEXT 1 0 5 "\nend:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "\n# DynkinDiagram returns a plot of the Dynkin diagram of the si mple Lie algebra" }{MPLTEXT 1 0 34 "\n# of the specified type and rank ." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 59 "\n# Note: These diagrams ca n be found in Humphreys, page 58." }{MPLTEXT 1 0 42 "\nDynkinDiagram : = proc(lie_type, lie_rank)" }{MPLTEXT 1 0 71 "\n local i,nds,lbl,lns ,dts,arrow_top,arrow_bottom,ul_corner,lr_corner;" }{MPLTEXT 1 0 4 "\n \+ " }{MPLTEXT 1 0 51 "\n if(SimpleLieTypeCheck(lie_type, lie_rank)) \+ then" }{MPLTEXT 1 0 31 "\n if (lie_type = 'A') then" }{MPLTEXT 1 0 33 "\n # Type A Dynkin Diagram" }{MPLTEXT 1 0 33 "\n \+ if (lie_rank = 1) then " }{MPLTEXT 1 0 67 "\n ul_corner := plottools[point]([0.5,0.5], color=white):" }{MPLTEXT 1 0 68 "\n \+ lr_corner := plottools[point]([1.5,-0.5], color=white):" } {MPLTEXT 1 0 66 "\n nds[1] := plottools[circle]([1,0], 0.1, color=black):" }{MPLTEXT 1 0 72 "\n lbl[1] := plots[textpl ot]([1,-0.1,1],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 97 "\n \+ plots[display](ul_corner, lr_corner, nds[1], lbl[1], axes=none, sca ling=constrained," }{MPLTEXT 1 0 84 "\n titl e = cat(\"Dynkin Diagram: Type \",lie_type,lie_rank));" }{MPLTEXT 1 0 36 "\n elif (lie_rank > 5) then " }{MPLTEXT 1 0 34 "\n \+ # abbreviated diagram" }{MPLTEXT 1 0 33 "\n for i from 1 to 5 do" }{MPLTEXT 1 0 69 "\n nds[i] := plottools[cir cle]([i,0], 0.1, color=black):" }{MPLTEXT 1 0 16 "\n od:" } {MPLTEXT 1 0 33 "\n for i from 1 to 3 do" }{MPLTEXT 1 0 75 "\n lbl[i] := plots[textplot]([i,-0.1,i],align=\{BELOW,C ENTER\}):" }{MPLTEXT 1 0 16 "\n od:" }{MPLTEXT 1 0 81 "\n \+ lbl[4] := plots[textplot]([4,-0.1,lie_rank-1],align=\{BELOW, CENTER\}):" }{MPLTEXT 1 0 79 "\n lbl[5] := plots[textplot]( [5,-0.1,lie_rank],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 69 "\n \+ lns[1] := plottools[line]([1.1,0],[1.9,0], color=black):" } {MPLTEXT 1 0 69 "\n lns[2] := plottools[line]([2.1,0],[2.9, 0], color=black):" }{MPLTEXT 1 0 33 "\n for i from 1 to 3 d o" }{MPLTEXT 1 0 66 "\n dts[i] := plots[textplot]([3.1 + 0.2*i,0.05,\".\"]):" }{MPLTEXT 1 0 16 "\n od:" }{MPLTEXT 1 0 69 "\n lns[3] := plottools[line]([4.1,0],[4.9,0], color =black):" }{MPLTEXT 1 0 69 "\n lns[4] := plottools[line]([3 .1,0],[3.2,0], color=black):" }{MPLTEXT 1 0 69 "\n lns[5] : = plottools[line]([3.8,0],[3.9,0], color=black):" }{MPLTEXT 1 0 68 "\n plots[display](seq(nds[i],i=1..5), seq(lbl[i],i=1..5), " } {MPLTEXT 1 0 68 "\n seq(lns[i],i=1..5), seq( dts[i],i=1..3), " }{MPLTEXT 1 0 59 "\n axes= none, scaling=constrained," }{MPLTEXT 1 0 84 "\n \+ title = cat(\"Dynkin Diagram: Type \",lie_type,lie_rank));" } {MPLTEXT 1 0 14 "\n else" }{MPLTEXT 1 0 28 "\n # fu ll diagram " }{MPLTEXT 1 0 40 "\n for i from 1 to lie_rank \+ do" }{MPLTEXT 1 0 69 "\n nds[i] := plottools[circle]([i, 0], 0.1, color=black):" }{MPLTEXT 1 0 75 "\n lbl[i] := p lots[textplot]([i,-0.1,i],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 16 " \n od:" }{MPLTEXT 1 0 42 "\n for i from 1 to lie _rank-1 do" }{MPLTEXT 1 0 76 "\n lns[i] := plottools[lin e]([i+0.1,0],[i+0.9,0], color=black):" }{MPLTEXT 1 0 16 "\n \+ od:" }{MPLTEXT 1 0 82 "\n plots[display](seq(nds[i],i=1..l ie_rank), seq(lbl[i],i=1..lie_rank), " }{MPLTEXT 1 0 88 "\n \+ seq(lns[i],i=1..lie_rank-1), axes=none, scaling=constr ained," }{MPLTEXT 1 0 84 "\n title = cat(\"D ynkin Diagram: Type \",lie_type,lie_rank));" }{MPLTEXT 1 0 17 "\n \+ end if;" }{MPLTEXT 1 0 51 "\n elif (lie_type = 'B' or lie_typ e = 'C') then" }{MPLTEXT 1 0 35 "\n # Type BC Coxeter Diagram" }{MPLTEXT 1 0 33 "\n if (lie_rank > 5) then " }{MPLTEXT 1 0 34 "\n # abbreviated diagram" }{MPLTEXT 1 0 33 "\n \+ for i from 1 to 5 do" }{MPLTEXT 1 0 69 "\n nds[i] := \+ plottools[circle]([i,0], 0.1, color=black):" }{MPLTEXT 1 0 16 "\n \+ od:" }{MPLTEXT 1 0 72 "\n lbl[1] := plots[textplot]( [1,-0.1,1],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 72 "\n l bl[2] := plots[textplot]([2,-0.1,2],align=\{BELOW,CENTER\}):" } {MPLTEXT 1 0 81 "\n lbl[3] := plots[textplot]([3,-0.1,lie_r ank-2],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 81 "\n lbl[4 ] := plots[textplot]([4,-0.1,lie_rank-1],align=\{BELOW,CENTER\}):" } {MPLTEXT 1 0 79 "\n lbl[5] := plots[textplot]([5,-0.1,lie_r ank],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 69 "\n lns[1] \+ := plottools[line]([1.1,0],[1.9,0], color=black):" }{MPLTEXT 1 0 33 " \n for i from 1 to 3 do" }{MPLTEXT 1 0 66 "\n \+ dts[i] := plots[textplot]([2.1 + 0.2*i,0.05,\".\"]):" }{MPLTEXT 1 0 16 "\n od:" }{MPLTEXT 1 0 69 "\n lns[2] := plott ools[line]([3.1,0],[3.9,0], color=black):" }{MPLTEXT 1 0 76 "\n \+ lns[3] := plottools[line]([4.05,0.1], [4.95,0.1], color=black):" }{MPLTEXT 1 0 78 "\n lns[4] := plottools[line]([4.05,-0.1], [4.95,-0.1], color=black):" }{MPLTEXT 1 0 69 "\n lns[5] := plottools[line]([2.1,0],[2.2,0], color=black):" }{MPLTEXT 1 0 69 "\n \+ lns[6] := plottools[line]([2.8,0],[2.9,0], color=black):" } {MPLTEXT 1 0 76 "\n # The only difference between B & C typ e Dynkin diagrams is the" }{MPLTEXT 1 0 42 "\n # direction \+ of their arrows. " }{MPLTEXT 1 0 49 "\n if (lie_type = 'B') then " }{MPLTEXT 1 0 79 "\n arrow_top := plo ttools[line]([4.6,0],[4.35,0.15], color=black):" }{MPLTEXT 1 0 83 "\n \+ arrow_bottom := plottools[line]([4.6,0],[4.35,-0.15], co lor=black):" }{MPLTEXT 1 0 17 "\n else" }{MPLTEXT 1 0 79 " \n arrow_top := plottools[line]([4.35,0],[4.6,0.15], col or=black):" }{MPLTEXT 1 0 83 "\n arrow_bottom := plottoo ls[line]([4.35,0],[4.6,-0.15], color=black):" }{MPLTEXT 1 0 20 "\n \+ end if;" }{MPLTEXT 1 0 68 "\n plots[display](seq(nd s[i],i=1..5), seq(lbl[i],i=1..5), " }{MPLTEXT 1 0 68 "\n \+ seq(lns[i],i=1..6), seq(dts[i],i=1..3), " }{MPLTEXT 1 0 85 "\n arrow_top, arrow_bottom, axes=none, s caling=constrained, " }{MPLTEXT 1 0 87 "\n t itle = cat(\"Dynkin Diagram: Type \",lie_type,lie_rank)): " } {MPLTEXT 1 0 14 "\n else" }{MPLTEXT 1 0 28 "\n # fu ll diagram " }{MPLTEXT 1 0 40 "\n for i from 1 to lie_rank \+ do" }{MPLTEXT 1 0 69 "\n nds[i] := plottools[circle]([i, 0], 0.1, color=black):" }{MPLTEXT 1 0 75 "\n lbl[i] := p lots[textplot]([i,-0.1,i],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 16 " \n od:" }{MPLTEXT 1 0 42 "\n for i from 1 to lie _rank-2 do" }{MPLTEXT 1 0 76 "\n lns[i] := plottools[lin e]([i+0.1,0],[i+0.9,0], color=black):" }{MPLTEXT 1 0 16 "\n \+ od:" }{MPLTEXT 1 0 103 "\n lns[lie_rank-1] := plottools[li ne]([lie_rank-0.95,0.1], [lie_rank-0.05,0.1], color=black):" } {MPLTEXT 1 0 103 "\n lns[lie_rank] := plottools[line]([lie_ rank-0.95,-0.1], [lie_rank-0.05,-0.1], color=black):" }{MPLTEXT 1 0 76 "\n # The only difference between B & C type Dynkin diag rams is the" }{MPLTEXT 1 0 42 "\n # direction of their arro ws. " }{MPLTEXT 1 0 49 "\n if (lie_type = 'B') then \+ " }{MPLTEXT 1 0 97 "\n arrow_top := plottools[line]( [lie_rank-0.4,0],[lie_rank-0.65,0.15], color=black):" }{MPLTEXT 1 0 101 "\n arrow_bottom := plottools[line]([lie_rank-0.4,0] ,[lie_rank-0.65,-0.15], color=black):" }{MPLTEXT 1 0 17 "\n \+ else" }{MPLTEXT 1 0 97 "\n arrow_top := plottools[line] ([lie_rank-0.65,0],[lie_rank-0.4,0.15], color=black):" }{MPLTEXT 1 0 101 "\n arrow_bottom := plottools[line]([lie_rank-0.65,0 ],[lie_rank-0.4,-0.15], color=black):" }{MPLTEXT 1 0 20 "\n \+ end if;" }{MPLTEXT 1 0 82 "\n plots[display](seq(nds[i],i= 1..lie_rank), seq(lbl[i],i=1..lie_rank), " }{MPLTEXT 1 0 92 "\n \+ seq(lns[i],i=1..lie_rank), arrow_top, arrow_bottom , axes=none, " }{MPLTEXT 1 0 49 "\n scaling =constrained, " }{MPLTEXT 1 0 87 "\n title = cat(\"Dynkin Diagram: Type \",lie_type,lie_rank)): " }{MPLTEXT 1 0 17 "\n end if;" }{MPLTEXT 1 0 33 "\n elif (lie_type = 'D' ) then" }{MPLTEXT 1 0 33 "\n # Type D Dynkin Diagram" } {MPLTEXT 1 0 33 "\n if (lie_rank > 6) then " }{MPLTEXT 1 0 34 "\n # abbreviated diagram" }{MPLTEXT 1 0 33 "\n \+ for i from 1 to 4 do" }{MPLTEXT 1 0 69 "\n nds[i] := plo ttools[circle]([i,0], 0.1, color=black):" }{MPLTEXT 1 0 16 "\n \+ od:" }{MPLTEXT 1 0 78 "\n nds[5] := plottools[circle]([ 4+1/2,sqrt(3)/2], 0.1, color=black):" }{MPLTEXT 1 0 79 "\n \+ nds[6] := plottools[circle]([4+1/2,-sqrt(3)/2], 0.1, color=black):" } {MPLTEXT 1 0 72 "\n lbl[1] := plots[textplot]([1,-0.1,1],al ign=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 72 "\n lbl[2] := plot s[textplot]([2,-0.1,2],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 81 "\n \+ lbl[3] := plots[textplot]([3,-0.1,lie_rank-3],align=\{BELOW ,CENTER\}):" }{MPLTEXT 1 0 79 "\n lbl[4] := plots[textplot] ([4,-0.1,lie_rank-2],align=\{BELOW,LEFT\}):" }{MPLTEXT 1 0 74 "\n \+ lbl[5] := plots[textplot]([4+1/2, sqrt(3)/2+0.1,lie_rank-1], " }{MPLTEXT 1 0 70 "\n ali gn=\{ABOVE,CENTER\}):" }{MPLTEXT 1 0 73 "\n lbl[6] := plots [textplot]([4+1/2, -sqrt(3)/2-0.1,lie_rank], " }{MPLTEXT 1 0 70 "\n \+ align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 69 "\n lns[1] := plottools[line]([1.1,0],[1.9 ,0], color=black):" }{MPLTEXT 1 0 69 "\n lns[2] := plottool s[line]([3.1,0],[3.9,0], color=black):" }{MPLTEXT 1 0 70 "\n \+ lns[3] := plottools[line]([4+0.1*(1/2),0.1*(sqrt(3)/2)], " } {MPLTEXT 1 0 83 "\n [4+0.9*(1/2), 0.9*(sqrt(3)/2)], color=black):" }{MPLTEXT 1 0 71 "\n lns[4 ] := plottools[line]([4+0.1*(1/2),-0.1*(sqrt(3)/2)], " }{MPLTEXT 1 0 84 "\n [4+0.9*(1/2),-0.9*(sqrt(3) /2)], color=black):" }{MPLTEXT 1 0 33 "\n for i from 1 to 3 do" }{MPLTEXT 1 0 66 "\n dts[i] := plots[textplot]([2.1 + 0.2*i,0.05,\".\"]):" }{MPLTEXT 1 0 16 "\n od:" } {MPLTEXT 1 0 69 "\n lns[5] := plottools[line]([2.1,0],[2.2, 0], color=black):" }{MPLTEXT 1 0 69 "\n lns[6] := plottools [line]([2.8,0],[2.9,0], color=black):" }{MPLTEXT 1 0 68 "\n \+ plots[display](seq(nds[i],i=1..6), seq(lbl[i],i=1..6), " }{MPLTEXT 1 0 68 "\n seq(lns[i],i=1..6), seq(dts[i],i=1. .3), " }{MPLTEXT 1 0 60 "\n axes=none, scali ng=constrained, " }{MPLTEXT 1 0 87 "\n title = cat(\"Dynkin Diagram: Type \",lie_type,lie_rank)): " }{MPLTEXT 1 0 14 "\n else" }{MPLTEXT 1 0 28 "\n # full diagram \+ " }{MPLTEXT 1 0 42 "\n for i from 1 to lie_rank-2 do" } {MPLTEXT 1 0 69 "\n nds[i] := plottools[circle]([i,0], 0 .1, color=black):" }{MPLTEXT 1 0 75 "\n lbl[i] := plots[ textplot]([i,-0.1,i],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 16 "\n \+ od:" }{MPLTEXT 1 0 96 "\n nds[lie_rank-1] := plott ools[circle]([lie_rank-2+1/2,sqrt(3)/2], 0.1, color=black):" } {MPLTEXT 1 0 92 "\n lbl[lie_rank-1] := plots[textplot]([lie _rank-2+1/2, sqrt(3)/2+0.1,lie_rank-1], " }{MPLTEXT 1 0 70 "\n \+ align=\{ABOVE,CENTER\}):" } {MPLTEXT 1 0 95 "\n nds[lie_rank] := plottools[circle]([lie _rank-2+1/2,-sqrt(3)/2], 0.1, color=black):" }{MPLTEXT 1 0 89 "\n \+ lbl[lie_rank] := plots[textplot]([lie_rank-2+1/2, -sqrt(3)/2-0. 1,lie_rank], " }{MPLTEXT 1 0 70 "\n \+ align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 42 "\n \+ for i from 1 to lie_rank-3 do" }{MPLTEXT 1 0 76 "\n lns[ i] := plottools[line]([i+0.1,0],[i+0.9,0], color=black):" }{MPLTEXT 1 0 16 "\n od:" }{MPLTEXT 1 0 88 "\n lns[lie_rank- 2] := plottools[line]([lie_rank-2+0.1*(1/2),0.1*(sqrt(3)/2)], " } {MPLTEXT 1 0 101 "\n [li e_rank-2+0.9*(1/2),0.9*(sqrt(3)/2)], color=black):" }{MPLTEXT 1 0 89 " \n lns[lie_rank-1] := plottools[line]([lie_rank-2+0.1*(1/2) ,-0.1*(sqrt(3)/2)], " }{MPLTEXT 1 0 102 "\n \+ [lie_rank-2+0.9*(1/2),-0.9*(sqrt(3)/2)], color=bla ck):" }{MPLTEXT 1 0 82 "\n plots[display](seq(nds[i],i=1..l ie_rank), seq(lbl[i],i=1..lie_rank), " }{MPLTEXT 1 0 89 "\n \+ seq(lns[i],i=1..lie_rank-1), axes=none, scaling=constr ained, " }{MPLTEXT 1 0 87 "\n title = cat(\" Dynkin Diagram: Type \",lie_type,lie_rank)): " }{MPLTEXT 1 0 17 "\n \+ end if;" }{MPLTEXT 1 0 33 "\n elif (lie_type = 'E') then" }{MPLTEXT 1 0 33 "\n # Type E Dynkin Diagram" }{MPLTEXT 1 0 39 "\n for i from 1 to lie_rank-1 do" }{MPLTEXT 1 0 66 "\n \+ nds[i] := plottools[circle]([i,0], 0.1, color=black):" } {MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 72 "\n nds[lie_ rank] := plottools[circle]([3,1], 0.1, color=black):" }{MPLTEXT 1 0 69 "\n lbl[1] := plots[textplot]([1,-0.1,1],align=\{BELOW,CENT ER\}):" }{MPLTEXT 1 0 68 "\n lbl[2] := plots[textplot]([3,1.1, 2],align=\{ABOVE,CENTER\}):" }{MPLTEXT 1 0 37 "\n for i from 3 to lie_rank do" }{MPLTEXT 1 0 74 "\n lbl[i] := plots[textp lot]([i-1,-0.1,i],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 13 "\n \+ od:" }{MPLTEXT 1 0 39 "\n for i from 1 to lie_rank-2 do" } {MPLTEXT 1 0 73 "\n lns[i] := plottools[line]([i+0.1,0],[i+ 0.9,0], color=black):" }{MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 75 "\n lns[lie_rank-1] := plottools[line]([3,0.1],[3,0.9], c olor=black):" }{MPLTEXT 1 0 79 "\n plots[display](seq(nds[i],i =1..lie_rank), seq(lbl[i],i=1..lie_rank), " }{MPLTEXT 1 0 91 "\n \+ seq(lns[i],i=1..(lie_rank-1)), axes=none, scaling=co nstrained, " }{MPLTEXT 1 0 84 "\n title = ca t(\"Dynkin Diagram: Type \",lie_type,lie_rank)): " }{MPLTEXT 1 0 33 "\n elif (lie_type = 'F') then" }{MPLTEXT 1 0 33 "\n # Ty pe F Dynkin Diagram" }{MPLTEXT 1 0 37 "\n for i from 1 to lie_ rank do" }{MPLTEXT 1 0 66 "\n nds[i] := plottools[circle]([ i,0], 0.1, color=black):" }{MPLTEXT 1 0 72 "\n lbl[i] := pl ots[textplot]([i,-0.1,i],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 13 " \n od:" }{MPLTEXT 1 0 66 "\n lns[1] := plottools[line] ([1.1,0],[1.9,0], color=black):" }{MPLTEXT 1 0 66 "\n lns[2] : = plottools[line]([3.1,0],[3.9,0], color=black):" }{MPLTEXT 1 0 66 "\n lns[3] := plottools[line]([2,0.1],[3,0.1], color=black):" } {MPLTEXT 1 0 68 "\n lns[4] := plottools[line]([2,-0.1],[3,-0.1 ], color=black):" }{MPLTEXT 1 0 73 "\n arrow_top := plottools[ line]([2.6,0],[2.35,0.15], color=black):" }{MPLTEXT 1 0 77 "\n \+ arrow_bottom := plottools[line]([2.6,0],[2.35,-0.15], color=black):" }{MPLTEXT 1 0 105 "\n plots[display](seq(nds[i],i=1..lie_rank) , seq(lbl[i],i=1..lie_rank), seq(lns[i],i=1..lie_rank)," }{MPLTEXT 1 0 81 "\n arrow_top, arrow_bottom, axes=none, sc aling=constrained," }{MPLTEXT 1 0 84 "\n title \+ = cat(\"Dynkin Diagram: Type \",lie_type,lie_rank)): " }{MPLTEXT 1 0 33 "\n elif (lie_type = 'G') then" }{MPLTEXT 1 0 33 "\n \+ # Type G Dynkin Diagram" }{MPLTEXT 1 0 63 "\n nds[1] := plott ools[circle]([1,0], 0.1, color=black):" }{MPLTEXT 1 0 69 "\n l bl[1] := plots[textplot]([1,-0.1,1],align=\{BELOW,CENTER\}):" } {MPLTEXT 1 0 63 "\n nds[2] := plottools[circle]([2,0], 0.1, co lor=black):" }{MPLTEXT 1 0 69 "\n lbl[2] := plots[textplot]([2 ,-0.1,2],align=\{BELOW,CENTER\}):" }{MPLTEXT 1 0 67 "\n lns[1] := plottools[line]([1,0.1], [2,0.1], color=black):" }{MPLTEXT 1 0 67 "\n lns[2] := plottools[line]([1.1,0], [1.9,0], color=black):" }{MPLTEXT 1 0 69 "\n lns[3] := plottools[line]([1,-0.1], [2,- 0.1], color=black):" }{MPLTEXT 1 0 73 "\n arrow_top := plottoo ls[line]([1.4,0],[1.65,0.15], color=black):" }{MPLTEXT 1 0 77 "\n \+ arrow_bottom := plottools[line]([1.4,0],[1.65,-0.15], color=black) :" }{MPLTEXT 1 0 90 "\n plots[display](nds[1], nds[2], lns[1], lns[2], lns[3], arrow_top, arrow_bottom, " }{MPLTEXT 1 0 72 "\n \+ lbl[1], lbl[2], axes=none, scaling=constrained," } {MPLTEXT 1 0 84 "\n title = cat(\"Dynkin Diagra m: Type \",lie_type,lie_rank)): " }{MPLTEXT 1 0 14 "\n end if;" }{MPLTEXT 1 0 11 "\n end if;" }{MPLTEXT 1 0 5 "\nend:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 96 "\n# SimpleRoots returns an array of simple \+ roots. If alpha = SimpleRoots(lie_type, lie_rank) then" }{MPLTEXT 1 0 95 "\n# \{alpha[1],alpha[2],...,alpha[lie_rank]\} is a base for the ro ot system of lie_type, lie_rank." }{MPLTEXT 1 0 90 "\n# For all bases \+ (except G2) have the properties: 1) Long roots have squared length 2 w ith" }{MPLTEXT 1 0 70 "\n# respect to the standard dot product. 2) Sho rt roots have length 1. " }{MPLTEXT 1 0 93 "\n# G2's long root has squ ared length 3 with respect to the standard dot product and its short" }{MPLTEXT 1 0 21 "\n# root has length 1." }{MPLTEXT 1 0 2 "\n#" } {MPLTEXT 1 0 74 "\n# Note: These bases can be found in Fulton and Harr is, pages 324,332,333." }{MPLTEXT 1 0 48 "\n# *** I have rescale d bases of type C ***" }{MPLTEXT 1 0 40 "\nSimpleRoots := proc(lie_typ e, lie_rank)" }{MPLTEXT 1 0 18 "\n local alpha,i;" }{MPLTEXT 1 0 4 " \n " }{MPLTEXT 1 0 52 "\n if (SimpleLieTypeCheck(lie_type, lie_ran k)) then" }{MPLTEXT 1 0 31 "\n if (lie_type = 'A') then" } {MPLTEXT 1 0 37 "\n for i from 1 to lie_rank do" }{MPLTEXT 1 0 78 "\n alpha[i] := [seq(`if`(j=i,1,`if`(j=i+1,-1,0)),j=1. .lie_rank+1)]; " }{MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 33 " \n elif (lie_type = 'B') then" }{MPLTEXT 1 0 39 "\n for i from 1 to lie_rank-1 do" }{MPLTEXT 1 0 76 "\n alpha[i] := \+ [seq(`if`(j=i,1,`if`(j=i+1,-1,0)),j=1..lie_rank)]; " }{MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 72 "\n alpha[lie_rank] := [seq( `if`(j=lie_rank,1,0),j=1..lie_rank)]: " }{MPLTEXT 1 0 33 "\n elif (lie_type = 'C') then" }{MPLTEXT 1 0 39 "\n for i from 1 to l ie_rank-1 do" }{MPLTEXT 1 0 92 "\n alpha[i] := [seq(`if`(j= i,1/sqrt(2),`if`(j=i+1,-1/sqrt(2),0)),j=1..lie_rank)]; " }{MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 78 "\n alpha[lie_rank] := \+ [seq(`if`(j=lie_rank,sqrt(2),0),j=1..lie_rank)]: " }{MPLTEXT 1 0 33 " \n elif (lie_type = 'D') then" }{MPLTEXT 1 0 39 "\n for i from 1 to lie_rank-1 do" }{MPLTEXT 1 0 76 "\n alpha[i] := \+ [seq(`if`(j=i,1,`if`(j=i+1,-1,0)),j=1..lie_rank)]; " }{MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 88 "\n alpha[lie_rank] := [seq( `if`(j=lie_rank or j=lie_rank-1,1,0),j=1..lie_rank)]: " }{MPLTEXT 1 0 33 "\n elif (lie_type = 'E') then" }{MPLTEXT 1 0 79 "\n i f (lie_rank=6) then alpha[1] := [1/2,-1/2,-1/2,-1/2,-1/2,sqrt(3)/2];" }{MPLTEXT 1 0 86 "\n elif (lie_rank=7) then alpha[1] := [1/2,- 1/2,-1/2,-1/2,-1/2,-1/2,sqrt(2)/2];" }{MPLTEXT 1 0 85 "\n elif (lie_rank=8) then alpha[1] := [1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2] ;" }{MPLTEXT 1 0 17 "\n end if:" }{MPLTEXT 1 0 66 "\n \+ alpha[2] := [seq(`if`(j=1 or j=2, 1, 0),j=1..lie_rank)];" }{MPLTEXT 1 0 37 "\n for i from 3 to lie_rank do" }{MPLTEXT 1 0 78 "\n \+ alpha[i] := [seq(`if`(j=i-1,1,`if`(j=i-2,-1,0)),j=1..lie_rank) ]; " }{MPLTEXT 1 0 13 "\n od:" }{MPLTEXT 1 0 33 "\n elif \+ (lie_type = 'F') then" }{MPLTEXT 1 0 33 "\n alpha[1] := [0,1,- 1,0]:" }{MPLTEXT 1 0 33 "\n alpha[2] := [0,0,1,-1]:" } {MPLTEXT 1 0 32 "\n alpha[3] := [0,0,0,1]:" }{MPLTEXT 1 0 43 " \n alpha[4] := [1/2,-1/2,-1/2,-1/2]:" }{MPLTEXT 1 0 33 "\n \+ elif (lie_type = 'G') then" }{MPLTEXT 1 0 28 "\n alpha[1] := [1,0]:" }{MPLTEXT 1 0 39 "\n alpha[2] := [-3/2,sqrt(3)/2]:" } {MPLTEXT 1 0 14 "\n end if;" }{MPLTEXT 1 0 13 "\n alpha;" } {MPLTEXT 1 0 11 "\n end if;" }{MPLTEXT 1 0 5 "\nend:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 73 "\n# FundamentalWeights returns an array conta ining the fundamental weights" }{MPLTEXT 1 0 64 "\n# of the simple Lie algebra specified by lie_type and lie_rank." }{MPLTEXT 1 0 67 "\n# Th at is if lambda := FundamentalWeights(lie_type, lie_rank) then" } {MPLTEXT 1 0 73 "\n# \{lambda[1],lambda[2],...,lambda[lie_rank]\} are \+ the fundamental weights" }{MPLTEXT 1 0 49 "\n# for the simple Lie alge bra lie_type, lie_rank." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 90 "\n# L et u.v be the dot product of u and v. Then if alpha := SimpleRoots(lie _type, lie_rank)" }{MPLTEXT 1 0 86 "\n# we have that 2*(lambda[i].alph a[j])/(alpha[j].alpha[j]) = 1 if i=j and 0 otherwise." }{MPLTEXT 1 0 49 "\nFundamentalWeights := proc (lie_type, lie_rank) " }{MPLTEXT 1 0 34 "\n local Inv_CM,i,j,alpha,lambda;" }{MPLTEXT 1 0 4 "\n " } {MPLTEXT 1 0 51 "\n if (SimpleLieTypeCheck(lie_type,lie_rank)) then" }{MPLTEXT 1 0 67 "\n Inv_CM := linalg[inverse](CartanMatrix(lie_ type, lie_rank));" }{MPLTEXT 1 0 48 "\n alpha := SimpleRoots(lie_ type, lie_rank);" }{MPLTEXT 1 0 34 "\n for i from 1 to lie_rank d o" }{MPLTEXT 1 0 44 "\n lambda[i] := Inv_CM[i,1]*alpha[1];" } {MPLTEXT 1 0 37 "\n for j from 2 to lie_rank do" }{MPLTEXT 1 0 59 "\n lambda[i] := lambda[i] + Inv_CM[i,j]*alpha[j];" } {MPLTEXT 1 0 13 "\n od;" }{MPLTEXT 1 0 10 "\n od:" } {MPLTEXT 1 0 14 "\n lambda;" }{MPLTEXT 1 0 11 "\n end if;" } {MPLTEXT 1 0 5 "\nend:" }}}}{SECT 1 {PARA 200 "" 0 "" {TEXT -1 28 "Wey l Group Action on Weights" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 71 "Re flection -- reflects lambda across the hyperplane orthogonal to alpha" }{TEXT 200 86 "\nMinusculeWeights -- returns the minuscule weights of the specified simple Lie algebra" }}{PARA 201 "" 0 "" {TEXT 200 118 " WeylOrbit -- computes the orbit (with respect to the Weyl group of the specified simple Lie algebra) of a given weight" }}{PARA 201 "" 0 "" {TEXT 200 118 "WeylGroupAction -- takes an orbit of weights and assign s a labels to each weight. The simple reflections permute these" }} {PARA 201 "" 0 "" {TEXT 200 120 " weight s. WeylGroupAction returns the permutations associated with the simple reflections." }}{PARA 201 "" 0 "" {TEXT 200 113 "CycleStructure -- re turns the cycle structures present in the subgroup generated by the gi ven set of permutations" }}{PARA 201 "" 0 "" {TEXT 200 112 "CycleStruc tureRandom -- Since CycleStructure is too inefficient for larger examp les, we provide a random version" }}{PARA 201 "" 0 "" {TEXT 200 111 " \+ which collects cycle structures \+ for N iterations (the default is 10000)." }}{PARA 201 "" 0 "" {TEXT 200 110 "RandElementConj -- Given a cycle structure and a generating s et of permutations, RandElementConj tries N times" }}{PARA 201 "" 0 "" {TEXT 200 124 " (the default is 10000) \+ to find an element with that cycle structure. \+ " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 70 "# Reflection reflects lambda across the hyperplane which is orthoginal" }{MPLTEXT 1 0 23 " \n# to the vector alpha." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 51 "\n# \+ If alpha := SimpleRoots(lie_type, lie_rank) then" }{MPLTEXT 1 0 90 "\n # \{Reflection(alpha[1],__), Reflection(alpha[2],__), ..., Reflection( alpha[lie_rank],__)\}" }{MPLTEXT 1 0 77 "\n# are the simple reflection s of the simple Lie algebra of lie_type,lie_rank." }{MPLTEXT 1 0 46 " \n# These reflections generate the Weyl group. " }{MPLTEXT 1 0 36 "\nR eflection := proc(alpha, lambda) " }{MPLTEXT 1 0 14 "\n local tmp;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "\n if (nops(alpha) = nops(la mbda)) then" }{MPLTEXT 1 0 87 "\n tmp := 2*ListTools[DotProduct]( lambda,alpha)/ListTools[DotProduct](alpha,alpha);" }{MPLTEXT 1 0 51 " \n tmp := [seq(tmp*alpha[j],j=1..nops(alpha))];" }{MPLTEXT 1 0 20 "\n lambda - tmp;" }{MPLTEXT 1 0 8 "\n else" }{MPLTEXT 1 0 60 "\n WARNING(\"Cannot reflect: list lengths don't match.\");" } {MPLTEXT 1 0 11 "\n end if;" }{MPLTEXT 1 0 6 "\nend: " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 63 "\n# MinusculeWeights returns the minuscule \+ weights of the simple" }{MPLTEXT 1 0 36 "\n# Lie algebra of lie_type,l ie_rank." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 87 "\n# A list of minusc ule weights can be found in Humpreys, page 72 (minimal = minuscule)." }{MPLTEXT 1 0 44 "\nMinusculeWeights := proc(lie_type,lie_rank)" } {MPLTEXT 1 0 17 "\n local lambda;" }{MPLTEXT 1 0 51 "\n if (Simple LieTypeCheck(lie_type,lie_rank)) then" }{MPLTEXT 1 0 57 "\n lambd a := FundamentalWeights(lie_type, lie_rank); " }{MPLTEXT 1 0 31 "\n \+ if (lie_type = 'A') then" }{MPLTEXT 1 0 42 "\n \{seq(lambda [i],i=1..lie_rank)\}; " }{MPLTEXT 1 0 33 "\n elif (lie_type = 'B' ) then" }{MPLTEXT 1 0 30 "\n \{lambda[lie_rank]\}; " } {MPLTEXT 1 0 33 "\n elif (lie_type = 'C') then" }{MPLTEXT 1 0 23 "\n \{lambda[1]\}; " }{MPLTEXT 1 0 33 "\n elif (lie_type \+ = 'D') then" }{MPLTEXT 1 0 61 "\n \{lambda[1], lambda[lie_rank -1], lambda[lie_rank]\}; " }{MPLTEXT 1 0 33 "\n elif (lie_type = \+ 'E') then" }{MPLTEXT 1 0 57 "\n if (lie_rank = 6) then \{lambd a[1], lambda[6]\}; " }{MPLTEXT 1 0 47 "\n elif (lie_rank = 7) \+ then \{lambda[7]\};" }{MPLTEXT 1 0 18 "\n else \{\};" } {MPLTEXT 1 0 17 "\n end if;" }{MPLTEXT 1 0 33 "\n elif (l ie_type = 'F') then" }{MPLTEXT 1 0 13 "\n \{\};" }{MPLTEXT 1 0 33 "\n elif (lie_type = 'G') then" }{MPLTEXT 1 0 13 "\n \+ \{\};" }{MPLTEXT 1 0 14 "\n end if;" }{MPLTEXT 1 0 11 "\n end \+ if:" }{MPLTEXT 1 0 5 "\nend:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 82 " \n# WeylOrbit returns the set of all weights in the orbit (with respec t to the Weyl" }{MPLTEXT 1 0 72 "\n# group of the simple Lie algebra l ie_type, lie_rank) of the weight wt." }{MPLTEXT 1 0 42 "\nWeylOrbit := proc(wt, lie_type, lie_rank)" }{MPLTEXT 1 0 46 "\n local chi,alpha, i,wt_stack,next_wt,new_wt;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "\n \+ if (SimpleLieTypeCheck(lie_type, lie_rank)) then " }{MPLTEXT 1 0 48 "\n alpha := SimpleRoots(lie_type, lie_rank);" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 42 "\n if (nops(wt) = nops(alpha[1])) then" } {MPLTEXT 1 0 35 "\n # wt is in its own orbit." }{MPLTEXT 1 0 22 "\n chi := \{wt\};" }{MPLTEXT 1 0 90 "\n # wt_stack keeps track of the weights which we need to apply simple reflections. " }{MPLTEXT 1 0 37 "\n wt_stack := stack[new](wt);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "\n while (not stack[empty](wt_sta ck)) do" }{MPLTEXT 1 0 45 "\n next_wt := stack[pop](wt_stac k);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "\n for i from 1 to lie_rank do" }{MPLTEXT 1 0 58 "\n new_wt := Reflecti on(alpha[i],next_wt); " }{MPLTEXT 1 0 46 "\n if (not ( new_wt in chi)) then " }{MPLTEXT 1 0 100 "\n # if new _wt is indeed new, add it to the set of weight and push it on the stac k. " }{MPLTEXT 1 0 45 "\n chi := chi union \{new_wt\} ;" }{MPLTEXT 1 0 49 "\n stack[push](new_wt, wt_stack) ;" }{MPLTEXT 1 0 23 "\n end if;" }{MPLTEXT 1 0 16 "\n \+ od;" }{MPLTEXT 1 0 13 "\n od;" }{MPLTEXT 1 0 36 "\n \+ chi := simplify(chi); " }{MPLTEXT 1 0 11 "\n else" } {MPLTEXT 1 0 63 "\n WARNING(\"Cannot reflect: list lengths don 't match.\");" }{MPLTEXT 1 0 14 "\n end if;" }{MPLTEXT 1 0 13 "\n end if; " }{MPLTEXT 1 0 5 "\nend:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 74 "\n# *** This procedure is used to impose a definite order on t he weights***" }{MPLTEXT 1 0 77 "\n# Without sorting the weights WeylG roupAction returns different permutations" }{MPLTEXT 1 0 79 "\n# each \+ time it is run (because the unordered set chi is turned into an ordere d" }{MPLTEXT 1 0 79 "\n# list chi_list). Without sorting chi_list is o rdered \"randomly\" from the set." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 72 "\n# listlex takes in two lists of equal length and returns true \+ or false." }{MPLTEXT 1 0 2 "\n#" }{MPLTEXT 1 0 71 "\n# listlex compare s two lists a & b and determines if a comes before b " }{MPLTEXT 1 0 64 "\n# in \"list lexicographic order\" (listlex returns true if a=b). " }{MPLTEXT 1 0 70 "\n# That is \"a\" \"listlex less than\" \"b\" iff at the first index where a " }{MPLTEXT 1 0 50 "\n# and b differ, a's \+ entry is less than b's entry." }{MPLTEXT 1 0 21 "\nlistlex := proc(a,b )" }{MPLTEXT 1 0 12 "\n local i;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 40 "\n # check that the list lengths match." }{MPLTEXT 1 0 72 "\n \+ if nops(a) <> nops(b) then error \"list sizes do not match.\"; end i f;" }{MPLTEXT 1 0 7 "\n " }{MPLTEXT 1 0 16 "\n if a=b then " } {MPLTEXT 1 0 67 "\n true; # if a=b be then a comes before b in li stlex ordering." }{MPLTEXT 1 0 8 "\n else" }{MPLTEXT 1 0 67 "\n \+ # find the first index where a and b differ (we already know" } {MPLTEXT 1 0 35 "\n # that a is not equal to b)." }{MPLTEXT 1 0 15 "\n i := 1; " }{MPLTEXT 1 0 38 "\n while(a[i]=b[i]) do i: =i+1; od;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 68 "\n # if a's ent ry is less than b's entry return true else false." }{MPLTEXT 1 0 72 " \n if (evalf(a[i]) < evalf(b[i])) then true; else false; end if; \+ " }{MPLTEXT 1 0 11 "\n end if;" }{MPLTEXT 1 0 10 "\nend proc:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 80 "\n# WeylGroupAction takes the or bit (chi) of some weight, assigns labels to each " }{MPLTEXT 1 0 79 " \n# weight (1,2,...,#(chi)), and represents the action of each simple \+ reflection" }{MPLTEXT 1 0 36 "\n# as a permutation of the weights. " } {MPLTEXT 1 0 49 "\nWeylGroupAction := proc(chi, lie_type, lie_rank)" } {MPLTEXT 1 0 40 "\n local alpha,chi_list,p,i,j,k,s_perm;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 52 "\n if (SimpleLieTypeCheck(lie_type, lie _rank)) then" }{MPLTEXT 1 0 48 "\n alpha := SimpleRoots(lie_type, lie_rank);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 76 "\n # Assign l abels -- convert the set of weights into a list of weights." } {MPLTEXT 1 0 75 "\n # impose the \"listlex\" ordering so that we \+ get the same permutations" }{MPLTEXT 1 0 45 "\n # even if this pr ocedure is restarted." }{MPLTEXT 1 0 45 "\n chi_list := sort([op( chi)], listlex); " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 50 "\n if(n ops(chi_list[1]) = nops(alpha[1])) then" }{MPLTEXT 1 0 46 "\n \+ # Define functions p(i,_) such that:" }{MPLTEXT 1 0 75 "\n # p (i,j) = k iff Reflection(alpha[i],chi_list[j]) = chi_list[k]" } {MPLTEXT 1 0 43 "\n for j from 1 to nops(chi_list) do" } {MPLTEXT 1 0 40 "\n for i from 1 to lie_rank do" }{MPLTEXT 1 0 49 "\n for k from 1 to nops(chi_list) do" }{MPLTEXT 1 0 93 "\n if (chi_list[k]=Reflection(alpha[i],chi_li st[j])) then p(i,j) := k end if;" }{MPLTEXT 1 0 19 "\n o d;" }{MPLTEXT 1 0 16 "\n od;" }{MPLTEXT 1 0 13 "\n \+ od;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "\n # Convert the p -functions into permutations." }{MPLTEXT 1 0 37 "\n for i from 1 to lie_rank do" }{MPLTEXT 1 0 79 "\n s_perm[i] := conver t([seq(p(i,j),j=1..nops(chi_list))],'disjcyc');" }{MPLTEXT 1 0 13 "\n \+ od;" }{MPLTEXT 1 0 17 "\n s_perm;" }{MPLTEXT 1 0 11 " \n else" }{MPLTEXT 1 0 63 "\n WARNING(\"Cannot reflect: l ist lengths don't match.\");" }{MPLTEXT 1 0 14 "\n end if;" } {MPLTEXT 1 0 11 "\n end if:" }{MPLTEXT 1 0 5 "\nend:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 86 "\n# CycleStructure gives the cycle structures of the elements of the subgroup generated" }{MPLTEXT 1 0 25 "\n# by t he set \"set_gen\". " }{MPLTEXT 1 0 77 "\n# Note: the 1-cycles are sup pressed i.e. [[1,2],[3,4,5],[6],[7]] ==> [3,2]. " }{MPLTEXT 1 0 32 "\n CycleStructure := proc(set_gen)" }{MPLTEXT 1 0 29 "\n local els,conj _classes,g;" }{MPLTEXT 1 0 29 "\n els := elements(set_gen);" } {MPLTEXT 1 0 23 "\n conj_classes := \{\};" }{MPLTEXT 1 0 19 "\n fo r g in els do" }{MPLTEXT 1 0 85 "\n conj_classes := conj_classes \+ union \{sort([seq(nops(g[i]),i=1..nops(g))],`>`)\};" }{MPLTEXT 1 0 7 " \n od;" }{MPLTEXT 1 0 17 "\n conj_classes;" }{MPLTEXT 1 0 5 "\nend :" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 90 "\n# CycleStructureRandom giv es the cycle structures of some random elements of the subgroup" } {MPLTEXT 1 0 86 "\n# generated by the set \"set_gen\". This procedure \+ is intended for \"large\" Weyl groups." }{MPLTEXT 1 0 77 "\n# Note: th e 1-cycles are suppressed i.e. [[1,2],[3,4,5],[6],[7]] ==> [3,2]. " } {MPLTEXT 1 0 38 "\nCycleStructureRandom := proc(set_gen)" }{MPLTEXT 1 0 37 "\n local i,j,max,conj_classes,g,G,N;" }{MPLTEXT 1 0 61 "\n N := 10000; # N = the number of random elements to check." }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "\n # find the maximum number occuring the generators." }{MPLTEXT 1 0 13 "\n max := 1;" }{MPLTEXT 1 0 23 "\n \+ for g in set_gen do" }{MPLTEXT 1 0 33 "\n for i from 1 to nops(g ) do" }{MPLTEXT 1 0 39 "\n for j from 1 to nops(g[i]) do" } {MPLTEXT 1 0 58 "\n if max < g[i][j] then max := g[i][j]; e nd if;" }{MPLTEXT 1 0 13 "\n od;" }{MPLTEXT 1 0 10 "\n od ;" }{MPLTEXT 1 0 7 "\n od;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 64 " \n # set_gen generates a subgroup of the symmetric group S_max." } {MPLTEXT 1 0 32 "\n G := permgroup(max,set_gen);" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 23 "\n conj_classes := \{\};" }{MPLTEXT 1 0 24 "\n for i from 1 to N do" }{MPLTEXT 1 0 27 "\n g := RandElement(G) ;" }{MPLTEXT 1 0 85 "\n conj_classes := conj_classes union \{sort ([seq(nops(g[i]),i=1..nops(g))],`>`)\};" }{MPLTEXT 1 0 7 "\n od;" } {MPLTEXT 1 0 17 "\n conj_classes;" }{MPLTEXT 1 0 5 "\nend:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 72 "\n# Given [conj_class,set_gen] = [a conjugacy class,a set of generators]," }{MPLTEXT 1 0 72 "\n# RandE lementConj tries N random elements from the subgroup G generated" } {MPLTEXT 1 0 77 "\n# by set_gen. RandElementConj returns false if no e lement from the conjugacy" }{MPLTEXT 1 0 70 "\n# class conj_class is f ound. Otherwise such an element is returned. " }{MPLTEXT 1 0 30 "\nRa ndElementConj := proc(inpt)" }{MPLTEXT 1 0 48 "\n local conj_class,s et_gen,N,max,G,i,j,g,flag;" }{MPLTEXT 1 0 26 "\n conj_class := inpt[ 1];" }{MPLTEXT 1 0 23 "\n set_gen := inpt[2];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 "\n N := 10000; # N = the number of random elements to check." }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "\n # find the max imum number occuring the generators." }{MPLTEXT 1 0 13 "\n max := 1; " }{MPLTEXT 1 0 23 "\n for g in set_gen do" }{MPLTEXT 1 0 33 "\n \+ for i from 1 to nops(g) do" }{MPLTEXT 1 0 39 "\n for j from \+ 1 to nops(g[i]) do" }{MPLTEXT 1 0 58 "\n if max < g[i][j] t hen max := g[i][j]; end if;" }{MPLTEXT 1 0 13 "\n od;" } {MPLTEXT 1 0 10 "\n od;" }{MPLTEXT 1 0 7 "\n od;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 64 "\n # set_gen generates a subgroup of the \+ symmetric group S_max." }{MPLTEXT 1 0 32 "\n G := permgroup(max,set_ gen);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "\n flag := true;" } {MPLTEXT 1 0 11 "\n i := 1;" }{MPLTEXT 1 0 27 "\n while flag and i < N do" }{MPLTEXT 1 0 27 "\n g := RandElement(G);" }{MPLTEXT 1 0 91 "\n if conj_class = sort([seq(nops(g[i]),i=1..nops(g))],`>`) then flag := false; end if;" }{MPLTEXT 1 0 16 "\n i := i+1;" } {MPLTEXT 1 0 7 "\n od;" }{MPLTEXT 1 0 4 "\n " }{MPLTEXT 1 0 39 "\n if flag then false; else g; end if;" }{MPLTEXT 1 0 5 "\nend:" }}}} {SECT 0 {PARA 200 "" 0 "" {TEXT -1 8 "Examples" }}{SECT 0 {PARA 203 "" 0 "" {TEXT -1 6 "Type A" }}{SECT 0 {PARA 204 "" 0 "" {TEXT -1 15 "A4: Master Demo" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 115 "Let us take a \+ look at A4 (that is sl(5)). Here we will demonstrate how to use each o f the procedures defined above." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 " " {TEXT 200 50 "To begin, let's print out the Cartan matrix of A4." }} }{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('A',4) );" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "Now A4's Dynkin diagram. " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('A',4); " }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 88 "Simple roots for A4 (their order corresponds to the labels of the Dynkin diagram above)." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 28 "alpha := SimpleRoots('A',4 ):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 22 "\nseq(alph a[i],i=1..4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 4 "The " }{TEXT 201 19 "fundamental weights" }{TEXT 200 83 " are weights \"dual\" to t he simple roots. That is, if \{alpha_i\} are the simple roots" }} {PARA 201 "" 0 "" {TEXT 200 107 "and \{lambda_j\} are the fundamental \+ weights, then 2*(lambda_j | alpha_i)/(alpha_i | alpha_i) is 1 if i=j a nd" }}{PARA 201 "" 0 "" {TEXT 200 103 "0 otherwise. One can easily obt ain the fundamental weights of a simple Lie algebra by using the inver se" }}{PARA 201 "" 0 "" {TEXT 200 21 "of the Cartan matrix." }{TEXT 200 1 "\n" }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 33 ": (al pha | beta) = inner product " }{TEXT 200 23 "(standard dot product) " }{TEXT 200 19 "of alpha and beta. " }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of A4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeights('A',4):" } {MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\nseq(lambda[i] ,i=1..4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 95 "Let V(lambda) be \+ the irreducible representation with highest weight lambda. \"lambda\" \+ is called " }{TEXT 201 9 "minuscule" }{TEXT 200 10 " if every " }} {PARA 201 "" 0 "" {TEXT 200 35 "weight of V(lambda) is conjugate to" } {TEXT 200 8 " lambda " }{TEXT 200 68 "under the Weyl group action. So \+ the weights of V(lambda) all lie in " }}{PARA 201 "" 0 "" {TEXT 200 36 "one orbit when lambda is minuscule. " }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 122 "If lambda is minuscule, it can be shown tha t each weight space of V(lambda) is one dimensional. So, for minuscule weights," }}{PARA 201 "" 0 "" {TEXT 200 119 "we can find all the weig hts of V(lambda) by acting on lambda with the Weyl group. Moreover, on ce we've found all of the" }{TEXT 200 92 "\nweights, we know that the \+ dimension of V(lambda) is the cardinality of the set of weights. " }} {PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 102 "The minuscule weig hts of A4 (all fundamental weights are minuscule for simple Lie algebr as of type A)." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "Minuscu leWeights('A',4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The weig hts in the orbit of lambda[1] (A4's 1st fundamental weight). Since lam bda[1] is minuscule, the weights all lie in" }}{PARA 201 "" 0 "" {TEXT 200 67 "one orbit, so this will give us all of the weights of V( lambda[1])." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 117 "The Weyl group acts on weights by reflection. If s_1, ..., s_4 are the si mple reflections associated with the simple " }}{PARA 201 "" 0 "" {TEXT 200 64 "roots alpha_1, ..., alpha_4, then s_2 acting on lambda[1 ] is... " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 31 "Reflection(al pha[2],lambda[1]);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "s_4(s_1( lambda[1]) is..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 52 "Refle ction(alpha[4],Reflection(alpha[1],lambda[1]));" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 200 48 "Now let's compute the entire orbit of lambda[1]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambd a[1],'A',4):" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (a fter sorting them)." }{MPLTEXT 1 0 48 "\n# [op(chi)] converts the set \+ \"chi\" into a list." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }} }{EXCHG {PARA 201 "" 0 "" {TEXT 200 52 "Now we know that the dimension of V(lambda[1]) is..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 33 "# the cardinality of the set chi." }{MPLTEXT 1 0 11 "\nnops(chi);" }} }{EXCHG {PARA 201 "" 0 "" {TEXT 200 116 "We know that the Weyl group p ermutes the weights. So we label each weight with a positive integer, \+ and then see what" }}{PARA 201 "" 0 "" {TEXT 200 29 "kinds of permutat ions we get." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 4 "Note " }{TEXT 200 119 ": The labels are assigned using the ordering of the \+ list above (the second weight in the listlex sorted list of weights" } }{PARA 201 "" 0 "" {TEXT 200 29 " gets the label \"2\")." }} {PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 125 "Since simple refle ctions generate the Weyl group, WeylGroupAction just returns the simpl e reflections viewed as permutations " }}{PARA 201 "" 0 "" {TEXT 200 12 "of weights. " }{TEXT 200 45 "The Weyl group action viewed as permu tations:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGrou pAction(chi,'A',4):" }{MPLTEXT 1 0 51 "\n# print out the simple reflec tions s[1], s[2], ..." }{MPLTEXT 1 0 31 "\nfor i from 1 to 4 do s[i]; \+ od;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 109 "So the subgroup of the symmetric group S_4 generated by \{s[i], i=1..4\} is isomorphic to th e Weyl group of A4." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 53 "The cycle structures present in this subgroup of S_4:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "CycleStructure(\{seq(s[i],i=1..4) \});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 203 9 "WARNING: " }{TEXT 200 96 "These are partitions of 5 (the dimension of V(lambda)), but the 1' s are suppressed. For example," }}{PARA 201 "" 0 "" {TEXT 200 59 " \+ the partition 2+2+1 is shown as [2,2]." }}{PARA 201 " " 0 "" }{PARA 201 "" 0 "" {TEXT 200 122 "Looking at the cycle structur es, we notice the partition [5]. This tells us that, given any weight \+ vector, we can generate" }}{PARA 201 "" 0 "" {TEXT 200 117 "the whole \+ representation using that vector and an element of the Weyl group with cycle structure [5]. In other words," }}{PARA 201 "" 0 "" {TEXT 200 72 "sometimes cycle structures can give us information about irreducib ility." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 78 "This part icular subgroup is fairly small. Let's print out all of its elements." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 29 "elements(\{seq(s[i],i= 1..4)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 117 "Sometimes we wis h to examine large subgroups of S_n. In those cases, CycleStructure is too computationally expensive." }}{PARA 201 "" 0 "" {TEXT 200 124 "Bu t it is still useful to examine a sampling of some of the cycle struct ures present in the subgroup. To do this, we use the" }}{PARA 201 "" 0 "" {TEXT 200 96 "procedure CycleStructureRandom. This procedure will \+ be particularly useful when dealing with E7." }}{PARA 201 "" 0 "" } {PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 125 ": By default Cycl eStructureRandom checks 10,000 elements at random. You will notice tha t in this particular case it is faster" }}{PARA 201 "" 0 "" {TEXT 200 32 " to run CycleStructure. " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 41 "CycleStructureRandom(\{seq(s[i],i=1..4)\});" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 125 "If there is a particular cycle structure we are interested in, we can ask RandElementConj to try to \+ find an element with that" }}{PARA 201 "" 0 "" {TEXT 200 133 "structur e for us. Although, if it fails, we can't be sure that there isn't an \+ element with that structure. We might have been unlucky" }}{PARA 201 " " 0 "" {TEXT 200 126 "because the conjugacy class was too small. For e xample, if we try to find something with cycle structure [] (only the \+ identity" }}{PARA 201 "" 0 "" {TEXT 200 76 "has this structure), we wi ll almost certainly fail if the subgroup is large." }}{PARA 201 "" 0 " " }{PARA 201 "" 0 "" {TEXT 200 83 "Let us find an element with a 3-cyc le and a transposition -- cycle structure [3,2]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 44 "RandElementConj([[3,2],\{seq(s[i],i=1..4)\} ]);" }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 8 "A3 (=D3)" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 84 "Let us take a look at sl(4) (that is A 3) which is isomorphic to so(6) (that is D3). " }}{PARA 201 "" 0 "" } {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of A3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('A',3));" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of D3." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('D',3)) ;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 22 "Their Dynkin diagrams." } }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('A',3);" } {MPLTEXT 1 0 22 "\nDynkinDiagram('D',3);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of A3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 39 "lambda[1] := FundamentalWeights('A',3):" } {MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 26 "\nseq(lambda[1] [i],i=1..3);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamenta l weights of D3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 39 "lambd a[2] := FundamentalWeights('D',3):" }{MPLTEXT 1 0 17 "\n# print them o ut" }{MPLTEXT 1 0 26 "\nseq(lambda[2][i],i=1..3);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule weights of A3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('A',3);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule weights of D3." } }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('D',3); " }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 71 "The weights in the orbit o f lambda[1][1] (A3's 1st fundamental weight)." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 106 ": Since lambda[1][1] is minuscule, \+ this gives us all of the weights of the irreducible representation wit h" }}{PARA 201 "" 0 "" {TEXT 200 37 " highest weight lambda[1] [1]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 37 "chi := WeylOrbit( lambda[1][1],'A',3):" }{MPLTEXT 1 0 56 "\n# print out the *list* of we ights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listle x);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action \+ viewed as permutations. (For example: 2 corresponds to the second weig ht in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "s[1] := WeylGroupAction(chi,'A',3):" }{MPLTEXT 1 0 17 "\n# pri nt them out" }{MPLTEXT 1 0 21 "\nfor i from 1 to 3 do" }{MPLTEXT 1 0 12 "\n s[1][i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 71 "The weights in the orbit of lambda[2][2] (D3's 2nd fund amental weight)." }{TEXT 202 5 "\nNote" }{TEXT 200 89 ": Looking at th e Dynkin diagrams, we see that node 1 in A3 is equivalent to node 2 in D3." }}{PARA 201 "" 0 "" {TEXT 200 81 " Again, lambda[2][2] i s minuscule so we get all the weights of the irrep." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 37 "chi := WeylOrbit(lambda[2][2],'D',3):" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (after sorting t hem)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permutation s. (for example: 2 corresponds to the second weight in the above sorte d list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "s[2] := WeylG roupAction(chi,'D',3):" }{MPLTEXT 1 0 17 "\n# print them out" } {MPLTEXT 1 0 21 "\nfor i from 1 to 3 do" }{MPLTEXT 1 0 12 "\n s[2][i ];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 58 " Notice that the actions are identical - as they should be." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 100 "The cycle structures pre sent in the subgroup of the symmetric group generated by these permuta tions." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 38 "CycleStructure( \{seq(s[1][i],i=1..3)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 " We get the cycle structure [4] (a 4-cycle). This cycle structure only \+ allows for invariant subspaces of dimensions 0 & 4." }}{PARA 201 "" 0 "" {TEXT 200 63 "Thus the cycle structure (in this case) implies irred ucibility." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 70 "Since this group is fairly small, let's print out all of its elements." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "elements(\{seq(s[1][i],i=1 ..3)\});" }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "An" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 123 "In this example, you can adjust the f ollowing parameters as needed to experiment with different A type simp le Lie algebras." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 54 "This will explore the An representation: V(lambda[k])." }{TEXT 200 75 "\n\"rank = n\" and \"highest weight = lambda[k] = the k-th fundame ntal weight.\"" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 15 "n := 5; k := 1;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Cartan matrix \+ for An." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMa trix('A',n));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 26 "The Dynkin di agram for An." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDi agram('A',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 31 "The fundament al weights for An." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lam bda := FundamentalWeights('A',n):" }{MPLTEXT 1 0 17 "\n# print them ou t" }{MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 29 "The minuscule weights for An." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('A',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[k]." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[k] ,'A',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 41 "The orbit of lambd a[k] has cardinality..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 73 "The Weyl group action (on the o rbit of lambda[k]) viewed as permutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,'A',n):" }{MPLTEXT 1 0 21 "\nfor i from 1 to n do" }{MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 33 "A sample of the \+ cycle structures." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 41 "Cycl eStructureRandom(\{seq(s[i],i=1..n)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 138 "Let's find an element with cycle structure [2,2] (two t ranspositions) if we can. This may be impossible depending on your cho ice of n & k." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 44 "RandElem entConj([[2,2],\{seq(s[i],i=1..n)\}]);" }}}}}{SECT 1 {PARA 203 "" 0 "" {TEXT -1 6 "Type B" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 114 "Type B \+ presents some interesting problems. By examining cycle structures, we \+ will \"see\" the irreducibility of the " }{TEXT 200 110 "\nminuscule r epresentations associated with types B2, B3, B5, and B7. However, we w ill also see that the cycle " }}{PARA 201 "" 0 "" {TEXT 200 51 "struct ures do not tell us enough in the case of B4." }}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 8 "B2 (=C2)" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 84 "L et us take a look at so(5) (that is B2) which is isomorphic to sp(4) ( that is C2). " }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 24 "Th e Cartan matrix of B2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('B',2));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of C2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('C',2));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 22 "Their Dynkin diagrams." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('B',2);" }{MPLTEXT 1 0 22 "\nDynkinDiag ram('C',2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of B2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 39 "lambda [1] := FundamentalWeights('B',2):" }{MPLTEXT 1 0 17 "\n# print them ou t" }{MPLTEXT 1 0 26 "\nseq(lambda[1][i],i=1..2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of C2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 39 "lambda[2] := FundamentalWeights('C',2): " }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 26 "\nseq(lambda [2][i],i=1..2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscu le weights of B2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "Minu sculeWeights('B',2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The mi nuscule weights of C2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('C',2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 71 "T he weights in the orbit of lambda[1][2] (B2's 2nd fundamental weight). " }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 106 ": Since lambd a[1][2] is minuscule, this gives us all of the weights of the irreduci ble representation with" }}{PARA 201 "" 0 "" {TEXT 200 37 " hi ghest weight lambda[1][2]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 37 "chi := WeylOrbit(lambda[1][2],'B',2):" }{MPLTEXT 1 0 56 "\n# pri nt out the *list* of weights (after sorting them)." }{MPLTEXT 1 0 25 " \nsort([op(chi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 107 "The Weyl group action viewed as permutations. (5 corresponds to the f ifth weight in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "s[1] := WeylGroupAction(chi,'B',2):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 21 "\nfor i from 1 to 2 do" } {MPLTEXT 1 0 12 "\n s[1][i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 71 "The weights in the orbit of lambda[2][ 1] (C2's 1st fundamental weight)." }}{PARA 201 "" 0 "" {TEXT 202 4 "No te" }{TEXT 200 89 ": Looking at the Dynkin diagrams, we see that node \+ 2 in B2 is equivalent to node 1 in C2." }}{PARA 201 "" 0 "" {TEXT 200 81 " Again, lambda[2][1] is minuscule so we get all the weight s of the irrep." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 37 "chi := WeylOrbit(lambda[2][1],'C',2):" }{MPLTEXT 1 0 56 "\n# print out the * list* of weights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(c hi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl gr oup action viewed as permutations. (For example: 2 corresponds to the \+ second weight in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 " " {MPLTEXT 1 0 35 "s[2] := WeylGroupAction(chi,'C',2):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 21 "\nfor i from 1 to 2 do" } {MPLTEXT 1 0 12 "\n s[2][i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 58 "Notice that the actions are identical \+ - as they should be." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 100 "The cycle structures present in the subgroup of the symmetric group generated by these permutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 38 "CycleStructure(\{seq(s[1][i],i=1..2)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 95 "The cycle structure [4] allows invaria nt subspaces of dimensions 0 and 4 -- trivial subspaces. " }}{PARA 201 "" 0 "" {TEXT 200 74 "Thus the cycle structure tells us that this \+ representation is irreducible." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 70 "Since this group is fairly small, let's print out all o f its elements." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "elemen ts(\{seq(s[1][i],i=1..2)\});" }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "B3" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of B 3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix( 'B',3));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of B3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram( 'B',3);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental wei ghts of B3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := \+ FundamentalWeights('B',3):" }{MPLTEXT 1 0 17 "\n# print them out" } {MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..3);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule weights of B3." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('B',3);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[3]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 108 ": Since lambda[3] is mi nuscule, this gives us all of the weights of the irrep with highest we ight lambda[3]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[3],'B',3):" }{MPLTEXT 1 0 56 "\n# print out the *lis t* of weights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi) ],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permutations. (For example: 2 corresponds to the sec ond weight in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,'B',3):" }{MPLTEXT 1 0 17 " \n# print them out" }{MPLTEXT 1 0 21 "\nfor i from 1 to 3 do" } {MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let's multiply all of the simple reflections together to find the Coxeter element." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply all of the generators together." } {MPLTEXT 1 0 13 "\ntmp := s[1]:" }{MPLTEXT 1 0 21 "\nfor i from 2 to 3 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i]):" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprint(tmp);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "The cycle structures present in the subgroup of the sy mmetric group generated by these permutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "CycleStructure(\{seq(s[i],i=1..3)\});" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 88 "The cycle structure [4,4] allow s for possible invariant subspaces of dimensions 0, 4, 8." }}{PARA 201 "" 0 "" {TEXT 200 91 "The cycle structure [6,2] allows for possibl e invariant subspaces of dimensions 0, 2, 6, 8." }}{PARA 201 "" 0 "" {TEXT 200 113 "Thus, these cycle structures together allow for only 0 \+ and 8 dimensional invariant subspaces (trivial subspaces)." }}{PARA 201 "" 0 "" {TEXT 200 54 "So again, the cycle structure gives us irred ucibility." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 129 "Let' s find an element which is the product of two disjoint 4 cycles (we kn ow there is one by the above list of cycle structures)." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 44 "RandElementConj([[4,4],\{seq(s[i] ,i=1..3)\}]);" }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "B4" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of B4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('B',4));" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of B4." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('B',4);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of B4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalW eights('B',4):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule weights of B4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('B',4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[4]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 108 ": Since lambda[4] is minuscule , this gives us all of the weights of the irrep with highest weight la mbda[4]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOr bit(lambda[4],'B',4):" }{MPLTEXT 1 0 56 "\n# print out the *list* of w eights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listl ex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permutations. (For example: 2 corresponds to the second wei ght in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,'B',4):" }{MPLTEXT 1 0 17 " \n# print them out" }{MPLTEXT 1 0 21 "\nfor i from 1 to 4 do" } {MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let's multiply all of the simple reflections together to find the Coxeter element." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply all of the generators together." } {MPLTEXT 1 0 13 "\ntmp := s[1]:" }{MPLTEXT 1 0 21 "\nfor i from 2 to 4 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i]):" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprint(tmp);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "The cycle structures present in the subgroup of the sy mmetric group generated by these permutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "CycleStructure(\{seq(s[i],i=1..4)\});" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 20 "We can see that the " }{TEXT 202 15 "cycle structure" }{TEXT 200 64 " leaves the possibility of an \+ 8-dimensional invariant subspace. " }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 117 ": Recall that the partitions r eturned by CycleStructure and CycleStructureRandom suppress 1's. Thus \+ [3,3,3,3] is the " }}{PARA 201 "" 0 "" {TEXT 200 109 " partiti on 3+3+3+3+1+1+1+1 (=16). So we have 3+3+1+1=8 (a possible 8-dimension al invariant subspace)." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 130 "It is known that the representation V(lambda[4]) is irreducib le, but unfortunately we cannot get this from cycle structures alone." }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "B5" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of B5." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('B',5));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of B5." }}}{EXCHG {PARA 202 " > " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('B',5);" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 200 30 "The fundamental weights of B5." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeights('B',5):" } {MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\nseq(lambda[i] ,i=1..5);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule wei ghts of B5." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeW eights('B',5);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights \+ in the orbit of lambda[5]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" } {TEXT 200 108 ": Since lambda[5] is minuscule, this gives us all of th e weights of the irrep with highest weight lambda[5]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[5],'B',5) :" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (after sortin g them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permu tations. (For example: 2 corresponds to the second weight in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := We ylGroupAction(chi,'B',5):" }{MPLTEXT 1 0 17 "\n# print them out" } {MPLTEXT 1 0 21 "\nfor i from 1 to 5 do" }{MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let' s multiply all of the simple reflections together to find the Coxeter \+ element." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply al l of the generators together." }{MPLTEXT 1 0 13 "\ntmp := s[1]:" } {MPLTEXT 1 0 21 "\nfor i from 2 to 5 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i]):" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprin t(tmp);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "The cycle structur es present in the subgroup of the symmetric group generated by these p ermutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "CycleStru cture(\{seq(s[i],i=1..5)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 113 "The cycle structure [10,10,10,2] allows for possible invariant su bspaces of dimensions 0, 2, 10, 12, 20, 22, 32. " }{TEXT 200 103 "\nTh e cycle structure [8,8,8,8] allows for possible invariant subspaces of dimensions 0, 8, 16, 24, 32. " }}{PARA 201 "" 0 "" {TEXT 200 114 "Thu s, these cycle structures together allow for only 0 and 32 dimensional invariant subspaces (trivial subspaces)." }}}}{EXCHG {PARA 201 "" 0 " " {TEXT 200 59 "Now for the one \"higher\" rank example and the genera l case." }}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "B7" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of B7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('B',7));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of B7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('B',7);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of B7." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeigh ts('B',7):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\n seq(lambda[i],i=1..7);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The \+ minuscule weights of B7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('B',7);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[7]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 108 ": Since lambda[7] is minuscule, this give s us all of the weights of the irrep with highest weight lambda[5]." } }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[ 7],'B',7):" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (aft er sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 35 "The dimension of V(lambda[7]) i s..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permutations. (For exam ple: 2 corresponds to the second weight in the above sorted list.)" }} {PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 7 "WARNING" }{TEXT 200 32 ": This may take a minute or two." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,'B',7):" }{MPLTEXT 1 0 17 " \n# print them out" }{MPLTEXT 1 0 21 "\nfor i from 1 to 6 do" } {MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let's multiply all of the simple reflections together to find the Coxeter element." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply all of the generators together." } {MPLTEXT 1 0 13 "\ntmp := s[1]:" }{MPLTEXT 1 0 21 "\nfor i from 2 to 7 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i]):" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprint(tmp);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 104 "At this point, it would be nice to find the cycle stru ctures present in the subgroup, but CycleStructure" }}{PARA 201 "" 0 " " {TEXT 200 58 "takes too long. So instead we can use CycleStructureRa ndom" }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 7 "WARNING" } {TEXT 200 32 ": This may take several minutes." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 41 "CycleStructureRandom(\{seq(s[i],i=1..7)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 76 "When I executed the above co mmands, I found the following cycle structures: " }{TEXT 200 104 "\n \+ [20,20,20,20,20,20,4,4], [14,14,14,14,14,14,14,14,14,2], and [8, 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8]." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 " " {TEXT 200 98 "The cycle structure [20,20,20,20,20,20,4,4] allows for possible invariant subspaces of dimensions:" }}{PARA 201 "" 0 "" {TEXT 200 30 " 0, 4, 8, 20, 24, 28, 4" }{TEXT 200 63 "0, 44, 48 , 60, 64, 68, 80, 84, 88, 100, 104, 108, 120, 124, 128" }{TEXT 200 2 " . " }{TEXT 200 25 "\nThe cycle structure [14," }{TEXT 200 25 "14,14,14 ,14,14,14,14,14,2" }{TEXT 200 56 "] allows for possible invariant subs paces of dimensions:" }}{PARA 201 "" 0 "" {TEXT 200 90 " 0, 2, \+ 14, 16, 28, 30, 42, 44, 56, 58, 70, 72, 84, 86, 98, 100, 112, 114, 126 , 128." }}{PARA 201 "" 0 "" {TEXT 200 108 "The cycle structure [8,8,8, 8,8,8,8,8,8,8,8,8,8,8,8,8] allows for possible invariant subspaces of \+ dimensions:" }}{PARA 201 "" 0 "" {TEXT 200 77 " 0, 8, 16, 24, 3 2, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 115 "Thus, these cycle structures t ogether allow for only 0 and 128 dimensional invariant subspaces (triv ial subspaces)." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 114 "Maybe those partitions didn't appear in your list. You can either tak e my word, try CycleStructureRandom again, or" }}{PARA 201 "" 0 "" {TEXT 200 27 "try the following commands:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 62 "RandElementConj([[20,20,20,20,20,20,4,4],\{seq(s[i ],i=1..7)\}]);" }{MPLTEXT 1 0 70 "\nRandElementConj([[14,14,14,14,14,1 4,14,14,14,2],\{seq(s[i],i=1..7)\}]);" }{MPLTEXT 1 0 73 "\nRandElement Conj([[8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8],\{seq(s[i],i=1..7)\}]);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 84 "Thus (unless you're very unluck y) we get elements with the desired cycle structures." }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "Bn" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 123 "In this example, you can adjust the following parameters as n eeded to experiment with different B type simple Lie algebras." }} {PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 54 "This will explore t he Bn representation: V(lambda[k])." }{TEXT 200 75 "\n\"rank = n\" and \"highest weight = lambda[k] = the k-th fundamental weight.\"" }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 15 "n := 6; k := 6;" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Cartan matrix for Bn." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('B',n)) ;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 26 "The Dynkin diagram for Bn ." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('B',n) ;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 31 "The fundamental weights f or Bn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := Funda mentalWeights('B',n):" }{MPLTEXT 1 0 17 "\n# print them out" } {MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 29 "The minuscule weights for Bn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('B',n);" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 200 38 "The weights in the orbit of lambda[k]." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[k] ,'B',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 41 "The orbit of lambd a[k] has cardinality..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 73 "The Weyl group action (on the o rbit of lambda[k]) viewed as permutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,'B',n):" }{MPLTEXT 1 0 21 "\nfor i from 1 to n do" }{MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 33 "A sample of the \+ cycle structures." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 41 "Cycl eStructureRandom(\{seq(s[i],i=1..n)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 138 "Let's find an element with cycle structure [2,2] (two t ranspositions) if we can. This may be impossible depending on your cho ice of n & k." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 44 "RandElem entConj([[2,2],\{seq(s[i],i=1..n)\}]);" }}}}}{SECT 1 {PARA 203 "" 0 "" {TEXT -1 6 "Type C" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 123 "In this example, you can adjust the following parameters as needed to experim ent with different C type simple Lie algebras." }}{PARA 201 "" 0 "" } {PARA 201 "" 0 "" {TEXT 200 54 "This will explore the Cn representatio n: V(lambda[k])." }{TEXT 200 75 "\n\"rank = n\" and \"highest weight = lambda[k] = the k-th fundamental weight.\"" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 15 "n := 5; k := 1;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Cartan matrix for Cn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('C',n));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 26 "The Dynkin diagram for Cn." }}}{EXCHG {PARA 202 "> \+ " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('C',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 31 "The fundamental weights for Cn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeights('C',n):" } {MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\nseq(lambda[i] ,i=1..n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 29 "The minuscule wei ghts for Cn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "Minuscule Weights('C',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[k]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[k],'C',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 41 "The orbit of lambda[k] has cardinality..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" } {MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 73 "The Weyl group action (on the orbit of lambda[k]) viewed as permut ations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroup Action(chi,'C',n):" }{MPLTEXT 1 0 21 "\nfor i from 1 to n do" } {MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 33 "A sample of the cycle structures." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 41 "CycleStructureRandom(\{seq (s[i],i=1..n)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 138 "Let's fi nd an element with cycle structure [2,2] (two transpositions) if we ca n. This may be impossible depending on your choice of n & k." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 44 "RandElementConj([[2,2],\{s eq(s[i],i=1..n)\}]);" }}}}{SECT 1 {PARA 203 "" 0 "" {TEXT -1 6 "Type D " }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 123 "In this example, you can a djust the following parameters as needed to experiment with different \+ D type simple Lie algebras." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 54 "This will explore the Dn representation: V(lambda[k])." }{TEXT 200 75 "\n\"rank = n\" and \"highest weight = lambda[k] = the k -th fundamental weight.\"" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 15 "n := 5; k := 1;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Car tan matrix for Dn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "pri nt(CartanMatrix('D',n));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 26 "Th e Dynkin diagram for Dn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('D',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 31 "T he fundamental weights for Dn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeights('D',n):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..n);" }} }{EXCHG {PARA 201 "" 0 "" {TEXT 200 29 "The minuscule weights for Dn." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('D',n );" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[k]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := \+ WeylOrbit(lambda[k],'D',n);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 41 "The orbit of lambda[k] has cardinality..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" }{MPLTEXT 1 0 11 "\nnop s(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 73 "The Weyl group acti on (on the orbit of lambda[k]) viewed as permutations." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,'D',n):" }{MPLTEXT 1 0 21 "\nfor i from 1 to n do" }{MPLTEXT 1 0 9 "\n s[i]; " }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 33 "A \+ sample of the cycle structures." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 41 "CycleStructureRandom(\{seq(s[i],i=1..n)\});" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 138 "Let's find an element with cyc le structure [2,2] (two transpositions) if we can. This may be impossi ble depending on your choice of n & k." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 44 "RandElementConj([[2,2],\{seq(s[i],i=1..n)\}]);" }}}} {SECT 0 {PARA 203 "" 0 "" {TEXT -1 6 "Type E" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 104 "Of the five exceptional algebras, only E6 and E7 hav e minuscule weights. You can still explore the other" }}{PARA 201 "" 0 "" {TEXT 200 101 "exceptional algebras, but a single orbit of weights will no longer capture the entire representation." }}}{SECT 0 {PARA 204 "" 0 "" {TEXT -1 2 "E6" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "T he Cartan matrix of E6." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('E',6));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of E6." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('E',6);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of E6." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeights('E',6):" } {MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\nseq(lambda[i] ,i=1..6);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule wei ghts of E6." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeW eights('E',6);" }}}{SECT 0 {PARA 205 "" 0 "" {TEXT 205 29 "Weight lamb da[1] is minuscule" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weigh ts in the orbit of lambda[1]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 108 ": Since lambda[1] is minuscule, this gives us all of t he weights of the irrep with highest weight lambda[1]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[1],'E',6) :" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (after sortin g them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 35 "The dimension of V(lambda[1]) is..." } }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 23 "# the cardnality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permutations. (For example: 2 corresponds to the second weight in the above sorted list.)" }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(chi,' E',6):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 21 "\nfor \+ i from 1 to 6 do" }{MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let's multiply all of the s imple reflections together to find the Coxeter element." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply all of the generators \+ together." }{MPLTEXT 1 0 13 "\ntmp := s[1]:" }{MPLTEXT 1 0 21 "\nfor i from 2 to 6 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i]):" } {MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprint(tmp);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "The cycle structures present in the s ubgroup of the symmetric group generated by these permutations." }} {PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 7 "WARNING" }{TEXT 200 32 ": This may take a minute or two." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "CycleStructure(\{seq(s[i],i=1..6)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 107 "The cycle structure [12,12,3] allows \+ for possible invariant subspaces of dimensions: 0, 3, 12, 15, 24, 27. \+ " }{TEXT 200 98 "\nThe cycle structure [9,9,9] allows for possible inv ariant subspaces of dimensions: 0, 9, 18, 27. " }}{PARA 201 "" 0 "" {TEXT 200 114 "Thus, these cycle structures together allow for only 0 \+ and 27 dimensional invariant subspaces (trivial subspaces)." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 95 "The Coxeter element has t wo 12-cycles and a 3-cycle. Let's find an element with three 9-cycles. " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 46 "RandElementConj([[9,9 ,9],\{seq(s[i],i=1..6)\}]);" }}}}{SECT 1 {PARA 205 "" 0 "" {TEXT 205 29 "Weight lambda[6] is minuscule" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[6]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 108 ": Since lambda[6] is minuscule, thi s gives us all of the weights of the irrep with highest weight lambda[ 6]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(l ambda[6],'E',6):" }{MPLTEXT 1 0 56 "\n# print out the *list* of weight s (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 35 "The dimension of V(lambda[6 ]) is..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardina lity of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permutations. (For e xample: 2 corresponds to the second weight in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := WeylGroupAction(c hi,'E',6):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 21 "\n for i from 1 to 6 do" }{MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\n od;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let's multiply all of t he simple reflections together to find the Coxeter element." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply all of the gene rators together." }{MPLTEXT 1 0 13 "\ntmp := s[1]:" }{MPLTEXT 1 0 21 " \nfor i from 2 to 6 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i ]):" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprint(tmp);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "The cycle structures present i n the subgroup of the symmetric group generated by these permutations. " }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 202 7 "WARNING" }{TEXT 200 32 ": This may take a minute or two." }}}{EXCHG {PARA 202 "> " 0 " " {MPLTEXT 1 0 35 "CycleStructure(\{seq(s[i],i=1..6)\});" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 107 "The cycle structure [12,12,3] allows \+ for possible invariant subspaces of dimensions: 0, 3, 12, 15, 24, 27. \+ " }{TEXT 200 98 "\nThe cycle structure [9,9,9] allows for possible inv ariant subspaces of dimensions: 0, 9, 18, 27. " }}{PARA 201 "" 0 "" {TEXT 200 114 "Thus, these cycle structures together allow for only 0 \+ and 27 dimensional invariant subspaces (trivial subspaces)." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 95 "The Coxeter element has t wo 12-cycles and a 3-cycle. Let's find an element with three 9-cycles. " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 46 "RandElementConj([[9,9 ,9],\{seq(s[i],i=1..6)\}]);" }}}}}{SECT 0 {PARA 204 "" 0 "" {TEXT -1 2 "E7" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of E 7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix( 'E',7));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of E7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram( 'E',7);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental wei ghts of E7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := \+ FundamentalWeights('E',7):" }{MPLTEXT 1 0 17 "\n# print them out" } {MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..7);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 28 "The minuscule weights of E7." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('E',7);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 38 "The weights in the orbit of lambda[7]." }}{PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 108 ": Since lambda[7] is mi nuscule, this gives us all of the weights of the irrep with highest we ight lambda[7]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(lambda[7],'E',7):" }{MPLTEXT 1 0 56 "\n# print out the *lis t* of weights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi) ],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 35 "The dimension o f V(lambda[7]) is..." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 121 "The Weyl group action viewed as permu tations. (For example: 2 corresponds to the second weight in the above sorted list.)" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "s := We ylGroupAction(chi,'E',7):" }{MPLTEXT 1 0 17 "\n# print them out" } {MPLTEXT 1 0 21 "\nfor i from 1 to 7 do" }{MPLTEXT 1 0 9 "\n s[i];" }{MPLTEXT 1 0 4 "\nod;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "Let' s multiply all of the simple reflections together to find the Coxeter \+ element." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 42 "# multiply al l of the generators together." }{MPLTEXT 1 0 13 "\ntmp := s[1]:" } {MPLTEXT 1 0 21 "\nfor i from 2 to 7 do" }{MPLTEXT 1 0 30 "\n tmp := mulperms(tmp,s[i]):" }{MPLTEXT 1 0 4 "\nod:" }{MPLTEXT 1 0 12 "\nprin t(tmp);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 53 "The Coxeter element has cycle structure [18,18,18,2]." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 94 "It would take too long (and too much memory) to run CycleStructure. So, we compromise and run " }}{PARA 201 "" 0 "" {TEXT 200 81 "CycleStructureRandom instead. Hopefully, we'll see the c ycle structures we need. " }{TEXT 202 1 "\n" }{TEXT 202 8 "\nWARNING" }{TEXT 200 30 ": This may take a few minutes." }}}{EXCHG {PARA 202 "> \+ " 0 "" {MPLTEXT 1 0 41 "CycleStructureRandom(\{seq(s[i],i=1..7)\});" } }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 105 "After running CycleStructure Random, (hopefully) we see that the cycle structure [14,14,14,14] is p resent." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 70 "Let's fi nd an element with this cycle structure (i.e. four 14-cycles)." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 52 "RandElementConj([[14,14,14 ,14],\{seq(s[i],i=1..7)\}]);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 114 "The cycle structure [18,18,18,2] allows for possible invariant su bspaces of dimensions: 0, 18, 20, 36, 38, 54, 56." }{TEXT 200 109 "\nT he cycle structure [14,14,14,14] allows for possible invariant subspac es of dimensions: 0, 14, 28, 42, 56. " }}{PARA 201 "" 0 "" {TEXT 200 114 "Thus, these cycle structures together allow for only 0 and 56 dim ensional invariant subspaces (trivial subspaces)." }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT -1 2 "E8" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 24 "T he Cartan matrix of E8." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('E',8));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of E8." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('E',8);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 20 "Simple roots for E8." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 28 "alpha := SimpleRoots('E',8):" }{MPLTEXT 1 0 17 "\n# p rint them out" }{MPLTEXT 1 0 22 "\nseq(alpha[i],i=1..8);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of E8." }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeigh ts('E',8):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 23 "\n seq(lambda[i],i=1..8);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 7 "E8 ha s " }{TEXT 202 2 "no" }{TEXT 200 19 " minuscule weights." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('E',8);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 61 "The highest long root of E8 is \+ given by Humphreys on page 66." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 12 "2*alpha[1]+3" }{MPLTEXT 1 0 76 "*alpha[2]+4*alpha[3]+ 6*alpha[4]+5*alpha[5]+4*alpha[6]+3*alpha[7]+2*alpha[8];" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 118 "Thus, the highest long root is better known as the last fundamental weight lambda[8]. Thus V(lambda[8]) is \+ the adjoint" }}{PARA 201 "" 0 "" {TEXT 200 50 "representation (irreduc ible because E8 is simple)." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 58 "Let's find all of the roots of E8 by reflecting lambda[8 ]." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 34 "chi := WeylOrbit(la mbda[8],'E',8):" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 101 "The dimension of E8 is (# o f negative roots) + (rank) + (# of positive roots) = 120 + 8 + 120 = 2 48. " }}{PARA 201 "" 0 "" {TEXT 200 52 "(see Humphreys page 66: E8 has 120 positive roots). " }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 67 "Since the cardinality of chi is 240, our list of roots is comp lete." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinalit y of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 202 4 "Note" }{TEXT 200 96 ": The weights of V(lambda[8]) are pr ecisely the set chi (the roots) union \{ [0,0,0,0,0,0,0,0] \}." }}}}} {SECT 1 {PARA 203 "" 0 "" {TEXT -1 6 "Type F" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 41 "The only type F simple Lie algebra is F4." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of F4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('F' ,4));" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 25 "The Dynkin diagram of F4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('F' ,4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 20 "Simple roots for F4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 28 "alpha := SimpleRoots('F ',4):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 22 "\nseq(a lpha[i],i=1..4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundam ental weights of F4." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "l ambda := FundamentalWeights('F',4):" }{MPLTEXT 1 0 17 "\n# print them \+ out" }{MPLTEXT 1 0 23 "\nseq(lambda[i],i=1..4);" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 200 7 "F4 has " }{TEXT 202 2 "no" }{TEXT 200 19 " minuscu le weights." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeW eights('F',4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "Let's find \+ all of the roots of F4. First we compute (and union) the orbits of alp ha[1] and alpha[4]." }}{PARA 202 "> " 0 "" {MPLTEXT 1 0 39 "chi := Wey lOrbit(alpha[1],'F',4) union " }{MPLTEXT 1 0 25 "WeylOrbit(alpha[4],'F ',4)" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 56 "\n# print out the *list* o f weights (after sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],li stlex);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 98 "The dimension of F4 is (# of negative roots) + (rank) + (# of positive roots) = 24 + 4 + \+ 24 = 52. " }}{PARA 201 "" 0 "" {TEXT 200 51 "(see Humphreys page 66: F 4 has 24 positive roots). " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the cardinality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 66 "Since the cardinality of chi is 48, our list of roots is complete." }}}}{SECT 1 {PARA 203 "" 0 "" {TEXT -1 6 "Type G" }}{EXCHG {PARA 201 "" 0 "" {TEXT 200 41 "The only \+ type G simple Lie algebra is G2." }}{PARA 201 "" 0 "" }{PARA 201 "" 0 "" {TEXT 200 24 "The Cartan matrix of G2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 27 "print(CartanMatrix('G',2));" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 200 25 "The Dynkin diagram of G2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "DynkinDiagram('G',2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 20 "Simple roots for G2." }}}{EXCHG {PARA 202 "> " 0 " " {MPLTEXT 1 0 28 "alpha := SimpleRoots('G',2):" }{MPLTEXT 1 0 17 "\n# print them out" }{MPLTEXT 1 0 10 "\nalpha[1]," }{MPLTEXT 1 0 8 "alpha [2]" }{MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 30 "The fundamental weights of G2." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 36 "lambda := FundamentalWeights('G',2):" }{MPLTEXT 1 0 17 "\n# prin t them out" }{MPLTEXT 1 0 11 "\nlambda[1]," }{MPLTEXT 1 0 10 "lambda[2 ];" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 7 "G2 has " }{TEXT 202 2 "no " }{TEXT 200 19 " minuscule weights." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "MinusculeWeights('G',2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 100 "Let's find all of the roots of G2. First we compute (a nd union) the orbits of alpha[1] and alpha[2]." }}{PARA 202 "> " 0 "" {MPLTEXT 1 0 65 "chi := WeylOrbit(alpha[1],'G',2) union WeylOrbit(alph a[2],'G',2):" }{MPLTEXT 1 0 56 "\n# print out the *list* of weights (a fter sorting them)." }{MPLTEXT 1 0 25 "\nsort([op(chi)],listlex);" }}} {EXCHG {PARA 201 "" 0 "" {TEXT 200 96 "The dimension of G2 is (# of ne gative roots) + (rank) + (# of positive roots) = 6 + 2 + 6 = 14. " }} {PARA 201 "" 0 "" {TEXT 200 50 "(see Humphreys page 66: G2 has 6 posit ive roots). " }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 24 "# the car dinality of chi" }{MPLTEXT 1 0 11 "\nnops(chi);" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 200 66 "Since the cardinality of chi is 12, our list of r oots is complete." }}}}}{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" } {PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" }{PARA 206 "" 0 "" } {PARA 206 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }