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path: root/Math/Combinatorics/.RootSystem.hs.swp
blob: f8ca37755be75b6725484ff8e39dec7d88137f53 (plain) (blame)
ofshex dumpascii
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1a00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1a20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1a40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1a60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1a80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1aa0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ac0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ae0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1b00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1b20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1b40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1b60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1b80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ba0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1bc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1be0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1c00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1c20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1c40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1c60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1c80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ca0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1cc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ce0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1d00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1d20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1d40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1d60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1d80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1da0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1dc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1de0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1e00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1e20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1e40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1e60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1e80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ea0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ec0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1ee0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1f00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1f20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1f40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1f60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1f80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1fa0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1fc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
1fe0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
2000 61 64 00 00 90 00 00 00 2c 02 00 00 00 10 00 00 60 00 00 00 00 00 00 00 ac 0f 00 00 6f 0f 00 00 ad......,.......`...........o...
2020 6e 0f 00 00 48 0f 00 00 47 0f 00 00 28 0f 00 00 27 0f 00 00 15 0f 00 00 04 0f 00 00 f2 0e 00 00 n...H...G...(...'...............
2040 d2 0e 00 00 b3 0e 00 00 b2 0e 00 00 90 0e 00 00 47 0e 00 00 18 0e 00 00 e6 0d 00 00 e5 0d 00 00 ................G...............
2060 ba 0d 00 00 b9 0d 00 00 85 0d 00 00 6b 0d 00 00 6a 0d 00 00 37 0d 00 00 36 0d 00 00 35 0d 00 00 ............k...j...7...6...5...
2080 23 0d 00 00 fb 0c 00 00 fa 0c 00 00 db 0c 00 00 8e 0c 00 00 54 0c 00 00 53 0c 00 00 01 0c 00 00 #...................T...S.......
20a0 c4 0b 00 00 7b 0b 00 00 26 0b 00 00 25 0b 00 00 c7 0a 00 00 9b 0a 00 00 5f 0a 00 00 42 0a 00 00 ....{...&...%..........._...B...
20c0 fb 09 00 00 e2 09 00 00 be 09 00 00 72 09 00 00 59 09 00 00 35 09 00 00 e2 08 00 00 c9 08 00 00 ............r...Y...5...........
20e0 a5 08 00 00 6c 08 00 00 53 08 00 00 f2 07 00 00 cc 07 00 00 90 07 00 00 35 07 00 00 1c 07 00 00 ....l...S...............5.......
2100 dc 06 00 00 c3 06 00 00 5a 06 00 00 59 06 00 00 49 06 00 00 0a 06 00 00 09 06 00 00 f9 05 00 00 ........Z...Y...I...............
2120 cb 05 00 00 aa 05 00 00 93 05 00 00 59 05 00 00 58 05 00 00 57 05 00 00 2a 05 00 00 d3 04 00 00 ............Y...X...W...*.......
2140 d2 04 00 00 d1 04 00 00 a9 04 00 00 79 04 00 00 6e 04 00 00 4a 04 00 00 17 04 00 00 f6 03 00 00 ............y...n...J...........
2160 ad 03 00 00 90 03 00 00 78 03 00 00 3f 03 00 00 fa 02 00 00 f1 02 00 00 cc 02 00 00 ad 02 00 00 ........x...?...................
2180 ac 02 00 00 ab 02 00 00 94 02 00 00 6d 02 00 00 2d 02 00 00 2c 02 00 00 a9 01 00 00 ac 01 00 00 ............m...-...,...........
21a0 00 00 00 00 00 00 00 00 00 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 73 73 20 3d 20 28 70 6f ..........positiveRoots.ss.=.(po
21c0 73 69 74 69 76 65 52 6f 6f 74 73 27 20 24 20 63 61 72 74 61 6e 4d 61 74 72 69 78 27 20 73 73 29 sitiveRoots'.$.cartanMatrix'.ss)
21e0 20 3c 3c 2a 3e 3e 20 73 73 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 3a 3a 20 53 69 6d 70 6c .<<*>>.ss.positiveRoots.::.Simpl
2200 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 2d 2d 20 7c 61 6c 6c 20 70 6f 73 69 74 69 76 eSystem.->.[[Q]].--.|all.positiv
2220 65 20 72 6f 6f 74 73 00 00 00 2d 2d 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 73 73 20 3d 20 e.roots...--.positiveRoots.ss.=.
2240 28 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 20 24 20 63 61 72 74 61 6e 4d 61 74 72 69 78 27 20 (positiveRoots'.$.cartanMatrix'.
2260 73 73 29 20 3c 3c 2a 3e 3e 20 73 73 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 3a 3a 20 53 69 ss).<<*>>.ss.positiveRoots.::.Si
2280 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 2d 2d 20 7c 61 6c 6c 20 70 6f 73 69 mpleSystem.->.[[Q]].--.|all.posi
22a0 74 69 76 65 20 72 6f 6f 74 73 00 00 00 2d 2d 6e 65 67 61 74 65 4d 20 3d 20 66 6d 61 70 20 28 66 tive.roots...--negateM.=.fmap.(f
22c0 6d 61 70 20 6e 65 67 61 74 65 29 00 2d 2d 6e 65 67 61 74 65 4d 20 3a 3a 20 4e 75 6d 20 61 20 3d map.negate).--negateM.::.Num.a.=
22e0 3e 20 5b 5b 61 5d 5d 20 2d 3e 20 5b 5b 61 5d 5d 00 20 20 20 20 20 20 20 20 00 20 20 20 20 20 20 >.[[a]].->.[[a]]................
2300 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 6c 6f 6e 67 65 73 74 45 6c 65 6d 65 6e ...................longestElemen
2320 74 27 20 28 61 6c 70 68 61 3a 78 73 29 20 28 73 20 61 6c 70 68 61 20 3c 24 3e 20 72 73 29 00 20 t'.(alpha:xs).(s.alpha.<$>.rs)..
2340 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 65 6c 73 65 20 6c 65 74 20 61 6c 70 68 61 20 3d 20 ...............else.let.alpha.=.
2360 28 68 65 61 64 20 28 53 2e 74 6f 4c 69 73 74 20 79 73 29 29 20 69 6e 00 20 20 20 20 20 20 20 20 (head.(S.toList.ys)).in.........
2380 20 20 20 20 20 20 20 20 74 68 65 6e 20 78 73 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 ........then.xs.................
23a0 69 66 20 53 2e 6e 75 6c 6c 20 79 73 00 20 20 20 20 20 20 20 20 20 20 20 20 6c 65 74 20 79 73 20 if.S.null.ys.............let.ys.
23c0 3d 20 28 53 2e 66 72 6f 6d 4c 69 73 74 20 73 73 29 20 60 53 2e 69 6e 74 65 72 73 65 63 74 69 6f =.(S.fromList.ss).`S.intersectio
23e0 6e 60 20 28 53 2e 66 72 6f 6d 4c 69 73 74 20 72 73 29 20 69 6e 00 20 20 20 20 20 20 20 20 6c 6f n`.(S.fromList.rs).in.........lo
2400 6e 67 65 73 74 45 6c 65 6d 65 6e 74 27 20 78 73 20 72 73 20 3d 20 00 20 20 20 20 20 20 20 20 6c ngestElement'.xs.rs.=..........l
2420 6f 6e 67 65 73 74 45 6c 65 6d 65 6e 74 27 20 3a 3a 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 5b 51 5d 5d ongestElement'.::.[[Q]].->.[[Q]]
2440 20 2d 3e 20 5b 5b 51 5d 5d 00 20 20 20 20 20 20 20 20 70 6f 73 52 6f 6f 74 73 20 3d 20 70 6f 73 .->.[[Q]].........posRoots.=.pos
2460 69 74 69 76 65 52 6f 6f 74 73 20 73 73 00 20 20 20 20 77 68 65 72 65 20 00 6c 6f 6e 67 65 73 74 itiveRoots.ss.....where..longest
2480 45 6c 65 6d 65 6e 74 20 73 73 20 3d 20 6c 6f 6e 67 65 73 74 45 6c 65 6d 65 6e 74 27 20 5b 5d 20 Element.ss.=.longestElement'.[].
24a0 70 6f 73 52 6f 6f 74 73 00 6c 6f 6e 67 65 73 74 45 6c 65 6d 65 6e 74 20 3a 3a 20 53 69 6d 70 6c posRoots.longestElement.::.Simpl
24c0 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 00 00 6c 6f 6e 67 65 73 74 45 6c 65 6d 65 6e eSystem.->.[[Q]]...longestElemen
24e0 74 49 6e 64 65 78 20 73 73 20 3d 20 28 2b 31 29 20 3c 24 3e 20 66 72 6f 6d 4a 75 73 74 20 3c 24 tIndex.ss.=.(+1).<$>.fromJust.<$
2500 3e 20 66 6c 69 70 20 65 6c 65 6d 49 6e 64 65 78 20 73 73 20 3c 24 3e 20 6c 6f 6e 67 65 73 74 45 >.flip.elemIndex.ss.<$>.longestE
2520 6c 65 6d 65 6e 74 20 73 73 00 6c 6f 6e 67 65 73 74 45 6c 65 6d 65 6e 74 49 6e 64 65 78 20 3a 3a lement.ss.longestElementIndex.::
2540 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 49 6e 74 5d 00 00 00 73 20 61 6c 70 68 61 .SimpleSystem.->.[Int]...s.alpha
2560 20 62 65 74 61 20 3d 20 62 65 74 61 20 3c 2d 3e 20 28 64 79 6e 6b 69 6e 49 6e 64 65 78 20 61 6c .beta.=.beta.<->.(dynkinIndex.al
2580 70 68 61 20 62 65 74 61 29 20 2a 3e 20 61 6c 70 68 61 00 73 20 3a 3a 20 5b 51 5d 20 2d 3e 20 5b pha.beta).*>.alpha.s.::.[Q].->.[
25a0 51 5d 20 2d 3e 20 5b 51 5d 00 2d 2d 20 73 20 61 6c 70 68 61 20 62 65 74 61 20 3d 20 73 5f 5c 61 Q].->.[Q].--.s.alpha.beta.=.s_\a
25c0 6c 70 68 61 20 5c 62 65 74 61 00 2d 2d 20 57 65 79 6c 20 67 72 6f 75 70 20 65 6c 65 6d 65 6e 74 lpha.\beta.--.Weyl.group.element
25e0 20 63 6f 72 72 65 73 70 6f 6e 64 69 6e 67 20 74 6f 20 61 20 72 6f 6f 74 00 2d 2d 20 48 75 6d 70 .corresponding.to.a.root.--.Hump
2600 68 72 65 79 73 20 70 33 00 00 2d 2d 20 43 61 6c 63 75 6c 61 74 69 6e 67 20 74 68 65 20 66 75 6c hreys.p3..--.Calculating.the.ful
2620 6c 20 72 6f 6f 74 20 73 79 73 74 65 6d 20 66 72 6f 6d 20 74 68 65 20 66 75 6e 64 61 6d 65 6e 74 l.root.system.from.the.fundament
2640 61 6c 20 72 6f 6f 74 73 00 2d 2d 20 52 4f 4f 54 20 53 59 53 54 45 4d 53 00 00 73 69 6d 70 6c 65 al.roots.--.ROOT.SYSTEMS..simple
2660 53 79 73 74 65 6d 20 74 20 6e 20 3d 20 65 72 72 6f 72 20 24 20 22 49 6e 76 61 6c 69 64 20 72 6f System.t.n.=.error.$."Invalid.ro
2680 6f 74 20 73 79 73 74 65 6d 20 6f 66 20 74 79 70 65 20 22 20 2b 2b 20 28 73 68 6f 77 20 74 29 20 ot.system.of.type.".++.(show.t).
26a0 2b 2b 20 22 20 61 6e 64 20 72 61 6e 6b 20 22 20 2b 2b 20 28 73 68 6f 77 20 6e 29 20 2b 2b 20 22 ++.".and.rank.".++.(show.n).++."
26c0 2e 22 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 33 00 73 69 6d 70 .".....where.e.=.basisElt.3.simp
26e0 6c 65 53 79 73 74 65 6d 20 47 20 32 20 3d 20 5b 65 20 31 20 3c 2d 3e 20 65 20 32 2c 20 28 28 2d leSystem.G.2.=.[e.1.<->.e.2,.((-
2700 32 29 20 2a 3e 20 65 20 31 29 20 3c 2b 3e 20 65 20 32 20 3c 2b 3e 20 65 20 33 5d 00 20 20 20 20 2).*>.e.1).<+>.e.2.<+>.e.3].....
2720 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 34 00 73 69 6d 70 6c 65 53 79 73 74 65 where.e.=.basisElt.4.simpleSyste
2740 6d 20 46 20 34 20 3d 20 5b 65 20 32 20 3c 2d 3e 20 65 20 33 2c 20 65 20 33 20 3c 2d 3e 20 65 20 m.F.4.=.[e.2.<->.e.3,.e.3.<->.e.
2760 34 2c 20 65 20 34 2c 20 28 31 2f 32 29 20 2a 3e 20 28 65 20 31 20 3c 2d 3e 20 65 20 32 20 3c 2d 4,.e.4,.(1/2).*>.(e.1.<->.e.2.<-
2780 3e 20 65 20 33 20 3c 2d 3e 20 65 20 34 29 5d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 >.e.3.<->.e.4)].................
27a0 20 20 20 20 20 20 3a 20 5b 65 20 28 69 2d 31 29 20 3c 2d 3e 20 65 20 28 69 2d 32 29 20 7c 20 69 ......:.[e.(i-1).<->.e.(i-2).|.i
27c0 20 3c 2d 20 5b 33 2e 2e 38 5d 5d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 .<-.[3..8]].....................
27e0 20 20 3a 20 28 65 20 31 20 3c 2b 3e 20 65 20 32 29 00 20 20 20 20 20 20 20 20 20 20 73 69 6d 70 ..:.(e.1.<+>.e.2)...........simp
2800 6c 65 72 6f 6f 74 73 20 3d 20 28 28 31 2f 32 29 20 2a 3e 20 28 65 20 31 20 3c 2d 3e 20 65 20 32 leroots.=.((1/2).*>.(e.1.<->.e.2
2820 20 3c 2d 3e 20 65 20 33 20 3c 2d 3e 20 65 20 34 20 3c 2d 3e 20 65 20 35 20 3c 2d 3e 20 65 20 36 .<->.e.3.<->.e.4.<->.e.5.<->.e.6
2840 20 3c 2d 3e 20 65 20 37 20 3c 2b 3e 20 65 20 38 29 29 00 20 20 20 20 77 68 65 72 65 20 65 20 3d .<->.e.7.<+>.e.8)).....where.e.=
2860 20 62 61 73 69 73 45 6c 74 20 38 00 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 45 20 6e 20 7c 20 6e .basisElt.8.simpleSystem.E.n.|.n
2880 20 60 65 6c 65 6d 60 20 5b 36 2c 37 2c 38 5d 20 3d 20 74 61 6b 65 20 6e 20 73 69 6d 70 6c 65 72 .`elem`.[6,7,8].=.take.n.simpler
28a0 6f 6f 74 73 00 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 44 20 33 20 3d 20 73 69 6d 70 6c 65 53 79 oots.simpleSystem.D.3.=.simpleSy
28c0 73 74 65 6d 20 41 20 33 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 stem.A.3.....where.e.=.basisElt.
28e0 6e 00 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 44 20 6e 20 7c 20 6e 20 3e 3d 20 34 20 3d 20 5b 65 n.simpleSystem.D.n.|.n.>=.4.=.[e
2900 20 69 20 3c 2d 3e 20 65 20 28 69 2b 31 29 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 2d 31 5d 5d 20 .i.<->.e.(i+1).|.i.<-.[1..n-1]].
2920 2b 2b 20 5b 65 20 28 6e 2d 31 29 20 3c 2b 3e 20 65 20 6e 5d 00 73 69 6d 70 6c 65 53 79 73 74 65 ++.[e.(n-1).<+>.e.n].simpleSyste
2940 6d 20 43 20 31 20 3d 20 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 41 20 31 00 20 20 20 20 77 68 65 m.C.1.=.simpleSystem.A.1.....whe
2960 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 6e 00 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 43 re.e.=.basisElt.n.simpleSystem.C
2980 20 6e 20 7c 20 6e 20 3e 3d 20 32 20 3d 20 5b 65 20 69 20 3c 2d 3e 20 65 20 28 69 2b 31 29 20 7c .n.|.n.>=.2.=.[e.i.<->.e.(i+1).|
29a0 20 69 20 3c 2d 20 5b 31 2e 2e 6e 2d 31 5d 5d 20 2b 2b 20 5b 32 20 2a 3e 20 65 20 6e 5d 00 73 69 .i.<-.[1..n-1]].++.[2.*>.e.n].si
29c0 6d 70 6c 65 53 79 73 74 65 6d 20 42 20 31 20 3d 20 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 41 20 mpleSystem.B.1.=.simpleSystem.A.
29e0 31 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 6e 00 73 69 6d 70 6c 1.....where.e.=.basisElt.n.simpl
2a00 65 53 79 73 74 65 6d 20 42 20 6e 20 7c 20 6e 20 3e 3d 20 32 20 3d 20 5b 65 20 69 20 3c 2d 3e 20 eSystem.B.n.|.n.>=.2.=.[e.i.<->.
2a20 65 20 28 69 2b 31 29 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 2d 31 5d 5d 20 2b 2b 20 5b 65 20 6e e.(i+1).|.i.<-.[1..n-1]].++.[e.n
2a40 5d 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 28 6e 2b 31 29 00 73 ].....where.e.=.basisElt.(n+1).s
2a60 69 6d 70 6c 65 53 79 73 74 65 6d 20 41 20 6e 20 7c 20 6e 20 3e 3d 20 31 20 3d 20 5b 65 20 69 20 impleSystem.A.n.|.n.>=.1.=.[e.i.
2a80 3c 2d 3e 20 65 20 28 69 2b 31 29 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 5d 00 73 69 6d 70 6c <->.e.(i+1).|.i.<-.[1..n]].simpl
2aa0 65 53 79 73 74 65 6d 20 3a 3a 20 54 79 70 65 20 2d 3e 20 49 6e 74 20 2d 3e 20 53 69 6d 70 6c 65 eSystem.::.Type.->.Int.->.Simple
2ac0 53 79 73 74 65 6d 00 2d 2d 20 41 20 73 69 6d 70 6c 65 20 73 79 73 74 65 6d 20 69 73 20 6c 69 6b System.--.A.simple.system.is.lik
2ae0 65 20 61 20 62 61 73 69 73 20 66 6f 72 20 74 68 65 20 72 6f 6f 74 20 73 79 73 74 65 6d 20 28 73 e.a.basis.for.the.root.system.(s
2b00 65 65 20 48 75 6d 70 68 72 65 79 73 20 70 38 20 66 6f 72 20 66 75 6c 6c 20 64 65 66 69 6e 69 74 ee.Humphreys.p8.for.full.definit
2b20 69 6f 6e 29 00 00 2d 2d 20 53 6f 20 6c 6f 6e 67 20 61 73 20 6f 75 72 20 73 69 6d 70 6c 65 20 73 ion)..--.So.long.as.our.simple.s
2b40 79 73 74 65 6d 73 20 61 72 65 20 72 61 74 69 6f 6e 61 6c 2c 20 74 68 65 6e 20 72 65 66 6c 65 63 ystems.are.rational,.then.reflec
2b60 74 69 6f 6e 20 6d 61 74 72 69 63 65 73 20 61 72 65 20 72 61 74 69 6f 6e 61 6c 00 2d 2d 20 57 65 tion.matrices.are.rational.--.We
2b80 20 6e 65 65 64 20 74 6f 20 77 6f 72 6b 20 6f 76 65 72 20 74 68 65 20 72 61 74 69 6f 6e 61 6c 73 .need.to.work.over.the.rationals
2ba0 20 74 6f 20 65 6e 73 75 72 65 20 74 68 61 74 20 61 72 69 74 68 6d 65 74 69 63 20 69 73 20 65 78 .to.ensure.that.arithmetic.is.ex
2bc0 61 63 74 00 2d 2d 62 61 73 69 73 45 6c 74 27 20 6e 20 69 20 3d 20 72 65 70 6c 69 63 61 74 65 20 act.--basisElt'.n.i.=.replicate.
2be0 28 69 2d 31 29 20 30 20 2b 2b 20 31 20 3a 20 72 65 70 6c 69 63 61 74 65 20 28 6e 2d 69 29 20 30 (i-1).0.++.1.:.replicate.(n-i).0
2c00 00 2d 2d 62 61 73 69 73 45 6c 74 27 20 3a 3a 20 49 6e 74 20 2d 3e 20 49 6e 74 20 2d 3e 20 5b 49 .--basisElt'.::.Int.->.Int.->.[I
2c20 6e 74 5d 20 2d 2d 20 74 68 69 73 20 74 79 70 65 20 73 69 67 6e 61 74 75 72 65 20 64 65 74 65 72 nt].--.this.type.signature.deter
2c40 6d 69 6e 65 73 20 61 6c 6c 20 74 68 65 20 72 65 73 74 00 00 62 61 73 69 73 45 6c 74 20 6e 20 69 mines.all.the.rest..basisElt.n.i
2c60 20 3d 20 72 65 70 6c 69 63 61 74 65 20 28 69 2d 31 29 20 30 20 2b 2b 20 31 20 3a 20 72 65 70 6c .=.replicate.(i-1).0.++.1.:.repl
2c80 69 63 61 74 65 20 28 6e 2d 69 29 20 30 00 62 61 73 69 73 45 6c 74 20 3a 3a 20 49 6e 74 20 2d 3e icate.(n-i).0.basisElt.::.Int.->
2ca0 20 49 6e 74 20 2d 3e 20 5b 51 5d 20 2d 2d 20 74 68 69 73 20 74 79 70 65 20 73 69 67 6e 61 74 75 .Int.->.[Q].--.this.type.signatu
2cc0 72 65 20 64 65 74 65 72 6d 69 6e 65 73 20 61 6c 6c 20 74 68 65 20 72 65 73 74 00 2d 2d 20 54 68 re.determines.all.the.rest.--.Th
2ce0 65 20 69 74 68 20 62 61 73 69 73 20 76 65 63 74 6f 72 20 69 6e 20 4b 5e 6e 00 00 2d 2d 20 73 6f e.ith.basis.vector.in.K^n..--.so
2d00 6d 65 74 69 6d 65 73 20 63 61 6c 6c 65 64 20 66 75 6e 64 61 6d 65 6e 74 61 6c 20 73 79 73 74 65 metimes.called.fundamental.syste
2d20 6d 73 00 2d 2d 20 53 49 4d 50 4c 45 20 53 59 53 54 45 4d 53 00 00 00 2d 2d 20 48 75 6d 70 68 72 ms.--.SIMPLE.SYSTEMS...--.Humphr
2d40 65 79 73 2c 20 52 65 66 6c 65 63 74 69 6f 6e 20 47 72 6f 75 70 73 20 61 6e 64 20 43 6f 78 65 74 eys,.Reflection.Groups.and.Coxet
2d60 65 72 20 47 72 6f 75 70 73 00 00 74 79 70 65 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 3d 20 5b er.Groups..type.SimpleSystem.=.[
2d80 5b 51 5d 5d 00 64 61 74 61 20 54 79 70 65 20 3d 20 41 20 7c 20 42 20 7c 20 43 20 7c 20 44 20 7c [Q]].data.Type.=.A.|.B.|.C.|.D.|
2da0 20 45 20 7c 20 46 20 7c 20 47 20 64 65 72 69 76 69 6e 67 20 53 68 6f 77 00 00 69 6d 70 6f 72 74 .E.|.F.|.G.deriving.Show..import
2dc0 20 4d 61 74 68 2e 41 6c 67 65 62 72 61 2e 46 69 65 6c 64 2e 42 61 73 65 20 28 51 29 2d 2d 20 66 .Math.Algebra.Field.Base.(Q)--.f
2de0 6f 72 20 51 00 00 2d 2d 69 6d 70 6f 72 74 20 4d 61 74 68 2e 41 6c 67 65 62 72 61 2e 47 72 6f 75 or.Q..--import.Math.Algebra.Grou
2e00 70 2e 53 74 72 69 6e 67 52 65 77 72 69 74 69 6e 67 20 61 73 20 53 47 00 2d 2d 69 6d 70 6f 72 74 p.StringRewriting.as.SG.--import
2e20 20 4d 61 74 68 2e 41 6c 67 65 62 72 61 2e 47 72 6f 75 70 2e 53 63 68 72 65 69 65 72 53 69 6d 73 .Math.Algebra.Group.SchreierSims
2e40 20 61 73 20 53 53 00 69 6d 70 6f 72 74 20 4d 61 74 68 2e 41 6c 67 65 62 72 61 2e 47 72 6f 75 70 .as.SS.import.Math.Algebra.Group
2e60 2e 50 65 72 6d 75 74 61 74 69 6f 6e 47 72 6f 75 70 20 68 69 64 69 6e 67 20 28 65 6c 74 73 2c 20 .PermutationGroup.hiding.(elts,.
2e80 6f 72 64 65 72 2c 20 63 6c 6f 73 75 72 65 29 00 69 6d 70 6f 72 74 20 4d 61 74 68 2e 41 6c 67 65 order,.closure).import.Math.Alge
2ea0 62 72 61 2e 4c 69 6e 65 61 72 41 6c 67 65 62 72 61 00 00 69 6d 70 6f 72 74 20 71 75 61 6c 69 66 bra.LinearAlgebra..import.qualif
2ec0 69 65 64 20 44 61 74 61 2e 53 65 74 20 61 73 20 53 00 69 6d 70 6f 72 74 20 71 75 61 6c 69 66 69 ied.Data.Set.as.S.import.qualifi
2ee0 65 64 20 44 61 74 61 2e 4c 69 73 74 20 61 73 20 4c 00 69 6d 70 6f 72 74 20 44 61 74 61 2e 4d 61 ed.Data.List.as.L.import.Data.Ma
2f00 79 62 65 00 69 6d 70 6f 72 74 20 44 61 74 61 2e 4c 69 73 74 00 69 6d 70 6f 72 74 20 44 61 74 61 ybe.import.Data.List.import.Data
2f20 2e 52 61 74 69 6f 00 00 69 6d 70 6f 72 74 20 50 72 65 6c 75 64 65 20 68 69 64 69 6e 67 20 28 20 .Ratio..import.Prelude.hiding.(.
2f40 28 2a 3e 29 20 29 00 00 6d 6f 64 75 6c 65 20 4d 61 74 68 2e 50 72 6f 6a 65 63 74 73 2e 52 6f 6f (*>).)..module.Math.Projects.Roo
2f60 74 53 79 73 74 65 6d 20 77 68 65 72 65 00 00 2d 2d 20 43 6f 70 79 72 69 67 68 74 20 28 63 29 20 tSystem.where..--.Copyright.(c).
2f80 44 61 76 69 64 20 41 6d 6f 73 2c 20 32 30 30 38 2d 32 30 31 35 2e 20 41 6c 6c 20 72 69 67 68 74 David.Amos,.2008-2015..All.right
2fa0 73 20 72 65 73 65 72 76 65 64 2e 00 2d 2d 20 43 6f 70 79 72 69 67 68 74 20 28 63 29 20 59 75 63 s.reserved..--.Copyright.(c).Yuc
2fc0 68 65 6e 20 50 65 69 2c 20 32 30 31 37 2e 20 28 41 64 64 65 64 20 70 6f 73 69 74 69 76 65 20 72 hen.Pei,.2017..(Added.positive.r
2fe0 6f 6f 74 73 20 61 6e 64 20 68 69 67 68 65 73 74 20 65 6c 65 6d 65 6e 74 73 20 65 74 63 2e 29 00 oots.and.highest.elements.etc.).
3000 61 64 00 00 0e 06 00 00 0a 07 00 00 00 10 00 00 38 00 00 00 00 00 00 00 b1 0f 00 00 94 0f 00 00 ad..............8...............
3020 62 0f 00 00 49 0f 00 00 1e 0f 00 00 ec 0e 00 00 aa 0e 00 00 68 0e 00 00 1f 0e 00 00 d6 0d 00 00 b...I...............h...........
3040 a4 0d 00 00 8b 0d 00 00 5b 0d 00 00 22 0d 00 00 e0 0c 00 00 9e 0c 00 00 55 0c 00 00 0c 0c 00 00 ........[..."...........U.......
3060 f2 0b 00 00 c3 0b 00 00 91 0b 00 00 58 0b 00 00 1f 0b 00 00 06 0b 00 00 dd 0a 00 00 c4 0a 00 00 ............X...................
3080 7c 0a 00 00 02 0a 00 00 01 0a 00 00 00 0a 00 00 ff 09 00 00 e0 09 00 00 b4 09 00 00 94 09 00 00 |...............................
30a0 72 09 00 00 50 09 00 00 2a 09 00 00 0c 09 00 00 e9 08 00 00 c4 08 00 00 aa 08 00 00 97 08 00 00 r...P...*.......................
30c0 96 08 00 00 95 08 00 00 6e 08 00 00 6d 08 00 00 6c 08 00 00 69 08 00 00 4d 08 00 00 02 08 00 00 ........n...m...l...i...M.......
30e0 ae 07 00 00 ad 07 00 00 6b 07 00 00 11 07 00 00 0e 07 00 00 0a 07 00 00 09 07 00 00 00 00 00 00 ........k.......................
3100 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3120 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3140 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3160 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3180 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
31a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
31c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
31e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3200 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3220 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3240 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3260 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3280 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
32a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
32c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
32e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3300 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3320 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3340 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3360 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3380 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
33a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
33c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
33e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3400 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3420 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3440 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3460 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3480 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
34a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
34c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
34e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3500 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3520 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3540 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3560 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3580 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
35a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
35c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
35e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3600 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3620 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3640 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3660 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3680 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
36a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
36c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
36e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3700 00 00 00 00 00 00 00 00 00 00 2d 2d 7d 00 2d 7d 00 20 20 20 20 5b 28 41 2c 33 29 2c 28 41 2c 34 ..........--}.-}.....[(A,3),(A,4
3720 29 2c 28 41 2c 35 29 2c 28 42 2c 33 29 2c 28 42 2c 34 29 2c 28 42 2c 35 29 2c 28 43 2c 33 29 2c ),(A,5),(B,3),(B,4),(B,5),(C,3),
3740 28 43 2c 34 29 2c 28 43 2c 35 29 2c 28 44 2c 34 29 2c 28 44 2c 35 29 2c 28 45 2c 36 29 2c 28 46 (C,4),(C,5),(D,4),(D,5),(E,6),(F
3760 2c 34 29 2c 28 47 2c 32 29 5d 00 74 65 73 74 32 20 3d 20 61 6c 6c 20 28 5c 28 74 2c 6e 29 20 2d ,4),(G,2)].test2.=.all.(\(t,n).-
3780 3e 20 6f 72 64 65 72 57 65 79 6c 20 74 20 6e 20 3d 3d 20 53 53 2e 6f 72 64 65 72 20 28 77 65 79 >.orderWeyl.t.n.==.SS.order.(wey
37a0 6c 50 65 72 6d 73 20 74 20 6e 29 29 00 00 20 20 20 20 5b 28 41 2c 33 29 2c 28 41 2c 34 29 2c 28 lPerms.t.n))......[(A,3),(A,4),(
37c0 41 2c 35 29 2c 28 42 2c 33 29 2c 28 42 2c 34 29 2c 28 42 2c 35 29 2c 28 43 2c 33 29 2c 28 43 2c A,5),(B,3),(B,4),(B,5),(C,3),(C,
37e0 34 29 2c 28 43 2c 35 29 2c 28 44 2c 34 29 2c 28 44 2c 35 29 2c 28 46 2c 34 29 2c 28 47 2c 32 29 4),(C,5),(D,4),(D,5),(F,4),(G,2)
3800 5d 00 74 65 73 74 31 20 3d 20 61 6c 6c 20 28 5c 28 74 2c 6e 29 20 2d 3e 20 6f 72 64 65 72 57 65 ].test1.=.all.(\(t,n).->.orderWe
3820 79 6c 20 74 20 6e 20 3d 3d 20 4c 2e 67 65 6e 65 72 69 63 4c 65 6e 67 74 68 20 28 65 6c 74 73 43 yl.t.n.==.L.genericLength.(eltsC
3840 6f 78 65 74 65 72 20 74 20 6e 29 29 00 2d 2d 20 6e 6f 77 20 6d 6f 76 65 64 20 74 6f 20 54 52 6f oxeter.t.n)).--.now.moved.to.TRo
3860 6f 74 53 79 73 74 65 6d 00 7b 2d 00 00 00 66 61 63 74 6f 72 69 61 6c 20 6e 20 3d 20 70 72 6f 64 otSystem.{-...factorial.n.=.prod
3880 75 63 74 20 5b 31 2e 2e 74 6f 49 6e 74 65 67 65 72 20 6e 5d 00 00 00 6f 72 64 65 72 57 65 79 6c uct.[1..toInteger.n]...orderWeyl
38a0 20 47 20 32 20 3d 20 31 32 00 6f 72 64 65 72 57 65 79 6c 20 46 20 34 20 3d 20 32 5e 37 20 2a 20 .G.2.=.12.orderWeyl.F.4.=.2^7.*.
38c0 33 5e 32 00 6f 72 64 65 72 57 65 79 6c 20 45 20 38 20 3d 20 32 5e 31 34 20 2a 20 33 5e 35 20 2a 3^2.orderWeyl.E.8.=.2^14.*.3^5.*
38e0 20 35 5e 32 20 2a 20 37 00 6f 72 64 65 72 57 65 79 6c 20 45 20 37 20 3d 20 32 5e 31 30 20 2a 20 .5^2.*.7.orderWeyl.E.7.=.2^10.*.
3900 33 5e 34 20 2a 20 35 20 2a 20 37 00 6f 72 64 65 72 57 65 79 6c 20 45 20 36 20 3d 20 32 5e 37 20 3^4.*.5.*.7.orderWeyl.E.6.=.2^7.
3920 2a 20 33 5e 34 20 2a 20 35 00 6f 72 64 65 72 57 65 79 6c 20 44 20 6e 20 3d 20 32 5e 28 6e 2d 31 *.3^4.*.5.orderWeyl.D.n.=.2^(n-1
3940 29 20 2a 20 66 61 63 74 6f 72 69 61 6c 20 6e 00 6f 72 64 65 72 57 65 79 6c 20 43 20 6e 20 3d 20 ).*.factorial.n.orderWeyl.C.n.=.
3960 32 5e 6e 20 2a 20 66 61 63 74 6f 72 69 61 6c 20 6e 00 6f 72 64 65 72 57 65 79 6c 20 42 20 6e 20 2^n.*.factorial.n.orderWeyl.B.n.
3980 3d 20 32 5e 6e 20 2a 20 66 61 63 74 6f 72 69 61 6c 20 6e 00 6f 72 64 65 72 57 65 79 6c 20 41 20 =.2^n.*.factorial.n.orderWeyl.A.
39a0 6e 20 3d 20 66 61 63 74 6f 72 69 61 6c 20 28 6e 2b 31 29 00 2d 2d 20 6f 72 64 65 72 57 65 79 6c n.=.factorial.(n+1).--.orderWeyl
39c0 20 74 20 6e 20 3d 3d 20 53 2e 6f 72 64 65 72 20 28 77 65 79 6c 50 65 72 6d 73 20 74 20 6e 29 00 .t.n.==.S.order.(weylPerms.t.n).
39e0 2d 2d 20 54 68 65 20 6f 72 64 65 72 20 6f 66 20 74 68 65 20 57 65 79 6c 20 67 72 6f 75 70 00 00 --.The.order.of.the.Weyl.group..
3a00 00 00 20 20 20 20 20 20 20 20 20 20 6c 6f 6e 67 52 6f 6f 74 73 20 3d 20 63 6f 6e 63 61 74 4d 61 ............longRoots.=.concatMa
3a20 70 20 28 5c 72 2d 3e 20 5b 72 2c 5b 5d 20 3c 2d 3e 20 72 5d 29 20 5b 32 20 2a 3e 20 65 20 69 20 p.(\r->.[r,[].<->.r]).[2.*>.e.i.
3a40 3c 2d 3e 20 65 20 6a 20 3c 2d 3e 20 65 20 6b 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 33 5d 2c 20 5b <->.e.j.<->.e.k.|.i.<-.[1..3],.[
3a60 6a 2c 6b 5d 20 3c 2d 20 5b 5b 31 2e 2e 33 5d 20 4c 2e 5c 5c 20 5b 69 5d 5d 20 5d 00 20 20 20 20 j,k].<-.[[1..3].L.\\.[i]].].....
3a80 20 20 20 20 20 20 73 68 6f 72 74 52 6f 6f 74 73 20 3d 20 5b 65 20 69 20 3c 2d 3e 20 65 20 6a 20 ......shortRoots.=.[e.i.<->.e.j.
3aa0 7c 20 69 20 3c 2d 20 5b 31 2e 2e 33 5d 2c 20 6a 20 3c 2d 20 5b 31 2e 2e 33 5d 2c 20 69 20 2f 3d |.i.<-.[1..3],.j.<-.[1..3],.i./=
3ac0 20 6a 5d 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 33 00 72 6f 6f .j].....where.e.=.basisElt.3.roo
3ae0 74 53 79 73 74 65 6d 20 47 20 32 20 3d 20 73 68 6f 72 74 52 6f 6f 74 73 20 2b 2b 20 6c 6f 6e 67 tSystem.G.2.=.shortRoots.++.long
3b00 52 6f 6f 74 73 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 6e 00 20 Roots.....where.e.=.basisElt.n..
3b20 20 20 20 2b 2b 20 5b 5b 5d 20 3c 2d 3e 20 65 20 69 20 3c 2d 3e 20 65 20 6a 20 7c 20 69 20 3c 2d ...++.[[].<->.e.i.<->.e.j.|.i.<-
3b40 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 20 20 20 2b 2b 20 5b .[1..n],.j.<-.[i+1..n]].....++.[
3b60 5b 5d 20 3c 2d 3e 20 65 20 69 20 3c 2b 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d [].<->.e.i.<+>.e.j.|.i.<-.[1..n]
3b80 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 20 20 20 2b 2b 20 5b 65 20 69 20 3c 2d 3e ,.j.<-.[i+1..n]].....++.[e.i.<->
3ba0 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e .e.j.|.i.<-.[1..n],.j.<-.[i+1..n
3bc0 5d 5d 00 20 20 20 20 5b 65 20 69 20 3c 2b 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e ]].....[e.i.<+>.e.j.|.i.<-.[1..n
3be0 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 72 6f 6f 74 53 79 73 74 65 6d 20 44 20 6e ],.j.<-.[i+1..n]].rootSystem.D.n
3c00 20 7c 20 6e 20 3e 3d 20 34 20 3d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 .|.n.>=.4.=.....................
3c20 2b 2b 20 5b 5b 5d 20 3c 2d 3e 20 65 20 69 20 3c 2d 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 ++.[[].<->.e.i.<->.e.j.|.i.<-.[1
3c40 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 20 20 20 20 20 20 20 20 20 20 ..n],.j.<-.[i+1..n]]............
3c60 20 20 20 20 20 20 20 20 20 2b 2b 20 5b 5b 5d 20 3c 2d 3e 20 65 20 69 20 3c 2b 3e 20 65 20 6a 20 .........++.[[].<->.e.i.<+>.e.j.
3c80 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 20 |.i.<-.[1..n],.j.<-.[i+1..n]]...
3ca0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2b 2b 20 5b 65 20 69 20 3c 2d 3e 20 65 20 ..................++.[e.i.<->.e.
3cc0 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 j.|.i.<-.[1..n],.j.<-.[i+1..n]].
3ce0 20 20 20 20 20 20 20 20 20 20 73 68 6f 72 74 52 6f 6f 74 73 20 3d 20 5b 65 20 69 20 3c 2b 3e 20 ..........shortRoots.=.[e.i.<+>.
3d00 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d e.j.|.i.<-.[1..n],.j.<-.[i+1..n]
3d20 5d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2b 2b 20 5b 5b 5d 20 3c 2d 3e ].....................++.[[].<->
3d40 20 28 32 20 2a 3e 20 65 20 69 29 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 5d 00 20 20 20 20 20 .(2.*>.e.i).|.i.<-.[1..n]]......
3d60 20 20 20 20 20 6c 6f 6e 67 52 6f 6f 74 73 20 20 3d 20 5b 32 20 2a 3e 20 65 20 69 20 7c 20 69 20 .....longRoots..=.[2.*>.e.i.|.i.
3d80 3c 2d 20 5b 31 2e 2e 6e 5d 5d 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c <-.[1..n]].....where.e.=.basisEl
3da0 74 20 6e 00 72 6f 6f 74 53 79 73 74 65 6d 20 43 20 6e 20 7c 20 6e 20 3e 3d 20 32 20 3d 20 6c 6f t.n.rootSystem.C.n.|.n.>=.2.=.lo
3dc0 6e 67 52 6f 6f 74 73 20 2b 2b 20 73 68 6f 72 74 52 6f 6f 74 73 00 20 20 20 20 20 20 20 20 20 20 ngRoots.++.shortRoots...........
3de0 20 20 20 20 20 20 20 20 20 20 2b 2b 20 5b 5b 5d 20 3c 2d 3e 20 65 20 69 20 3c 2d 3e 20 65 20 6a ..........++.[[].<->.e.i.<->.e.j
3e00 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 .|.i.<-.[1..n],.j.<-.[i+1..n]]..
3e20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2b 2b 20 5b 5b 5d 20 3c 2d 3e 20 65 20 ...................++.[[].<->.e.
3e40 69 20 3c 2b 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 5b 69 i.<+>.e.j.|.i.<-.[1..n],.j.<-.[i
3e60 2b 31 2e 2e 6e 5d 5d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2b 2b 20 5b +1..n]].....................++.[
3e80 65 20 69 20 3c 2d 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c 2d 20 e.i.<->.e.j.|.i.<-.[1..n],.j.<-.
3ea0 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 20 20 20 20 20 20 20 20 20 6c 6f 6e 67 52 6f 6f 74 73 20 20 3d [i+1..n]]...........longRoots..=
3ec0 20 5b 65 20 69 20 3c 2b 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 2c 20 6a 20 3c .[e.i.<+>.e.j.|.i.<-.[1..n],.j.<
3ee0 2d 20 5b 69 2b 31 2e 2e 6e 5d 5d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 -.[i+1..n]].....................
3f00 2b 2b 20 5b 5b 5d 20 3c 2d 3e 20 65 20 69 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e 6e 5d 5d 00 20 20 ++.[[].<->.e.i.|.i.<-.[1..n]]...
3f20 20 20 20 20 20 20 20 20 73 68 6f 72 74 52 6f 6f 74 73 20 3d 20 5b 65 20 69 20 7c 20 69 20 3c 2d ........shortRoots.=.[e.i.|.i.<-
3f40 20 5b 31 2e 2e 6e 5d 5d 00 20 20 20 20 77 68 65 72 65 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 .[1..n]].....where.e.=.basisElt.
3f60 6e 00 72 6f 6f 74 53 79 73 74 65 6d 20 42 20 6e 20 7c 20 6e 20 3e 3d 20 32 20 3d 20 73 68 6f 72 n.rootSystem.B.n.|.n.>=.2.=.shor
3f80 74 52 6f 6f 74 73 20 2b 2b 20 6c 6f 6e 67 52 6f 6f 74 73 00 20 20 20 20 77 68 65 72 65 20 65 20 tRoots.++.longRoots.....where.e.
3fa0 3d 20 62 61 73 69 73 45 6c 74 20 28 6e 2b 31 29 00 72 6f 6f 74 53 79 73 74 65 6d 20 41 20 6e 20 =.basisElt.(n+1).rootSystem.A.n.
3fc0 7c 20 6e 20 3e 3d 20 31 20 3d 20 5b 65 20 69 20 3c 2d 3e 20 65 20 6a 20 7c 20 69 20 3c 2d 20 5b |.n.>=.1.=.[e.i.<->.e.j.|.i.<-.[
3fe0 31 2e 2e 6e 2b 31 5d 2c 20 6a 20 3c 2d 20 5b 31 2e 2e 6e 2b 31 5d 2c 20 69 20 2f 3d 20 6a 5d 00 1..n+1],.j.<-.[1..n+1],.i./=.j].
4000 61 64 00 00 10 00 00 00 c4 01 00 00 00 10 00 00 66 00 00 00 00 00 00 00 c1 0f 00 00 77 0f 00 00 ad..............f...........w...
4020 76 0f 00 00 75 0f 00 00 44 0f 00 00 43 0f 00 00 f2 0e 00 00 cf 0e 00 00 8f 0e 00 00 70 0e 00 00 v...u...D...C...............p...
4040 29 0e 00 00 e4 0d 00 00 93 0d 00 00 92 0d 00 00 52 0d 00 00 fa 0c 00 00 e4 0c 00 00 e3 0c 00 00 )...............R...............
4060 c0 0c 00 00 68 0c 00 00 4c 0c 00 00 fb 0b 00 00 fa 0b 00 00 c2 0b 00 00 c1 0b 00 00 49 0b 00 00 ....h...L...................I...
4080 29 0b 00 00 0c 0b 00 00 ef 0a 00 00 d2 0a 00 00 b5 0a 00 00 7f 0a 00 00 52 0a 00 00 51 0a 00 00 ).......................R...Q...
40a0 ff 09 00 00 c5 09 00 00 c4 09 00 00 c3 09 00 00 69 09 00 00 2f 09 00 00 0a 09 00 00 f8 08 00 00 ................i.../...........
40c0 e1 08 00 00 cd 08 00 00 b9 08 00 00 3e 08 00 00 fd 07 00 00 fc 07 00 00 b7 07 00 00 91 07 00 00 ............>...................
40e0 7f 07 00 00 68 07 00 00 54 07 00 00 40 07 00 00 c5 06 00 00 70 06 00 00 6f 06 00 00 6e 06 00 00 ....h...T...@.......p...o...n...
4100 6d 06 00 00 2d 06 00 00 2c 06 00 00 e9 05 00 00 ae 05 00 00 ad 05 00 00 66 05 00 00 65 05 00 00 m...-...,...............f...e...
4120 64 05 00 00 54 05 00 00 53 05 00 00 17 05 00 00 fb 04 00 00 e2 04 00 00 e1 04 00 00 e0 04 00 00 d...T...S.......................
4140 c8 04 00 00 c7 04 00 00 a7 04 00 00 67 04 00 00 24 04 00 00 23 04 00 00 05 04 00 00 ef 03 00 00 ............g...$...#...........
4160 d2 03 00 00 ad 03 00 00 8c 03 00 00 67 03 00 00 46 03 00 00 2d 03 00 00 0d 03 00 00 0c 03 00 00 ............g...F...-...........
4180 0b 03 00 00 00 03 00 00 b0 02 00 00 80 02 00 00 7f 02 00 00 51 02 00 00 50 02 00 00 3d 02 00 00 ....................Q...P...=...
41a0 3c 02 00 00 24 02 00 00 eb 01 00 00 c4 01 00 00 c3 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 <...$...........................
41c0 00 00 00 00 2d 2d 20 72 6f 6f 74 53 79 73 74 65 6d 20 3a 3a 20 54 79 70 65 20 2d 3e 20 49 6e 74 ....--.rootSystem.::.Type.->.Int
41e0 20 2d 3e 20 5b 5b 51 51 5d 5d 00 2d 2d 20 4c 2e 73 6f 72 74 20 28 72 6f 6f 74 53 79 73 74 65 6d .->.[[QQ]].--.L.sort.(rootSystem
4200 20 74 20 6e 29 20 3d 3d 20 63 6c 6f 73 75 72 65 20 28 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 .t.n).==.closure.(simpleSystem.t
4220 20 6e 29 00 2d 2d 20 54 68 65 20 66 75 6c 6c 20 72 6f 6f 74 20 73 79 73 74 65 6d 00 00 2d 2d 20 .n).--.The.full.root.system..--.
4240 48 75 6d 70 68 72 65 79 73 20 70 34 31 66 66 00 00 2d 2d 20 21 21 20 4e 6f 74 20 79 65 74 20 67 Humphreys.p41ff..--.!!.Not.yet.g
4260 6f 74 20 72 6f 6f 74 20 73 79 73 74 65 6d 73 20 66 6f 72 20 45 36 2c 37 2c 38 2c 20 46 34 00 00 ot.root.systems.for.E6,7,8,.F4..
4280 2d 2d 20 66 6f 72 20 63 6f 6d 70 61 72 69 73 6f 6e 20 61 67 61 69 6e 73 74 20 74 68 65 20 63 61 --.for.comparison.against.the.ca
42a0 6c 63 75 6c 61 74 65 64 20 76 61 6c 75 65 73 00 2d 2d 20 54 68 65 20 65 78 70 65 63 74 65 64 20 lculated.values.--.The.expected.
42c0 76 61 6c 75 65 73 20 6f 66 20 74 68 65 20 72 6f 6f 74 20 73 79 73 74 65 6d 2c 20 6e 75 6d 62 65 values.of.the.root.system,.numbe
42e0 72 20 6f 66 20 72 6f 6f 74 73 2c 20 6f 72 64 65 72 20 6f 66 20 57 65 79 6c 20 67 72 6f 75 70 00 r.of.roots,.order.of.Weyl.group.
4300 2d 2d 20 54 45 53 54 49 4e 47 00 00 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 28 69 64 27 --.TESTING..................(id'
4320 20 20 2b 7c 2b 20 7a 4d 78 20 6e 29 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 ..+|+.zMx.n)....................
4340 20 20 2b 2d 2b 00 20 20 20 20 20 20 20 20 20 20 20 69 6e 20 28 7a 4d 78 20 6e 20 2b 7c 2b 20 69 ..+-+............in.(zMx.n.+|+.i
4360 64 4d 78 20 6e 29 00 66 6f 72 6d 20 42 20 6e 20 3d 20 6c 65 74 20 69 64 27 20 3d 20 28 2d 31 29 dMx.n).form.B.n.=.let.id'.=.(-1)
4380 20 2a 3e 3e 20 69 64 4d 78 20 6e 00 20 20 20 20 20 20 20 20 20 20 20 28 6d 61 70 20 28 30 3a 29 .*>>.idMx.n............(map.(0:)
43a0 20 28 66 6f 72 6d 20 44 20 6e 29 29 00 66 6f 72 6d 20 43 20 6e 20 3d 20 28 32 20 3a 20 72 65 70 .(form.D.n)).form.C.n.=.(2.:.rep
43c0 6c 69 63 61 74 65 20 28 32 2a 6e 29 20 30 29 20 3a 00 20 20 20 20 20 20 20 20 20 20 28 69 64 4d licate.(2*n).0).:...........(idM
43e0 78 20 6e 20 2b 7c 2b 20 7a 4d 78 20 6e 29 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 x.n.+|+.zMx.n)..................
4400 20 2b 2d 2b 00 66 6f 72 6d 20 44 20 6e 20 3d 20 28 7a 4d 78 20 6e 20 2b 7c 2b 20 69 64 4d 78 20 .+-+.form.D.n.=.(zMx.n.+|+.idMx.
4420 6e 29 00 00 28 2b 2d 2b 29 20 3d 20 28 2b 2b 29 20 20 20 20 20 20 20 20 20 2d 2d 20 67 6c 75 65 n)..(+-+).=.(++).........--.glue
4440 20 74 77 6f 20 6d 61 74 72 69 63 65 73 20 74 6f 67 65 74 68 65 72 20 61 62 6f 76 65 20 61 6e 64 .two.matrices.together.above.and
4460 20 62 65 6c 6f 77 00 28 2b 7c 2b 29 20 3d 20 7a 69 70 57 69 74 68 20 28 2b 2b 29 20 2d 2d 20 67 .below.(+|+).=.zipWith.(++).--.g
4480 6c 75 65 20 74 77 6f 20 6d 61 74 72 69 63 65 73 20 74 6f 67 65 74 68 65 72 20 73 69 64 65 20 62 lue.two.matrices.together.side.b
44a0 79 20 73 69 64 65 00 2d 2d 20 66 6f 72 20 67 6c 75 69 6e 67 20 6d 61 74 72 69 63 65 73 20 74 6f y.side.--.for.gluing.matrices.to
44c0 67 65 74 68 65 72 00 00 6c 69 65 4d 75 6c 74 20 78 20 79 20 3d 20 78 2a 79 20 2d 20 79 2a 78 00 gether..lieMult.x.y.=.x*y.-.y*x.
44e0 00 00 20 20 20 20 20 20 20 20 20 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 6e 00 20 20 20 20 77 ............e.=.basisElt.n.....w
4500 68 65 72 65 20 7a 20 3d 20 72 65 70 6c 69 63 61 74 65 20 6e 20 30 00 65 6c 65 6d 4d 78 20 6e 20 here.z.=.replicate.n.0.elemMx.n.
4520 69 20 6a 20 3d 20 72 65 70 6c 69 63 61 74 65 20 28 69 2d 31 29 20 7a 20 2b 2b 20 65 20 6a 20 3a i.j.=.replicate.(i-1).z.++.e.j.:
4540 20 72 65 70 6c 69 63 61 74 65 20 28 6e 2d 69 29 20 7a 00 00 2d 2d 20 4c 49 45 20 41 4c 47 45 42 .replicate.(n-i).z..--.LIE.ALGEB
4560 52 41 53 00 00 00 70 6f 69 6e 63 61 72 65 50 6f 6c 79 20 74 20 6e 20 3d 20 6d 61 70 20 6c 65 6e RAS...poincarePoly.t.n.=.map.len
4580 67 74 68 20 24 20 4c 2e 67 72 6f 75 70 20 24 20 6d 61 70 20 6c 65 6e 67 74 68 20 24 20 65 6c 74 gth.$.L.group.$.map.length.$.elt
45a0 73 43 6f 78 65 74 65 72 20 74 20 6e 00 00 2d 2d 20 69 74 27 73 20 6a 75 73 74 20 73 6c 69 67 68 sCoxeter.t.n..--.it's.just.sligh
45c0 74 6c 79 20 66 61 73 74 65 72 20 74 6f 20 75 73 65 20 74 68 65 20 62 72 61 69 64 20 70 72 65 73 tly.faster.to.use.the.braid.pres
45e0 65 6e 74 61 74 69 6f 6e 00 65 6c 74 73 43 6f 78 65 74 65 72 20 74 20 6e 20 3d 20 53 47 2e 65 6c entation.eltsCoxeter.t.n.=.SG.el
4600 74 73 20 24 20 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 69 78 32 20 24 20 63 6f 78 65 74 65 ts.$.fromCoxeterMatrix2.$.coxete
4620 72 4d 61 74 72 69 78 20 74 20 6e 00 00 63 6f 78 65 74 65 72 50 72 65 73 65 6e 74 61 74 69 6f 6e rMatrix.t.n..coxeterPresentation
4640 20 74 20 6e 20 3d 20 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 69 78 20 24 20 63 6f 78 65 74 .t.n.=.fromCoxeterMatrix.$.coxet
4660 65 72 4d 61 74 72 69 78 20 74 20 6e 00 00 00 00 20 20 20 20 62 72 61 69 64 52 65 6c 61 74 69 6f erMatrix.t.n........braidRelatio
4680 6e 20 69 20 6a 20 6d 20 3d 20 28 74 61 6b 65 20 6d 20 24 20 63 79 63 6c 65 20 5b 73 5f 20 6a 2c n.i.j.m.=.(take.m.$.cycle.[s_.j,
46a0 20 73 5f 20 69 5d 2c 20 74 61 6b 65 20 6d 20 24 20 63 79 63 6c 65 20 5b 73 5f 20 69 2c 20 73 5f .s_.i],.take.m.$.cycle.[s_.i,.s_
46c0 20 6a 5d 29 00 20 20 20 20 72 75 6c 65 73 20 28 28 31 3a 78 73 29 3a 72 73 29 20 69 20 3d 20 28 .j]).....rules.((1:xs):rs).i.=.(
46e0 5b 73 5f 20 69 2c 20 73 5f 20 69 5d 2c 5b 5d 29 20 3a 20 5b 62 72 61 69 64 52 65 6c 61 74 69 6f [s_.i,.s_.i],[]).:.[braidRelatio
4700 6e 20 69 20 6a 20 6d 20 7c 20 28 6a 2c 6d 29 20 3c 2d 20 7a 69 70 20 5b 69 2b 31 2e 2e 5d 20 78 n.i.j.m.|.(j,m).<-.zip.[i+1..].x
4720 73 5d 20 2b 2b 20 72 75 6c 65 73 20 28 6d 61 70 20 74 61 69 6c 20 72 73 29 20 28 69 2b 31 29 00 s].++.rules.(map.tail.rs).(i+1).
4740 20 20 20 20 72 75 6c 65 73 20 5b 5d 20 5f 20 3d 20 5b 5d 00 20 20 20 20 72 73 20 3d 20 72 75 6c ....rules.[]._.=.[].....rs.=.rul
4760 65 73 20 6d 78 20 31 00 20 20 20 20 67 73 20 3d 20 6d 61 70 20 73 5f 20 5b 31 2e 2e 6e 5d 00 20 es.mx.1.....gs.=.map.s_.[1..n]..
4780 20 20 20 6e 20 3d 20 6c 65 6e 67 74 68 20 6d 78 00 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 ...n.=.length.mx.fromCoxeterMatr
47a0 69 78 32 20 6d 78 20 3d 20 28 67 73 2c 72 73 29 20 77 68 65 72 65 00 2d 2d 20 41 6e 6f 74 68 65 ix2.mx.=.(gs,rs).where.--.Anothe
47c0 72 20 70 72 65 73 65 6e 74 61 74 69 6f 6e 20 66 6f 72 20 74 68 65 20 43 6f 78 65 74 65 72 20 67 r.presentation.for.the.Coxeter.g
47e0 72 6f 75 70 2c 20 75 73 69 6e 67 20 62 72 61 69 64 20 72 65 6c 61 74 69 6f 6e 73 00 00 20 20 20 roup,.using.braid.relations.....
4800 20 70 6f 77 65 72 52 65 6c 61 74 69 6f 6e 20 69 20 6a 20 6d 20 3d 20 28 63 6f 6e 63 61 74 20 24 .powerRelation.i.j.m.=.(concat.$
4820 20 72 65 70 6c 69 63 61 74 65 20 6d 20 5b 73 5f 20 69 2c 20 73 5f 20 6a 5d 2c 5b 5d 29 00 20 20 .replicate.m.[s_.i,.s_.j],[])...
4840 20 20 72 75 6c 65 73 20 28 28 31 3a 78 73 29 3a 72 73 29 20 69 20 3d 20 28 5b 73 5f 20 69 2c 20 ..rules.((1:xs):rs).i.=.([s_.i,.
4860 73 5f 20 69 5d 2c 5b 5d 29 20 3a 20 5b 70 6f 77 65 72 52 65 6c 61 74 69 6f 6e 20 69 20 6a 20 6d s_.i],[]).:.[powerRelation.i.j.m
4880 20 7c 20 28 6a 2c 6d 29 20 3c 2d 20 7a 69 70 20 5b 69 2b 31 2e 2e 5d 20 78 73 5d 20 2b 2b 20 72 .|.(j,m).<-.zip.[i+1..].xs].++.r
48a0 75 6c 65 73 20 28 6d 61 70 20 74 61 69 6c 20 72 73 29 20 28 69 2b 31 29 00 20 20 20 20 72 75 6c ules.(map.tail.rs).(i+1).....rul
48c0 65 73 20 5b 5d 20 5f 20 3d 20 5b 5d 00 20 20 20 20 72 73 20 3d 20 72 75 6c 65 73 20 6d 78 20 31 es.[]._.=.[].....rs.=.rules.mx.1
48e0 00 20 20 20 20 67 73 20 3d 20 6d 61 70 20 73 5f 20 5b 31 2e 2e 6e 5d 00 20 20 20 20 6e 20 3d 20 .....gs.=.map.s_.[1..n].....n.=.
4900 6c 65 6e 67 74 68 20 6d 78 00 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 69 78 20 6d 78 20 3d length.mx.fromCoxeterMatrix.mx.=
4920 20 28 67 73 2c 72 73 29 20 77 68 65 72 65 00 2d 2d 20 57 65 20 61 73 73 75 6d 65 20 62 75 74 20 .(gs,rs).where.--.We.assume.but.
4940 64 6f 6e 27 74 20 63 68 65 63 6b 20 74 68 61 74 20 6d 69 69 20 3d 3d 20 31 20 61 6e 64 20 6d 69 don't.check.that.mii.==.1.and.mi
4960 6a 20 3d 3d 20 6d 6a 69 00 2d 2d 20 47 69 76 65 6e 20 74 68 65 20 6d 61 74 72 69 78 20 6f 66 20 j.==.mji.--.Given.the.matrix.of.
4980 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6d 69 6a 2c 20 72 65 74 75 72 6e 20 74 68 65 20 43 6f 78 coefficients.mij,.return.the.Cox
49a0 65 74 65 72 20 67 72 6f 75 70 20 3c 73 69 20 7c 20 73 69 5e 32 2c 20 28 73 69 73 6a 29 5e 6d 69 eter.group.<si.|.si^2,.(sisj)^mi
49c0 6a 3e 00 00 00 63 6f 78 65 74 65 72 4d 61 74 72 69 78 20 74 20 6e 20 3d 20 63 6f 78 65 74 65 72 j>...coxeterMatrix.t.n.=.coxeter
49e0 46 72 6f 6d 44 79 6e 6b 69 6e 20 24 20 64 79 6e 6b 69 6e 44 69 61 67 72 61 6d 20 74 20 6e 00 2d FromDynkin.$.dynkinDiagram.t.n.-
4a00 2d 20 54 68 65 20 6d 69 6a 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6f 66 20 74 68 65 20 43 6f -.The.mij.coefficients.of.the.Co
4a20 78 65 74 65 72 20 67 72 6f 75 70 20 3c 73 69 20 7c 20 73 69 5e 32 2c 20 28 73 69 73 6a 29 5e 6d xeter.group.<si.|.si^2,.(sisj)^m
4a40 69 6a 3e 2c 20 61 73 20 61 20 6d 61 74 72 69 78 00 00 20 20 20 20 77 68 65 72 65 20 66 20 30 20 ij>,.as.a.matrix......where.f.0.
4a60 3d 20 32 3b 20 66 20 31 20 3d 20 33 3b 20 66 20 32 20 3d 20 34 3b 20 66 20 33 20 3d 20 36 00 63 =.2;.f.1.=.3;.f.2.=.4;.f.3.=.6.c
4a80 6f 78 65 74 65 72 46 72 6f 6d 44 79 6e 6b 69 6e 20 6e 69 6a 20 3d 20 73 65 74 44 69 61 67 20 31 oxeterFromDynkin.nij.=.setDiag.1
4aa0 20 24 20 28 6d 61 70 20 2e 20 6d 61 70 29 20 66 20 6e 69 6a 00 2d 2d 20 6e 69 6a 20 3d 3d 20 33 .$.(map...map).f.nij.--.nij.==.3
4ac0 20 3c 3d 3e 20 74 68 65 74 61 20 3d 20 70 69 2f 36 00 2d 2d 20 6e 69 6a 20 3d 3d 20 32 20 3c 3d .<=>.theta.=.pi/6.--.nij.==.2.<=
4ae0 3e 20 74 68 65 74 61 20 3d 20 70 69 2f 34 00 2d 2d 20 6e 69 6a 20 3d 3d 20 31 20 3c 3d 3e 20 74 >.theta.=.pi/4.--.nij.==.1.<=>.t
4b00 68 65 74 61 20 3d 20 70 69 2f 33 00 2d 2d 20 6e 69 6a 20 3d 3d 20 30 20 3c 3d 3e 20 74 68 65 74 heta.=.pi/3.--.nij.==.0.<=>.thet
4b20 61 20 3d 20 70 69 2f 32 00 2d 2d 20 75 73 69 6e 67 20 6e 69 6a 20 3d 20 34 20 63 6f 73 5e 32 20 a.=.pi/2.--.using.nij.=.4.cos^2.
4b40 74 68 65 74 61 5f 69 6a 00 2d 2d 20 67 69 76 65 6e 20 74 68 65 20 44 79 6e 6b 69 6e 20 64 69 61 theta_ij.--.given.the.Dynkin.dia
4b60 67 72 61 6d 20 6e 69 6a 2c 20 64 65 72 69 76 65 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 gram.nij,.derive.the.coefficient
4b80 73 20 6d 69 6a 20 6f 66 20 74 68 65 20 43 6f 78 65 74 65 72 20 67 72 6f 75 70 20 3c 73 69 20 7c s.mij.of.the.Coxeter.group.<si.|
4ba0 20 73 69 5e 32 2c 20 28 73 69 73 6a 29 5e 6d 69 6a 3e 20 28 73 6f 20 6d 69 69 20 3d 3d 20 31 29 .si^2,.(sisj)^mij>.(so.mii.==.1)
4bc0 00 00 64 79 6e 6b 69 6e 44 69 61 67 72 61 6d 20 74 20 6e 20 3d 20 64 79 6e 6b 69 6e 46 72 6f 6d ..dynkinDiagram.t.n.=.dynkinFrom
4be0 43 61 72 74 61 6e 20 24 20 63 61 72 74 61 6e 4d 61 74 72 69 78 20 74 20 6e 00 00 64 79 6e 6b 69 Cartan.$.cartanMatrix.t.n..dynki
4c00 6e 46 72 6f 6d 43 61 72 74 61 6e 20 61 69 6a 20 3d 20 73 65 74 44 69 61 67 20 30 20 24 20 28 7a nFromCartan.aij.=.setDiag.0.$.(z
4c20 69 70 57 69 74 68 20 2e 20 7a 69 70 57 69 74 68 29 20 28 2a 29 20 61 69 6a 20 28 4c 2e 74 72 61 ipWith...zipWith).(*).aij.(L.tra
4c40 6e 73 70 6f 73 65 20 61 69 6a 29 00 2d 2d 20 6e 69 6a 20 3d 20 41 69 6a 20 2a 20 41 6a 69 2c 20 nspose.aij).--.nij.=.Aij.*.Aji,.
4c60 6e 69 69 20 3d 20 30 00 2d 2d 20 67 69 76 65 6e 20 61 20 43 61 72 74 61 6e 20 6d 61 74 72 69 78 nii.=.0.--.given.a.Cartan.matrix
4c80 2c 20 64 65 72 69 76 65 20 74 68 65 20 63 6f 72 72 65 73 70 6f 6e 64 69 6e 67 20 6d 61 74 72 69 ,.derive.the.corresponding.matri
4ca0 78 20 64 65 73 63 72 69 62 69 6e 67 20 74 68 65 20 44 79 6e 6b 69 6e 20 64 69 61 67 72 61 6d 00 x.describing.the.Dynkin.diagram.
4cc0 2d 2d 20 43 61 72 74 65 72 2c 20 53 65 67 61 6c 2c 20 4d 61 63 64 6f 6e 61 6c 64 20 70 31 37 2d --.Carter,.Segal,.Macdonald.p17-
4ce0 31 38 00 00 73 65 74 44 69 61 67 20 5f 20 5b 5b 5d 5d 20 3d 20 5b 5b 5d 5d 00 73 65 74 44 69 61 18..setDiag._.[[]].=.[[]].setDia
4d00 67 20 63 20 6d 78 40 28 28 78 3a 78 73 29 3a 72 73 29 20 3d 20 28 63 3a 78 73 29 20 3a 20 7a 69 g.c.mx@((x:xs):rs).=.(c:xs).:.zi
4d20 70 57 69 74 68 20 28 3a 29 20 28 6d 61 70 20 68 65 61 64 20 72 73 29 20 28 73 65 74 44 69 61 67 pWith.(:).(map.head.rs).(setDiag
4d40 20 63 20 24 20 6d 61 70 20 74 61 69 6c 20 72 73 29 00 2d 2d 20 73 65 74 20 74 68 65 20 64 69 61 .c.$.map.tail.rs).--.set.the.dia
4d60 67 6f 6e 61 6c 20 65 6e 74 72 69 65 73 20 6f 66 20 28 73 71 75 61 72 65 29 20 6d 61 74 72 69 78 gonal.entries.of.(square).matrix
4d80 20 6d 78 20 74 6f 20 63 6f 6e 73 74 61 6e 74 20 63 00 00 2d 2d 20 28 53 6f 20 70 72 6f 62 61 62 .mx.to.constant.c..--.(So.probab
4da0 6c 79 20 43 61 72 74 65 72 20 64 65 66 69 6e 65 73 20 74 68 65 20 72 6f 6f 74 73 20 6f 66 20 47 ly.Carter.defines.the.roots.of.G
4dc0 32 20 74 68 65 20 6f 74 68 65 72 20 77 61 79 20 72 6f 75 6e 64 20 74 6f 20 48 75 6d 70 68 72 65 2.the.other.way.round.to.Humphre
4de0 79 73 29 00 2d 2d 20 54 68 65 79 20 61 67 72 65 65 20 77 69 74 68 20 6f 75 72 20 61 6e 73 77 65 ys).--.They.agree.with.our.answe
4e00 72 73 20 65 78 63 65 70 74 20 66 6f 72 20 47 32 2c 20 77 68 69 63 68 20 69 73 20 74 68 65 20 74 rs.except.for.G2,.which.is.the.t
4e20 72 61 6e 73 70 6f 73 65 00 2d 2d 20 43 61 72 74 65 72 2c 20 53 69 6d 70 6c 65 20 47 72 6f 75 70 ranspose.--.Carter,.Simple.Group
4e40 73 20 6f 66 20 4c 69 65 20 54 79 70 65 2c 20 70 34 34 2d 35 20 67 69 76 65 73 20 74 68 65 20 65 s.of.Lie.Type,.p44-5.gives.the.e
4e60 78 70 65 63 74 65 64 20 61 6e 73 77 65 72 73 00 2d 2d 20 54 68 6f 73 65 20 6f 66 20 42 2c 20 43 xpected.answers.--.Those.of.B,.C
4e80 2c 20 46 2c 20 47 20 61 72 65 20 6e 6f 74 00 2d 2d 20 4e 6f 74 65 3a 20 54 68 65 20 43 61 72 74 ,.F,.G.are.not.--.Note:.The.Cart
4ea0 61 6e 20 6d 61 74 72 69 63 65 73 20 66 6f 72 20 41 2c 20 44 2c 20 45 20 73 79 73 74 65 6d 73 20 an.matrices.for.A,.D,.E.systems.
4ec0 61 72 65 20 73 79 6d 6d 65 74 72 69 63 2e 00 20 20 20 20 77 68 65 72 65 20 72 6f 6f 74 73 20 3d are.symmetric......where.roots.=
4ee0 20 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 20 6e 00 63 61 72 74 61 6e 4d 61 74 72 69 78 20 74 .simpleSystem.t.n.cartanMatrix.t
4f00 20 6e 20 3d 20 5b 5b 32 20 2a 20 28 61 69 20 3c 2e 3e 20 61 6a 29 20 2f 20 28 61 69 20 3c 2e 3e .n.=.[[2.*.(ai.<.>.aj)./.(ai.<.>
4f20 20 61 69 29 20 7c 20 61 6a 20 3c 2d 20 72 6f 6f 74 73 5d 20 7c 20 61 69 20 3c 2d 20 72 6f 6f 74 .ai).|.aj.<-.roots].|.ai.<-.root
4f40 73 5d 00 00 2d 2d 20 43 41 52 54 41 4e 20 4d 41 54 52 49 58 2c 20 44 59 4e 4b 49 4e 20 44 49 41 s]..--.CARTAN.MATRIX,.DYNKIN.DIA
4f60 47 52 41 4d 2c 20 43 4f 58 45 54 45 52 20 53 59 53 54 45 4d 00 00 00 2d 2d 20 68 6f 77 65 76 65 GRAM,.COXETER.SYSTEM...--.howeve
4f80 72 2c 20 72 65 66 6c 65 63 74 69 6f 6e 20 6d 61 74 72 69 63 65 73 20 61 72 65 20 73 79 6d 6d 65 r,.reflection.matrices.are.symme
4fa0 74 72 69 63 2c 20 73 6f 20 74 68 65 79 20 61 6c 73 6f 20 66 6f 72 6d 20 74 68 65 20 72 6f 77 73 tric,.so.they.also.form.the.rows
4fc0 00 2d 2d 20 74 68 65 20 69 6d 61 67 65 73 20 6f 66 20 74 68 65 20 62 61 73 69 73 20 65 6c 74 73 .--.the.images.of.the.basis.elts
4fe0 20 66 6f 72 6d 20 74 68 65 20 63 6f 6c 75 6d 6e 73 20 6f 66 20 74 68 65 20 6d 61 74 72 69 78 00 .form.the.columns.of.the.matrix.
5000 61 64 00 00 2a 06 00 00 66 07 00 00 00 10 00 00 48 00 00 00 00 00 00 00 d2 0f 00 00 b1 0f 00 00 ad..*...f.......H...............
5020 78 0f 00 00 43 0f 00 00 22 0f 00 00 04 0f 00 00 76 0e 00 00 75 0e 00 00 4f 0e 00 00 1e 0e 00 00 x...C...".......v...u...O.......
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6120 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6140 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6160 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6180 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
61a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
61c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
61e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6200 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6220 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6240 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6260 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6280 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
62a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
62c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
62e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6300 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6320 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6340 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6360 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6380 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
63a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
63c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
63e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6400 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6420 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6440 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6460 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6480 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
64a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
64c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
64e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6500 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6520 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6540 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6560 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6580 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
65a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
65c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
65e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6600 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6620 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6640 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6660 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6680 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
66a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
66c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
66e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6700 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6720 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6740 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6760 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6780 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
67a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
67c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
67e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6800 00 00 00 00 00 20 20 20 20 20 20 20 20 20 20 20 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 64 00 .................e.=.basisElt.d.
6820 20 20 20 20 77 68 65 72 65 20 64 20 3d 20 6c 65 6e 20 20 20 20 20 20 20 20 20 20 65 20 3d 20 62 ....where.d.=.len..........e.=.b
6840 61 73 69 73 45 6c 74 20 64 00 20 20 20 20 77 68 65 72 65 20 20 20 20 20 20 20 20 20 20 20 65 20 asisElt.d.....where...........e.
6860 3d 20 62 61 73 69 73 45 6c 74 20 64 00 20 20 20 20 77 68 65 72 65 20 64 20 3d 20 6c 65 6e 67 74 =.basisElt.d.....where.d.=.lengt
6880 68 20 72 20 2d 20 20 20 20 20 20 20 20 20 20 65 20 3d 20 62 61 73 69 73 45 6c 74 20 64 00 20 20 h.r.-..........e.=.basisElt.d...
68a0 20 20 77 68 65 72 65 20 64 20 3d 20 6c 65 6e 67 74 68 20 20 20 20 20 20 20 20 20 20 20 20 65 20 ..where.d.=.length............e.
68c0 3d 20 62 61 73 69 73 45 6c 74 20 64 00 20 20 20 20 77 68 65 72 65 20 64 20 3d 20 6c 65 6e 67 74 =.basisElt.d.....where.d.=.lengt
68e0 68 20 72 20 2d 2d 20 64 69 6d 65 6e 73 69 6f 6e 20 6f 66 20 74 68 65 20 73 70 61 63 65 00 77 4d h.r.--.dimension.of.the.space.wM
6900 78 20 72 20 3d 20 6d 61 70 20 28 77 20 72 29 20 5b 65 20 69 20 7c 20 69 20 3c 2d 20 5b 31 2e 2e x.r.=.map.(w.r).[e.i.|.i.<-.[1..
6920 64 5d 5d 20 2d 2d 20 6d 61 74 72 69 78 20 66 6f 72 20 72 65 66 6c 65 63 74 69 6f 6e 20 69 6e 20 d]].--.matrix.for.reflection.in.
6940 68 79 70 65 72 70 6c 61 6e 65 20 6f 72 74 68 6f 67 6f 6e 61 6c 20 74 6f 20 72 00 2d 2d 20 54 68 hyperplane.orthogonal.to.r.--.Th
6960 65 20 57 65 79 6c 20 67 72 6f 75 70 20 65 6c 65 6d 65 6e 74 20 63 6f 72 72 65 73 70 6f 6e 64 69 e.Weyl.group.element.correspondi
6980 6e 67 20 74 6f 20 61 20 72 6f 6f 74 2c 20 72 65 70 72 65 73 65 6e 74 65 64 20 61 73 20 61 20 6d ng.to.a.root,.represented.as.a.m
69a0 61 74 72 69 78 00 00 77 65 79 6c 4d 61 74 72 69 63 65 73 20 74 20 6e 20 3d 20 6d 61 70 20 77 4d atrix..weylMatrices.t.n.=.map.wM
69c0 78 20 28 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 20 6e 29 00 2d 2d 20 47 65 6e 65 72 61 74 6f x.(simpleSystem.t.n).--.Generato
69e0 72 73 20 6f 66 20 74 68 65 20 57 65 79 6c 20 67 72 6f 75 70 20 61 73 20 61 20 6d 61 74 72 69 78 rs.of.the.Weyl.group.as.a.matrix
6a00 20 67 72 6f 75 70 00 20 20 20 20 00 20 20 20 20 69 6e 20 6d 61 70 20 74 6f 50 65 72 6d 20 72 73 .group..........in.map.toPerm.rs
6a20 00 20 20 20 20 20 20 20 20 74 6f 50 65 72 6d 20 72 20 3d 20 66 72 6f 6d 50 61 69 72 73 20 5b 28 .........toPerm.r.=.fromPairs.[(
6a40 78 2c 20 77 20 72 20 78 29 20 7c 20 78 20 3c 2d 20 78 73 5d 00 20 20 20 20 20 20 20 20 78 73 20 x,.w.r.x).|.x.<-.xs].........xs.
6a60 3d 20 63 6c 6f 73 75 72 65 20 72 73 00 20 20 20 20 6c 65 74 20 72 73 20 3d 20 73 69 6d 70 6c 65 =.closure.rs.....let.rs.=.simple
6a80 53 79 73 74 65 6d 20 74 20 6e 00 77 65 79 6c 50 65 72 6d 73 20 74 20 6e 20 3d 00 2d 2d 20 47 65 System.t.n.weylPerms.t.n.=.--.Ge
6aa0 6e 65 72 61 74 6f 72 73 20 6f 66 20 74 68 65 20 57 65 79 6c 20 67 72 6f 75 70 20 61 73 20 70 65 nerators.of.the.Weyl.group.as.pe
6ac0 72 6d 75 74 61 74 69 6f 6e 20 67 72 6f 75 70 20 6f 6e 20 74 68 65 20 72 6f 6f 74 73 00 00 2d 2d rmutation.group.on.the.roots..--
6ae0 20 54 68 65 20 66 69 6e 69 74 65 20 72 65 66 6c 65 63 74 69 6f 6e 20 67 72 6f 75 70 20 67 65 6e .The.finite.reflection.group.gen
6b00 65 72 61 74 65 64 20 62 79 20 74 68 65 20 72 6f 6f 74 20 73 79 73 74 65 6d 00 2d 2d 20 57 45 59 erated.by.the.root.system.--.WEY
6b20 4c 20 47 52 4f 55 50 00 7b 2d 2d 00 00 00 20 20 20 20 20 20 20 20 20 20 20 20 69 6e 20 63 6c 6f L.GROUP.{--...............in.clo
6b40 73 75 72 65 20 69 6e 74 65 72 69 6f 72 27 20 62 6f 75 6e 64 61 72 79 27 00 20 20 20 20 20 20 20 sure.interior'.boundary'........
6b60 20 20 20 20 20 20 20 20 20 62 6f 75 6e 64 61 72 79 27 20 3d 20 53 2e 66 72 6f 6d 4c 69 73 74 20 .........boundary'.=.S.fromList.
6b80 5b 73 20 61 6c 70 68 61 20 62 65 74 61 20 7c 20 61 6c 70 68 61 20 3c 2d 20 73 73 2c 20 62 65 74 [s.alpha.beta.|.alpha.<-.ss,.bet
6ba0 61 20 3c 2d 20 53 2e 74 6f 4c 69 73 74 20 62 6f 75 6e 64 61 72 79 5d 20 53 2e 5c 5c 20 69 6e 74 a.<-.S.toList.boundary].S.\\.int
6bc0 65 72 69 6f 72 27 00 20 20 20 20 20 20 20 20 20 20 20 20 6c 65 74 20 69 6e 74 65 72 69 6f 72 27 erior'.............let.interior'
6be0 20 3d 20 53 2e 75 6e 69 6f 6e 20 69 6e 74 65 72 69 6f 72 20 62 6f 75 6e 64 61 72 79 00 20 20 20 .=.S.union.interior.boundary....
6c00 20 20 20 20 20 7c 20 6f 74 68 65 72 77 69 73 65 20 3d 00 20 20 20 20 20 20 20 20 7c 20 53 2e 6e .....|.otherwise.=.........|.S.n
6c20 75 6c 6c 20 62 6f 75 6e 64 61 72 79 20 3d 20 69 6e 74 65 72 69 6f 72 00 20 20 20 20 63 6c 6f 73 ull.boundary.=.interior.....clos
6c40 75 72 65 20 69 6e 74 65 72 69 6f 72 20 62 6f 75 6e 64 61 72 79 00 61 6c 6c 52 6f 6f 74 73 20 73 ure.interior.boundary.allRoots.s
6c60 73 20 3d 20 53 2e 74 6f 4c 69 73 74 20 24 20 63 6c 6f 73 75 72 65 20 53 2e 65 6d 70 74 79 20 28 s.=.S.toList.$.closure.S.empty.(
6c80 53 2e 66 72 6f 6d 4c 69 73 74 20 73 73 29 20 77 68 65 72 65 00 61 6c 6c 52 6f 6f 74 73 20 3a 3a S.fromList.ss).where.allRoots.::
6ca0 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 2d 2d 20 54 68 65 20 63 6c .SimpleSystem.->.[[Q]].--.The.cl
6cc0 6f 73 75 72 65 20 6f 66 20 61 20 73 65 74 20 6f 66 20 72 6f 6f 74 73 20 75 6e 64 65 72 20 72 65 osure.of.a.set.of.roots.under.re
6ce0 66 6c 65 63 74 69 6f 6e 00 2d 2d 20 47 69 76 65 6e 20 61 20 73 69 6d 70 6c 65 20 73 79 73 74 65 flection.--.Given.a.simple.syste
6d00 6d 2c 20 72 65 74 75 72 6e 20 74 68 65 20 66 75 6c 6c 20 72 6f 6f 74 20 73 79 73 74 65 6d 00 00 m,.return.the.full.root.system..
6d20 20 20 20 20 20 20 20 20 28 53 2e 66 72 6f 6d 4c 69 73 74 20 24 20 70 72 20 2b 2b 20 28 28 2d 31 ........(S.fromList.$.pr.++.((-1
6d40 29 20 2a 3e 3e 20 70 72 29 29 20 3d 3d 20 28 53 2e 66 72 6f 6d 4c 69 73 74 20 24 20 61 6c 6c 52 ).*>>.pr)).==.(S.fromList.$.allR
6d60 6f 6f 74 73 20 28 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 20 6e 29 29 00 20 20 20 20 6c 65 74 oots.(simpleSystem.t.n)).....let
6d80 20 70 72 20 3d 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 28 73 69 6d 70 6c 65 53 79 73 74 65 .pr.=.positiveRoots.(simpleSyste
6da0 6d 20 74 20 6e 29 20 69 6e 00 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 31 27 20 74 m.t.n).in.prop_positiveRoots1'.t
6dc0 20 6e 20 3d 00 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 31 27 20 3a 3a 20 54 79 70 .n.=.prop_positiveRoots1'.::.Typ
6de0 65 20 2d 3e 20 49 6e 74 20 2d 3e 20 42 6f 6f 6c 00 00 00 20 20 20 20 77 68 65 72 65 20 28 74 2c e.->.Int.->.Bool.......where.(t,
6e00 20 6d 29 20 3d 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 5f 74 72 61 6e 73 66 6f 72 6d 20 6e 00 .m).=.positiveRoots_transform.n.
6e20 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 31 20 6e 20 3d 20 70 72 6f 70 5f 70 6f 73 prop_positiveRoots1.n.=.prop_pos
6e40 69 74 69 76 65 52 6f 6f 74 73 31 27 20 74 20 6d 00 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 52 6f itiveRoots1'.t.m.prop_positiveRo
6e60 6f 74 73 31 20 3a 3a 20 49 6e 74 20 2d 3e 20 42 6f 6f 6c 00 2d 2d 20 7c 20 74 65 73 74 20 74 68 ots1.::.Int.->.Bool.--.|.test.th
6e80 65 20 70 6f 73 69 74 69 76 65 20 72 6f 6f 74 73 20 61 6e 64 20 6e 65 67 61 74 69 76 65 20 72 6f e.positive.roots.and.negative.ro
6ea0 6f 74 73 20 66 6f 72 6d 20 61 6c 6c 20 74 68 65 20 72 6f 6f 74 73 00 00 00 70 72 6f 70 5f 70 6f ots.form.all.the.roots...prop_po
6ec0 73 69 74 69 76 65 52 6f 6f 74 73 27 20 74 20 6e 20 3d 20 28 6c 65 6e 67 74 68 20 24 20 70 6f 73 sitiveRoots'.t.n.=.(length.$.pos
6ee0 69 74 69 76 65 52 6f 6f 74 73 20 28 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 20 6e 29 29 20 2a itiveRoots.(simpleSystem.t.n)).*
6f00 20 32 20 3d 3d 20 6e 75 6d 52 6f 6f 74 73 20 74 20 6e 00 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 .2.==.numRoots.t.n.prop_positive
6f20 52 6f 6f 74 73 27 20 3a 3a 20 54 79 70 65 20 2d 3e 20 49 6e 74 20 2d 3e 20 42 6f 6f 6c 00 00 20 Roots'.::.Type.->.Int.->.Bool...
6f40 20 20 20 77 68 65 72 65 20 28 74 2c 20 6d 29 20 3d 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 5f ...where.(t,.m).=.positiveRoots_
6f60 74 72 61 6e 73 66 6f 72 6d 20 6e 00 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 6e transform.n.prop_positiveRoots.n
6f80 20 3d 20 70 72 6f 70 5f 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 20 74 20 6d 00 70 72 6f 70 5f .=.prop_positiveRoots'.t.m.prop_
6fa0 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 3a 3a 20 49 6e 74 20 2d 3e 20 42 6f 6f 6c 00 2d 2d 20 positiveRoots.::.Int.->.Bool.--.
6fc0 7c 20 74 65 73 74 20 74 68 65 20 6e 75 6d 62 65 72 20 6f 66 20 70 6f 73 69 74 69 76 65 20 72 6f |.test.the.number.of.positive.ro
6fe0 6f 74 73 20 69 73 20 68 61 6c 66 20 74 68 61 74 20 6f 66 20 61 6c 6c 20 72 6f 6f 74 73 00 00 00 ots.is.half.that.of.all.roots...
generated by cgit v1.2.3 (git 2.25.1) at 2024-04-25 14:48:54 +0000