aboutsummaryrefslogtreecommitdiff
path: root/Math/Combinatorics/.RootSystem.hs.swp
blob: b31575fc0de4c34e3b9194a49cc601a62a1b85e9 (plain) (blame)
ofshex dumpascii
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0060 00 00 00 00 00 00 00 00 00 00 00 00 7e 62 61 63 6f 6e 2f 63 6f 64 65 73 2f 4d 61 74 68 6b 65 6c ............~bacon/codes/Mathkel
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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 57 00 00 00 e3 01 00 00 00 10 00 00 5c 00 00 00 00 00 00 00 dc 0f 00 00 97 0f 00 00 ad..W...........\...............
2020 94 0f 00 00 57 0f 00 00 56 0f 00 00 30 0f 00 00 2f 0f 00 00 10 0f 00 00 0f 0f 00 00 fd 0e 00 00 ....W...V...0.../...............
2040 ec 0e 00 00 cc 0e 00 00 ad 0e 00 00 ac 0e 00 00 8a 0e 00 00 41 0e 00 00 12 0e 00 00 e0 0d 00 00 ....................A...........
2060 df 0d 00 00 b4 0d 00 00 b3 0d 00 00 7f 0d 00 00 65 0d 00 00 64 0d 00 00 31 0d 00 00 30 0d 00 00 ................e...d...1...0...
2080 2f 0d 00 00 1d 0d 00 00 f5 0c 00 00 f4 0c 00 00 d5 0c 00 00 88 0c 00 00 4e 0c 00 00 4d 0c 00 00 /.......................N...M...
20a0 fb 0b 00 00 be 0b 00 00 75 0b 00 00 20 0b 00 00 1f 0b 00 00 c1 0a 00 00 95 0a 00 00 59 0a 00 00 ........u...................Y...
20c0 3c 0a 00 00 f5 09 00 00 dc 09 00 00 90 09 00 00 77 09 00 00 24 09 00 00 0b 09 00 00 d2 08 00 00 <...............w...$...........
20e0 b9 08 00 00 58 08 00 00 32 08 00 00 f6 07 00 00 9b 07 00 00 82 07 00 00 42 07 00 00 29 07 00 00 ....X...2...............B...)...
2100 c0 06 00 00 bf 06 00 00 af 06 00 00 70 06 00 00 6f 06 00 00 5f 06 00 00 31 06 00 00 10 06 00 00 ............p...o..._...1.......
2120 f9 05 00 00 bf 05 00 00 be 05 00 00 98 05 00 00 79 05 00 00 78 05 00 00 51 05 00 00 12 05 00 00 ................y...x...Q.......
2140 11 05 00 00 e3 04 00 00 c2 04 00 00 6f 04 00 00 3a 04 00 00 19 04 00 00 fb 03 00 00 6d 03 00 00 ............o...:...........m...
2160 6c 03 00 00 4a 03 00 00 3c 03 00 00 28 03 00 00 ad 02 00 00 7a 02 00 00 67 02 00 00 41 02 00 00 l...J...<...(.......z...g...A...
2180 10 02 00 00 e3 01 00 00 b7 01 00 00 94 01 00 00 a1 01 00 00 6d 49 6e 64 65 78 20 72 6f 6f 74 49 ....................mIndex.rootI
21a0 6e 64 65 78 20 61 6c 6c 52 6f 6f 74 49 6e 64 69 63 65 73 20 3d 20 00 6d 49 6e 64 65 78 20 3a 3a ndex.allRootIndices.=..mIndex.::
21c0 20 5b 51 5d 20 2d 3e 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 51 5d 00 00 70 49 6e 20 20 20 20 77 68 65 .[Q].->.[[Q]].->.[Q]..pIn....whe
21e0 72 65 20 20 20 20 20 77 68 65 72 65 20 67 6f 20 3a 3a 20 5b 51 5d 20 2d 3e 20 5b 49 6e 74 5d 20 re.....where.go.::.[Q].->.[Int].
2200 2d 3e 20 49 6e 74 20 2d 3e 20 5b 5b 51 5d 5d 00 6e 65 77 52 6f 6f 74 49 6e 64 69 63 65 73 20 72 ->.Int.->.[[Q]].newRootIndices.r
2220 69 20 70 69 20 3d 20 28 72 69 20 3c 2b 3e 29 20 3c 24 3e 20 28 67 6f 20 70 69 20 5b 5d 20 31 29 i.pi.=.(ri.<+>).<$>.(go.pi.[].1)
2240 00 6e 65 77 52 6f 6f 74 49 6e 64 69 63 65 73 20 3a 3a 20 5b 51 5d 20 2d 3e 20 5b 51 5d 20 2d 3e .newRootIndices.::.[Q].->.[Q].->
2260 20 5b 5b 51 5d 5d 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 00 20 20 20 20 20 20 .[[Q]]..........................
2280 20 20 77 68 65 72 65 20 63 6d 20 3d 20 63 61 72 74 61 6e 4d 61 74 72 69 78 20 28 73 69 6d 70 6c ..where.cm.=.cartanMatrix.(simpl
22a0 65 53 79 73 74 65 6d 20 47 20 32 29 00 20 20 20 20 7c 20 6f 74 68 65 72 77 69 73 65 20 3d 20 74 eSystem.G.2).....|.otherwise.=.t
22c0 65 73 74 27 27 20 28 70 72 20 2b 2b 20 6e 70 72 29 20 28 53 2e 74 6f 4c 69 73 74 20 2e 20 53 2e est''.(pr.++.npr).(S.toList...S.
22e0 66 72 6f 6d 4c 69 73 74 20 24 20 6d 63 6f 6e 63 61 74 20 24 20 7a 69 70 57 69 74 68 20 6e 65 77 fromList.$.mconcat.$.zipWith.new
2300 52 6f 6f 74 49 6e 64 69 63 65 73 20 6e 70 72 20 28 70 49 6e 64 65 78 20 70 72 20 63 6d 20 3c 24 RootIndices.npr.(pIndex.pr.cm.<$
2320 3e 20 6e 70 72 29 29 00 20 20 20 20 7c 20 6e 75 6c 6c 20 6e 70 72 20 3d 20 70 72 00 74 65 73 74 >.npr)).....|.null.npr.=.pr.test
2340 27 27 20 70 72 20 6e 70 72 00 74 65 73 74 27 27 20 3a 3a 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 5b 51 ''.pr.npr.test''.::.[[Q]].->.[[Q
2360 5d 5d 20 2d 3e 20 5b 5b 51 5d 5d 00 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 7c 20 6f 74 68 ]].->.[[Q]]................|.oth
2380 65 72 77 69 73 65 20 3d 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 27 20 28 70 72 20 2b 2b 20 erwise.=.positiveRoots''.(pr.++.
23a0 6e 70 72 29 20 28 53 2e 74 6f 4c 69 73 74 20 2e 20 53 2e 66 72 6f 6d 4c 69 73 74 20 24 20 6d 63 npr).(S.toList...S.fromList.$.mc
23c0 6f 6e 63 61 74 20 24 20 7a 69 70 57 69 74 68 20 6e 65 77 52 6f 6f 74 49 6e 64 69 63 65 73 20 6e oncat.$.zipWith.newRootIndices.n
23e0 70 72 20 28 70 49 6e 64 65 78 20 70 72 20 63 6d 20 3c 24 3e 20 6e 70 72 29 29 00 20 20 20 20 20 pr.(pIndex.pr.cm.<$>.npr))......
2400 20 20 20 20 20 20 20 20 20 7c 20 6e 75 6c 6c 20 6e 70 72 20 3d 20 70 72 00 20 20 20 20 20 20 20 .........|.null.npr.=.pr........
2420 20 20 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 27 20 70 72 20 6e 70 72 00 20 20 20 20 77 68 ...positiveRoots''.pr.npr.....wh
2440 65 72 65 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 27 20 3a 3a 20 5b 5b 51 5d 5d 20 2d 3e 20 ere.positiveRoots''.::.[[Q]].->.
2460 5b 5b 51 5d 5d 20 2d 3e 20 5b 5b 51 5d 5d 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 20 63 6d [[Q]].->.[[Q]].positiveRoots'.cm
2480 20 3d 20 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 27 20 5b 5d 20 28 28 62 61 73 69 73 45 6c 74 .=.positiveRoots''.[].((basisElt
24a0 20 24 20 6c 65 6e 67 74 68 20 63 6d 29 20 3c 24 3e 20 5b 31 2e 2e 6c 65 6e 67 74 68 20 63 6d 5d .$.length.cm).<$>.[1..length.cm]
24c0 29 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 20 3a 3a 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 5b 51 ).positiveRoots'.::.[[Q]].->.[[Q
24e0 5d 5d 00 2d 2d 20 7c 72 65 74 75 72 6e 20 72 6f 6f 74 20 69 6e 64 69 63 65 73 20 6f 66 20 61 6c ]].--.|return.root.indices.of.al
2500 6c 20 70 6f 73 69 74 69 76 65 20 72 6f 6f 74 73 00 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 l.positive.roots..positiveRoots.
2520 73 73 20 3d 20 28 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 27 20 24 20 63 61 72 74 61 6e 4d 61 74 ss.=.(positiveRoots'.$.cartanMat
2540 72 69 78 20 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 rix.ss).<<*>>.ss.positiveRoots.:
2560 3a 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 00 2d 2d 63 61 72 74 61 :.SimpleSystem.->.[[Q]]..--carta
2580 6e 52 6f 77 73 20 73 73 20 3d 20 63 61 72 74 61 6e 52 6f 77 73 27 20 00 2d 2d 63 61 72 74 61 6e nRows.ss.=.cartanRows'..--cartan
25a0 52 6f 77 73 20 3a 3a 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 00 73 Rows.::.SimpleSystem.->.[[Q]]..s
25c0 20 61 6c 70 68 61 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 .alpha.beta.=.beta.<->.(dynkinIn
25e0 64 65 78 20 61 6c 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 dex.alpha.beta).*>.alpha.s.::.[Q
2600 5d 20 2d 3e 20 5b 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 ].->.[Q].->.[Q].--.s.alpha.beta.
2620 3d 20 73 5f 5c 61 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 =.s_\alpha.\beta.--.Weyl.group.e
2640 6c 65 6d 65 6e 74 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 lement.corresponding.to.a.root.-
2660 2d 20 48 75 6d 70 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 -.Humphreys.p3..--.Calculating.t
2680 68 65 20 66 75 6c 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 he.full.root.system.from.the.fun
26a0 64 61 6d 65 6e 74 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 damental.roots.--.ROOT.SYSTEMS..
26c0 73 69 6d 70 6c 65 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 simpleSystem.t.n.=.error.$."Inva
26e0 6c 69 64 20 72 6f 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 lid.root.system.of.type.".++.(sh
2700 6f 77 20 74 29 20 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 ow.t).++.".and.rank.".++.(show.n
2720 29 20 2b 2b 20 22 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 ).++.".".....where.e.=.basisElt.
2740 33 00 73 69 6d 70 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 3.simpleSystem.G.2.=.[e.1.<->.e.
2760 32 2c 20 28 28 2d 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 2,.((-2).*>.e.1).<+>.e.2.<+>.e.3
2780 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 34 00 73 69 6d 70 6c ].....where.e.=.basisElt.4.simpl
27a0 65 53 79 73 74 65 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 eSystem.F.4.=.[e.2.<->.e.3,.e.3.
27c0 3c 2d 3e 20 65 20 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 <->.e.4,.e.4,.(1/2).*>.(e.1.<->.
27e0 65 20 32 20 3c 2d 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 e.2.<->.e.3.<->.e.4)]...........
2800 20 20 20 20 20 20 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 ............:.[e.(i-1).<->.e.(i-
2820 32 29 20 7c 20 69 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 2).|.i.<-.[3..8]]...............
2840 20 20 20 20 20 20 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 ........:.(e.1.<+>.e.2).........
2860 20 20 73 69 6d 70 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 ..simpleroots.=.((1/2).*>.(e.1.<
2880 2d 3e 20 65 20 32 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 ->.e.2.<->.e.3.<->.e.4.<->.e.5.<
28a0 2d 3e 20 65 20 36 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 ->.e.6.<->.e.7.<+>.e.8)).....whe
28c0 72 65 20 65 20 3d 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 re.e.=.basisElt.8.simpleSystem.E
28e0 20 6e 20 7c 20 6e 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 .n.|.n.`elem`.[6,7,8].=.take.n.s
2900 69 6d 70 6c 65 72 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 impleroots.....where.e.=.basisEl
2920 74 20 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 t.n.simpleSystem.D.n.|.n.>=.4.=.
2940 5b 65 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 [e.i.<->.e.(i+1).|.i.<-.[1..n-1]
2960 5d 20 2b 2b 20 5b 65 20 28 6e 2d 31 29 20 3c 2b 3e 20 65 20 6e 5d 00 20 20 20 20 77 68 65 72 65 ].++.[e.(n-1).<+>.e.n].....where
2980 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 20 6e .e.=.basisElt.n.simpleSystem.C.n
29a0 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 20 69 .|.n.>=.2.=.[e.i.<->.e.(i+1).|.i
29c0 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 20 20 20 20 .<-.[1..n-1]].++.[2.*>.e.n].....
29e0 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 65 53 79 73 74 65 where.e.=.basisElt.n.simpleSyste
2a00 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 65 20 28 69 2b 31 m.B.n.|.n.>=.2.=.[e.i.<->.e.(i+1
2a20 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 5d 00 20 20 20 20 ).|.i.<-.[1..n-1]].++.[e.n].....
2a40 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 69 6d 70 6c 65 53 where.e.=.basisElt.(n+1).simpleS
2a60 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 3c 2d 3e 20 65 20 ystem.A.n.|.n.>=.1.=.[e.i.<->.e.
2a80 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 65 53 79 73 74 65 (i+1).|.i.<-.[1..n]].simpleSyste
2aa0 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 53 79 73 74 65 6d m.::.Type.->.Int.->.SimpleSystem
2ac0 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 65 20 61 20 62 61 .--.A.simple.system.is.like.a.ba
2ae0 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 65 65 20 48 75 6d sis.for.the.root.system.(see.Hum
2b00 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 69 6f 6e 29 00 00 phreys.p8.for.full.definition)..
2b20 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 79 73 74 65 6d 73 --.So.long.as.our.simple.systems
2b40 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 74 69 6f 6e 20 6d .are.rational,.then.reflection.m
2b60 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 20 6e 65 65 64 20 atrices.are.rational.--.We.need.
2b80 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 20 74 6f 20 65 6e to.work.over.the.rationals.to.en
2ba0 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 61 63 74 00 2d 2d sure.that.arithmetic.is.exact.--
2bc0 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 28 69 2d 31 29 20 basisElt'.n.i.=.replicate.(i-1).
2be0 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 00 2d 2d 62 61 73 0.++.1.:.replicate.(n-i).0.--bas
2c00 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 6e 74 5d 20 2d 2d isElt'.::.Int.->.Int.->.[Int].--
2c20 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 6d 69 6e 65 73 20 .this.type.signature.determines.
2c40 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 20 3d 20 72 65 70 all.the.rest..basisElt.n.i.=.rep
2c60 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 69 63 61 74 65 20 licate.(i-1).0.++.1.:.replicate.
2c80 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 20 49 6e 74 20 2d (n-i).0.basisElt.::.Int.->.Int.-
2ca0 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 72 65 20 64 65 74 >.[Q].--.this.type.signature.det
2cc0 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 65 20 69 74 68 20 ermines.all.the.rest.--.The.ith.
2ce0 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 6d 65 74 69 6d 65 basis.vector.in.K^n..--.sometime
2d00 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 6d 73 00 2d 2d 20 s.called.fundamental.systems.--.
2d20 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 65 79 73 2c 20 52 SIMPLE.SYSTEMS...--.Humphreys,.R
2d40 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 65 72 20 47 72 6f eflection.Groups.and.Coxeter.Gro
2d60 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 5b 51 5d 5d 00 64 ups..type.SimpleSystem.=.[[Q]].d
2d80 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 20 45 20 7c 20 46 ata.Type.=.A.|.B.|.C.|.D.|.E.|.F
2da0 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 20 4d 61 74 68 2e .|.G.deriving.Show..import.Math.
2dc0 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 6f 72 20 51 00 00 Algebra.Field.Base.(Q)--.for.Q..
2de0 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 70 2e 53 74 72 69 --import.Math.Algebra.Group.Stri
2e00 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 20 4d 61 74 68 2e ngRewriting.as.SG.--import.Math.
2e20 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 20 61 73 20 53 53 Algebra.Group.SchreierSims.as.SS
2e40 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 2e 50 65 72 6d 75 .import.Math.Algebra.Group.Permu
2e60 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 6f 72 64 65 72 2c tationGroup.hiding.(elts,.order,
2e80 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 62 72 61 2e 4c 69 .closure).import.Math.Algebra.Li
2ea0 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 69 65 64 20 44 61 nearAlgebra..import.qualified.Da
2ec0 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 65 64 20 44 61 74 ta.Set.as.S.import.qualified.Dat
2ee0 61 2e 4c 69 73 74 20 61 73 20 4c 00 69 6d 70 6f 72 74 20 44 61 74 61 2e 4c 69 73 74 00 69 6d 70 a.List.as.L.import.Data.List.imp
2f00 6f 72 74 20 44 61 74 61 2e 52 61 74 69 6f 00 00 69 6d 70 6f 72 74 20 50 72 65 6c 75 64 65 20 68 ort.Data.Ratio..import.Prelude.h
2f20 69 64 69 6e 67 20 28 20 28 2a 3e 29 20 29 00 00 6d 6f 64 75 6c 65 20 4d 61 74 68 2e 50 72 6f 6a iding.(.(*>).)..module.Math.Proj
2f40 65 63 74 73 2e 52 6f 6f 74 53 79 73 74 65 6d 20 77 68 65 72 65 00 00 2d 2d 20 43 6f 70 79 72 69 ects.RootSystem.where..--.Copyri
2f60 67 68 74 20 28 63 29 20 44 61 76 69 64 20 41 6d 6f 73 2c 20 32 30 30 38 2d 32 30 31 35 2e 20 41 ght.(c).David.Amos,.2008-2015..A
2f80 6c 6c 20 72 69 67 68 74 73 20 72 65 73 65 72 76 65 64 2e 00 2d 2d 00 2d 2d 20 52 65 6c 65 61 73 ll.rights.reserved..--.--.Releas
2fa0 65 64 20 75 6e 64 65 72 20 74 68 65 20 47 4e 55 20 47 65 6e 65 72 61 6c 20 50 75 62 6c 69 63 20 ed.under.the.GNU.General.Public.
2fc0 4c 69 63 65 6e 73 65 20 76 65 72 73 69 6f 6e 20 33 20 6f 72 20 6c 61 74 65 72 2e 00 2d 2d 20 43 License.version.3.or.later..--.C
2fe0 6f 70 79 72 69 67 68 74 20 28 63 29 20 59 75 63 68 65 6e 20 50 65 69 2c 20 32 30 31 37 2e 20 00 opyright.(c).Yuchen.Pei,.2017...
3000 61 64 00 00 6d 0b 00 00 19 0c 00 00 00 10 00 00 24 00 00 00 00 00 00 00 c9 0f 00 00 b2 0f 00 00 ad..m...........$...............
3020 9d 0f 00 00 88 0f 00 00 6f 0f 00 00 5d 0f 00 00 4a 0f 00 00 33 0f 00 00 21 0f 00 00 0f 0f 00 00 ........o...]...J...3...!.......
3040 0e 0f 00 00 ef 0e 00 00 c3 0e 00 00 a3 0e 00 00 81 0e 00 00 5f 0e 00 00 39 0e 00 00 1b 0e 00 00 ...................._...9.......
3060 f8 0d 00 00 d3 0d 00 00 b9 0d 00 00 a6 0d 00 00 a5 0d 00 00 a4 0d 00 00 7d 0d 00 00 7c 0d 00 00 ........................}...|...
3080 7b 0d 00 00 78 0d 00 00 5c 0d 00 00 11 0d 00 00 bd 0c 00 00 bc 0c 00 00 7a 0c 00 00 20 0c 00 00 {...x...\...............z.......
30a0 1d 0c 00 00 19 0c 00 00 18 0c 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
30c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
30e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3720 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3740 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3760 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3780 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
37a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
37c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
37e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3800 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3820 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3840 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3860 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3880 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
38a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
38c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
38e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3900 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3920 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3940 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3960 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3980 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
39a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
39c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
39e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3a00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3a20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3a40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3a60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3a80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3aa0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3ac0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3ae0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3b00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3b20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3b40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3b60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3b80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3ba0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3bc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3be0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
3c00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 2d 2d 7d 00 2d 7d 00 .........................--}.-}.
3c20 20 20 20 20 5b 28 41 2c 33 29 2c 28 41 2c 34 29 2c 28 41 2c 35 29 2c 28 42 2c 33 29 2c 28 42 2c ....[(A,3),(A,4),(A,5),(B,3),(B,
3c40 34 29 2c 28 42 2c 35 29 2c 28 43 2c 33 29 2c 28 43 2c 34 29 2c 28 43 2c 35 29 2c 28 44 2c 34 29 4),(B,5),(C,3),(C,4),(C,5),(D,4)
3c60 2c 28 44 2c 35 29 2c 28 45 2c 36 29 2c 28 46 2c 34 29 2c 28 47 2c 32 29 5d 00 74 65 73 74 32 20 ,(D,5),(E,6),(F,4),(G,2)].test2.
3c80 3d 20 61 6c 6c 20 28 5c 28 74 2c 6e 29 20 2d 3e 20 6f 72 64 65 72 57 65 79 6c 20 74 20 6e 20 3d =.all.(\(t,n).->.orderWeyl.t.n.=
3ca0 3d 20 53 53 2e 6f 72 64 65 72 20 28 77 65 79 6c 50 65 72 6d 73 20 74 20 6e 29 29 00 00 20 20 20 =.SS.order.(weylPerms.t.n)).....
3cc0 20 5b 28 41 2c 33 29 2c 28 41 2c 34 29 2c 28 41 2c 35 29 2c 28 42 2c 33 29 2c 28 42 2c 34 29 2c .[(A,3),(A,4),(A,5),(B,3),(B,4),
3ce0 28 42 2c 35 29 2c 28 43 2c 33 29 2c 28 43 2c 34 29 2c 28 43 2c 35 29 2c 28 44 2c 34 29 2c 28 44 (B,5),(C,3),(C,4),(C,5),(D,4),(D
3d00 2c 35 29 2c 28 46 2c 34 29 2c 28 47 2c 32 29 5d 00 74 65 73 74 31 20 3d 20 61 6c 6c 20 28 5c 28 ,5),(F,4),(G,2)].test1.=.all.(\(
3d20 74 2c 6e 29 20 2d 3e 20 6f 72 64 65 72 57 65 79 6c 20 74 20 6e 20 3d 3d 20 4c 2e 67 65 6e 65 72 t,n).->.orderWeyl.t.n.==.L.gener
3d40 69 63 4c 65 6e 67 74 68 20 28 65 6c 74 73 43 6f 78 65 74 65 72 20 74 20 6e 29 29 00 2d 2d 20 6e icLength.(eltsCoxeter.t.n)).--.n
3d60 6f 77 20 6d 6f 76 65 64 20 74 6f 20 54 52 6f 6f 74 53 79 73 74 65 6d 00 7b 2d 00 00 00 66 61 63 ow.moved.to.TRootSystem.{-...fac
3d80 74 6f 72 69 61 6c 20 6e 20 3d 20 70 72 6f 64 75 63 74 20 5b 31 2e 2e 74 6f 49 6e 74 65 67 65 72 torial.n.=.product.[1..toInteger
3da0 20 6e 5d 00 00 00 6f 72 64 65 72 57 65 79 6c 20 47 20 32 20 3d 20 31 32 00 6f 72 64 65 72 57 65 .n]...orderWeyl.G.2.=.12.orderWe
3dc0 79 6c 20 46 20 34 20 3d 20 32 5e 37 20 2a 20 33 5e 32 00 6f 72 64 65 72 57 65 79 6c 20 45 20 38 yl.F.4.=.2^7.*.3^2.orderWeyl.E.8
3de0 20 3d 20 32 5e 31 34 20 2a 20 33 5e 35 20 2a 20 35 5e 32 20 2a 20 37 00 6f 72 64 65 72 57 65 79 .=.2^14.*.3^5.*.5^2.*.7.orderWey
3e00 6c 20 45 20 37 20 3d 20 32 5e 31 30 20 2a 20 33 5e 34 20 2a 20 35 20 2a 20 37 00 6f 72 64 65 72 l.E.7.=.2^10.*.3^4.*.5.*.7.order
3e20 57 65 79 6c 20 45 20 36 20 3d 20 32 5e 37 20 2a 20 33 5e 34 20 2a 20 35 00 6f 72 64 65 72 57 65 Weyl.E.6.=.2^7.*.3^4.*.5.orderWe
3e40 79 6c 20 44 20 6e 20 3d 20 32 5e 28 6e 2d 31 29 20 2a 20 66 61 63 74 6f 72 69 61 6c 20 6e 00 6f yl.D.n.=.2^(n-1).*.factorial.n.o
3e60 72 64 65 72 57 65 79 6c 20 43 20 6e 20 3d 20 32 5e 6e 20 2a 20 66 61 63 74 6f 72 69 61 6c 20 6e rderWeyl.C.n.=.2^n.*.factorial.n
3e80 00 6f 72 64 65 72 57 65 79 6c 20 42 20 6e 20 3d 20 32 5e 6e 20 2a 20 66 61 63 74 6f 72 69 61 6c .orderWeyl.B.n.=.2^n.*.factorial
3ea0 20 6e 00 6f 72 64 65 72 57 65 79 6c 20 41 20 6e 20 3d 20 66 61 63 74 6f 72 69 61 6c 20 28 6e 2b .n.orderWeyl.A.n.=.factorial.(n+
3ec0 31 29 00 2d 2d 20 6f 72 64 65 72 57 65 79 6c 20 74 20 6e 20 3d 3d 20 53 2e 6f 72 64 65 72 20 28 1).--.orderWeyl.t.n.==.S.order.(
3ee0 77 65 79 6c 50 65 72 6d 73 20 74 20 6e 29 00 2d 2d 20 54 68 65 20 6f 72 64 65 72 20 6f 66 20 74 weylPerms.t.n).--.The.order.of.t
3f00 68 65 20 57 65 79 6c 20 67 72 6f 75 70 00 00 6e 75 6d 52 6f 6f 74 73 20 47 20 32 20 3d 20 31 32 he.Weyl.group..numRoots.G.2.=.12
3f20 00 6e 75 6d 52 6f 6f 74 73 20 46 20 34 20 3d 20 34 38 00 6e 75 6d 52 6f 6f 74 73 20 45 20 38 20 .numRoots.F.4.=.48.numRoots.E.8.
3f40 3d 20 32 34 30 20 20 20 20 00 6e 75 6d 52 6f 6f 74 73 20 45 20 37 20 3d 20 31 32 36 00 6e 75 6d =.240.....numRoots.E.7.=.126.num
3f60 52 6f 6f 74 73 20 45 20 36 20 3d 20 37 32 00 6e 75 6d 52 6f 6f 74 73 20 44 20 6e 20 3d 20 32 2a Roots.E.6.=.72.numRoots.D.n.=.2*
3f80 6e 2a 28 6e 2d 31 29 00 6e 75 6d 52 6f 6f 74 73 20 43 20 6e 20 3d 20 32 2a 6e 2a 6e 00 6e 75 6d n*(n-1).numRoots.C.n.=.2*n*n.num
3fa0 52 6f 6f 74 73 20 42 20 6e 20 3d 20 32 2a 6e 2a 6e 00 6e 75 6d 52 6f 6f 74 73 20 41 20 6e 20 3d Roots.B.n.=.2*n*n.numRoots.A.n.=
3fc0 20 6e 2a 28 6e 2b 31 29 00 2d 2d 20 6e 75 6d 52 6f 6f 74 73 20 74 20 6e 20 3d 3d 20 6c 65 6e 67 .n*(n+1).--.numRoots.t.n.==.leng
3fe0 74 68 20 28 63 6c 6f 73 75 72 65 20 24 20 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 20 6e 29 00 th.(closure.$.simpleSystem.t.n).
4000 61 64 00 00 25 00 00 00 c5 01 00 00 00 10 00 00 61 00 00 00 00 00 00 00 c6 0f 00 00 c5 0f 00 00 ad..%...........a...............
4020 c4 0f 00 00 6a 0f 00 00 30 0f 00 00 0b 0f 00 00 f9 0e 00 00 e2 0e 00 00 ce 0e 00 00 ba 0e 00 00 ....j...0.......................
4040 3f 0e 00 00 fe 0d 00 00 fd 0d 00 00 b8 0d 00 00 92 0d 00 00 80 0d 00 00 69 0d 00 00 55 0d 00 00 ?.......................i...U...
4060 41 0d 00 00 c6 0c 00 00 71 0c 00 00 70 0c 00 00 6f 0c 00 00 6e 0c 00 00 2e 0c 00 00 2d 0c 00 00 A.......q...p...o...n.......-...
4080 ea 0b 00 00 af 0b 00 00 ae 0b 00 00 67 0b 00 00 66 0b 00 00 65 0b 00 00 55 0b 00 00 54 0b 00 00 ............g...f...e...U...T...
40a0 18 0b 00 00 fc 0a 00 00 e3 0a 00 00 e2 0a 00 00 e1 0a 00 00 c9 0a 00 00 c8 0a 00 00 a8 0a 00 00 ................................
40c0 68 0a 00 00 25 0a 00 00 24 0a 00 00 06 0a 00 00 f0 09 00 00 d3 09 00 00 ae 09 00 00 8d 09 00 00 h...%...$.......................
40e0 68 09 00 00 47 09 00 00 2e 09 00 00 0e 09 00 00 0d 09 00 00 0c 09 00 00 01 09 00 00 b1 08 00 00 h...G...........................
4100 81 08 00 00 80 08 00 00 52 08 00 00 51 08 00 00 3e 08 00 00 3d 08 00 00 25 08 00 00 ec 07 00 00 ........R...Q...>...=...%.......
4120 c5 07 00 00 76 07 00 00 59 07 00 00 27 07 00 00 0e 07 00 00 e3 06 00 00 b1 06 00 00 6f 06 00 00 ....v...Y...'...............o...
4140 2d 06 00 00 e4 05 00 00 9b 05 00 00 69 05 00 00 50 05 00 00 20 05 00 00 e7 04 00 00 a5 04 00 00 -...........i...P...............
4160 63 04 00 00 1a 04 00 00 d1 03 00 00 b7 03 00 00 88 03 00 00 56 03 00 00 1d 03 00 00 e4 02 00 00 c...................V...........
4180 cb 02 00 00 a2 02 00 00 89 02 00 00 41 02 00 00 c7 01 00 00 c6 01 00 00 c5 01 00 00 c4 01 00 00 ............A...................
41a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
41c0 00 00 00 00 00 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 .................longRoots.=.con
41e0 63 61 74 4d 61 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 catMap.(\r->.[r,[].<->.r]).[2.*>
4200 20 65 20 69 20 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 .e.i.<->.e.j.<->.e.k.|.i.<-.[1..
4220 33 5d 2c 20 5b 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 3],.[j,k].<-.[[1..3].L.\\.[i]].]
4240 00 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 2d 3e ...........shortRoots.=.[e.i.<->
4260 20 65 20 6a 20 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 .e.j.|.i.<-.[1..3],.j.<-.[1..3],
4280 20 69 20 2f 3d 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 .i./=.j].....where.e.=.basisElt.
42a0 33 00 72 6f 6f 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 3.rootSystem.G.2.=.shortRoots.++
42c0 20 6c 6f 6e 67 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 .longRoots.....where.e.=.basisEl
42e0 74 20 6e 00 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 20 7c t.n.....++.[[].<->.e.i.<->.e.j.|
4300 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 20 .i.<-.[1..n],.j.<-.[i+1..n]]....
4320 20 2b 2b 20 5b 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 .++.[[].<->.e.i.<+>.e.j.|.i.<-.[
4340 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 65 20 1..n],.j.<-.[i+1..n]].....++.[e.
4360 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 5b 69 i.<->.e.j.|.i.<-.[1..n],.j.<-.[i
4380 2b 31 2e 2e 6e 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 +1..n]].....[e.i.<+>.e.j.|.i.<-.
43a0 5b 31 2e 2e 6e 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 [1..n],.j.<-.[i+1..n]].rootSyste
43c0 6d 20 44 20 6e 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 m.D.n.|.n.>=.4.=................
43e0 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 20 7c 20 69 20 .....++.[[].<->.e.i.<->.e.j.|.i.
4400 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 20 20 20 20 <-.[1..n],.j.<-.[i+1..n]].......
4420 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 69 20 3c 2b 3e ..............++.[[].<->.e.i.<+>
4440 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
4460 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 65 20 69 20 3c ]].....................++.[e.i.<
4480 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 5b 69 2b 31 2e ->.e.j.|.i.<-.[1..n],.j.<-.[i+1.
44a0 2e 6e 5d 5d 00 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 .n]]...........shortRoots.=.[e.i
44c0 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 2b .<+>.e.j.|.i.<-.[1..n],.j.<-.[i+
44e0 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 5b 1..n]].....................++.[[
4500 5d 20 3c 2d 3e 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 ].<->.(2.*>.e.i).|.i.<-.[1..n]].
4520 20 20 20 20 20 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 ..........longRoots..=.[2.*>.e.i
4540 20 7c 20 69 20 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 .|.i.<-.[1..n]].....where.e.=.ba
4560 73 69 73 45 6c 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 sisElt.n.rootSystem.C.n.|.n.>=.2
4580 20 3d 20 6c 6f 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 .=.longRoots.++.shortRoots......
45a0 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 69 20 3c 2d ...............++.[[].<->.e.i.<-
45c0 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 2b 31 2e 2e >.e.j.|.i.<-.[1..n],.j.<-.[i+1..
45e0 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 5b 5d 20 3c n]].....................++.[[].<
4600 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 2c 20 6a 20 ->.e.i.<+>.e.j.|.i.<-.[1..n],.j.
4620 3c 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 <-.[i+1..n]]....................
4640 20 2b 2b 20 5b 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 .++.[e.i.<->.e.j.|.i.<-.[1..n],.
4660 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 6c 6f 6e 67 52 6f 6f j.<-.[i+1..n]]...........longRoo
4680 74 73 20 20 3d 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 ts..=.[e.i.<+>.e.j.|.i.<-.[1..n]
46a0 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 20 20 20 20 ,.j.<-.[i+1..n]]................
46c0 20 20 20 20 20 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 .....++.[[].<->.e.i.|.i.<-.[1..n
46e0 5d 5d 00 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 7c ]]...........shortRoots.=.[e.i.|
4700 20 69 20 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 .i.<-.[1..n]].....where.e.=.basi
4720 73 45 6c 74 20 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 sElt.n.rootSystem.B.n.|.n.>=.2.=
4740 20 73 68 6f 72 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 .shortRoots.++.longRoots.....whe
4760 72 65 20 65 20 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 re.e.=.basisElt.(n+1).rootSystem
4780 20 41 20 6e 20 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 .A.n.|.n.>=.1.=.[e.i.<->.e.j.|.i
47a0 20 3c 2d 20 5b 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 .<-.[1..n+1],.j.<-.[1..n+1],.i./
47c0 3d 20 6a 5d 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 =.j].--.rootSystem.::.Type.->.In
47e0 74 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 t.->.[[QQ]].--.L.sort.(rootSyste
4800 6d 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 m.t.n).==.closure.(simpleSystem.
4820 74 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 t.n).--.The.full.root.system..--
4840 20 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 .Humphreys.p41ff..--.!!.Not.yet.
4860 67 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 got.root.systems.for.E6,7,8,.F4.
4880 00 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 .--.for.comparison.against.the.c
48a0 61 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 alculated.values.--.The.expected
48c0 20 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 .values.of.the.root.system,.numb
48e0 65 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 er.of.roots,.order.of.Weyl.group
4900 00 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 .--.TESTING..................(id
4920 27 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 '..+|+.zMx.n)...................
4940 20 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 ...+-+............in.(zMx.n.+|+.
4960 69 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 idMx.n).form.B.n.=.let.id'.=.(-1
4980 29 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 ).*>>.idMx.n............(map.(0:
49a0 29 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 ).(form.D.n)).form.C.n.=.(2.:.re
49c0 70 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 plicate.(2*n).0).:...........(id
49e0 4d 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 Mx.n.+|+.zMx.n).................
4a00 20 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 ..+-+.form.D.n.=.(zMx.n.+|+.idMx
4a20 20 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 .n)..(+-+).=.(++).........--.glu
4a40 65 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 e.two.matrices.together.above.an
4a60 64 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 d.below.(+|+).=.zipWith.(++).--.
4a80 67 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 glue.two.matrices.together.side.
4aa0 62 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 by.side.--.for.gluing.matrices.t
4ac0 6f 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 ogether..lieMult.x.y.=.x*y.-.y*x
4ae0 00 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 .............e.=.basisElt.n.....
4b00 77 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 where.z.=.replicate.n.0.elemMx.n
4b20 20 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 .i.j.=.replicate.(i-1).z.++.e.j.
4b40 3a 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 :.replicate.(n-i).z..--.LIE.ALGE
4b60 42 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 BRAS...poincarePoly.t.n.=.map.le
4b80 6e 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 ngth.$.L.group.$.map.length.$.el
4ba0 74 73 43 6f 78 65 74 65 72 20 74 20 6e 00 00 2d 2d 20 69 74 27 73 20 6a 75 73 74 20 73 6c 69 67 tsCoxeter.t.n..--.it's.just.slig
4bc0 68 74 6c 79 20 66 61 73 74 65 72 20 74 6f 20 75 73 65 20 74 68 65 20 62 72 61 69 64 20 70 72 65 htly.faster.to.use.the.braid.pre
4be0 73 65 6e 74 61 74 69 6f 6e 00 65 6c 74 73 43 6f 78 65 74 65 72 20 74 20 6e 20 3d 20 53 47 2e 65 sentation.eltsCoxeter.t.n.=.SG.e
4c00 6c 74 73 20 24 20 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 69 78 32 20 24 20 63 6f 78 65 74 lts.$.fromCoxeterMatrix2.$.coxet
4c20 65 72 4d 61 74 72 69 78 20 74 20 6e 00 00 63 6f 78 65 74 65 72 50 72 65 73 65 6e 74 61 74 69 6f erMatrix.t.n..coxeterPresentatio
4c40 6e 20 74 20 6e 20 3d 20 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 69 78 20 24 20 63 6f 78 65 n.t.n.=.fromCoxeterMatrix.$.coxe
4c60 74 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 terMatrix.t.n........braidRelati
4c80 6f 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 on.i.j.m.=.(take.m.$.cycle.[s_.j
4ca0 2c 20 73 5f 20 69 5d 2c 20 74 61 6b 65 20 6d 20 24 20 63 79 63 6c 65 20 5b 73 5f 20 69 2c 20 73 ,.s_.i],.take.m.$.cycle.[s_.i,.s
4cc0 5f 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 _.j]).....rules.((1:xs):rs).i.=.
4ce0 28 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 ([s_.i,.s_.i],[]).:.[braidRelati
4d00 6f 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 on.i.j.m.|.(j,m).<-.zip.[i+1..].
4d20 78 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 xs].++.rules.(map.tail.rs).(i+1)
4d40 00 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 .....rules.[]._.=.[].....rs.=.ru
4d60 6c 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 les.mx.1.....gs.=.map.s_.[1..n].
4d80 20 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 ....n.=.length.mx.fromCoxeterMat
4da0 72 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 rix2.mx.=.(gs,rs).where.--.Anoth
4dc0 65 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 er.presentation.for.the.Coxeter.
4de0 67 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 group,.using.braid.relations....
4e00 20 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 ..powerRelation.i.j.m.=.(concat.
4e20 24 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 $.replicate.m.[s_.i,.s_.j],[])..
4e40 20 20 20 72 75 6c 65 73 20 28 28 31 3a 78 73 29 3a 72 73 29 20 69 20 3d 20 28 5b 73 5f 20 69 2c ...rules.((1:xs):rs).i.=.([s_.i,
4e60 20 73 5f 20 69 5d 2c 5b 5d 29 20 3a 20 5b 70 6f 77 65 72 52 65 6c 61 74 69 6f 6e 20 69 20 6a 20 .s_.i],[]).:.[powerRelation.i.j.
4e80 6d 20 7c 20 28 6a 2c 6d 29 20 3c 2d 20 7a 69 70 20 5b 69 2b 31 2e 2e 5d 20 78 73 5d 20 2b 2b 20 m.|.(j,m).<-.zip.[i+1..].xs].++.
4ea0 72 75 6c 65 73 20 28 6d 61 70 20 74 61 69 6c 20 72 73 29 20 28 69 2b 31 29 00 20 20 20 20 72 75 rules.(map.tail.rs).(i+1).....ru
4ec0 6c 65 73 20 5b 5d 20 5f 20 3d 20 5b 5d 00 20 20 20 20 72 73 20 3d 20 72 75 6c 65 73 20 6d 78 20 les.[]._.=.[].....rs.=.rules.mx.
4ee0 31 00 20 20 20 20 67 73 20 3d 20 6d 61 70 20 73 5f 20 5b 31 2e 2e 6e 5d 00 20 20 20 20 6e 20 3d 1.....gs.=.map.s_.[1..n].....n.=
4f00 20 6c 65 6e 67 74 68 20 6d 78 00 66 72 6f 6d 43 6f 78 65 74 65 72 4d 61 74 72 69 78 20 6d 78 20 .length.mx.fromCoxeterMatrix.mx.
4f20 3d 20 28 67 73 2c 72 73 29 20 77 68 65 72 65 00 2d 2d 20 57 65 20 61 73 73 75 6d 65 20 62 75 74 =.(gs,rs).where.--.We.assume.but
4f40 20 64 6f 6e 27 74 20 63 68 65 63 6b 20 74 68 61 74 20 6d 69 69 20 3d 3d 20 31 20 61 6e 64 20 6d .don't.check.that.mii.==.1.and.m
4f60 69 6a 20 3d 3d 20 6d 6a 69 00 2d 2d 20 47 69 76 65 6e 20 74 68 65 20 6d 61 74 72 69 78 20 6f 66 ij.==.mji.--.Given.the.matrix.of
4f80 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6d 69 6a 2c 20 72 65 74 75 72 6e 20 74 68 65 20 43 6f .coefficients.mij,.return.the.Co
4fa0 78 65 74 65 72 20 67 72 6f 75 70 20 3c 73 69 20 7c 20 73 69 5e 32 2c 20 28 73 69 73 6a 29 5e 6d xeter.group.<si.|.si^2,.(sisj)^m
4fc0 69 6a 3e 00 00 00 63 6f 78 65 74 65 72 4d 61 74 72 69 78 20 74 20 6e 20 3d 20 63 6f 78 65 74 65 ij>...coxeterMatrix.t.n.=.coxete
4fe0 72 46 72 6f 6d 44 79 6e 6b 69 6e 20 24 20 64 79 6e 6b 69 6e 44 69 61 67 72 61 6d 20 74 20 6e 00 rFromDynkin.$.dynkinDiagram.t.n.
5000 61 64 00 00 ff 0e 00 00 33 0f 00 00 00 10 00 00 06 00 00 00 00 00 00 00 e2 0f 00 00 c6 0f 00 00 ad......3.......................
5020 77 0f 00 00 76 0f 00 00 58 0f 00 00 33 0f 00 00 4b 0f 00 00 33 0f 00 00 1f 0f 00 00 1e 0f 00 00 w...v...X...3...K...3...........
5040 f8 0e 00 00 c5 0e 00 00 c4 0e 00 00 a9 0e 00 00 7a 0e 00 00 79 0e 00 00 51 0e 00 00 2b 0e 00 00 ................z...y...Q...+...
5060 2a 0e 00 00 0b 0e 00 00 df 0d 00 00 de 0d 00 00 ba 0d 00 00 9a 0d 00 00 99 0d 00 00 63 0d 00 00 *...........................c...
5080 31 0d 00 00 0c 0d 00 00 cd 0c 00 00 af 0c 00 00 8a 0c 00 00 74 0c 00 00 3e 0c 00 00 d0 0b 00 00 1...................t...>.......
50a0 a5 0b 00 00 95 0b 00 00 94 0b 00 00 6a 0b 00 00 2b 0b 00 00 1b 0b 00 00 ff 0a 00 00 e9 0a 00 00 ............j...+...............
50c0 c2 0a 00 00 53 0a 00 00 36 0a 00 00 26 0a 00 00 25 0a 00 00 21 0a 00 00 13 0a 00 00 d7 09 00 00 ....S...6...&...%...!...........
50e0 d6 09 00 00 94 09 00 00 84 09 00 00 66 09 00 00 4e 09 00 00 1a 09 00 00 05 09 00 00 00 09 00 00 ............f...N...............
5100 ce 08 00 00 a0 08 00 00 9f 08 00 00 54 08 00 00 f7 07 00 00 c6 07 00 00 ad 07 00 00 6e 07 00 00 ............T...............n...
5120 24 07 00 00 23 07 00 00 22 07 00 00 f1 06 00 00 f0 06 00 00 9f 06 00 00 7c 06 00 00 3c 06 00 00 $...#..."...............|...<...
5140 1d 06 00 00 d6 05 00 00 91 05 00 00 40 05 00 00 3f 05 00 00 ff 04 00 00 a7 04 00 00 91 04 00 00 ............@...?...............
5160 90 04 00 00 6d 04 00 00 15 04 00 00 f9 03 00 00 a8 03 00 00 a7 03 00 00 6f 03 00 00 6e 03 00 00 ....m...................o...n...
5180 f6 02 00 00 d6 02 00 00 b9 02 00 00 9c 02 00 00 7f 02 00 00 62 02 00 00 2c 02 00 00 ff 01 00 00 ....................b...,.......
51a0 fe 01 00 00 ac 01 00 00 00 00 00 00 2d 2d 20 54 68 65 20 6d 69 6a 20 63 6f 65 66 66 69 63 69 65 ............--.The.mij.coefficie
51c0 6e 74 73 20 6f 66 20 74 68 65 20 43 6f 78 65 74 65 72 20 67 72 6f 75 70 20 3c 73 69 20 7c 20 73 nts.of.the.Coxeter.group.<si.|.s
51e0 69 5e 32 2c 20 28 73 69 73 6a 29 5e 6d 69 6a 3e 2c 20 61 73 20 61 20 6d 61 74 72 69 78 00 00 20 i^2,.(sisj)^mij>,.as.a.matrix...
5200 20 20 20 77 68 65 72 65 20 66 20 30 20 3d 20 32 3b 20 66 20 31 20 3d 20 33 3b 20 66 20 32 20 3d ...where.f.0.=.2;.f.1.=.3;.f.2.=
5220 20 34 3b 20 66 20 33 20 3d 20 36 00 63 6f 78 65 74 65 72 46 72 6f 6d 44 79 6e 6b 69 6e 20 6e 69 .4;.f.3.=.6.coxeterFromDynkin.ni
5240 6a 20 3d 20 73 65 74 44 69 61 67 20 31 20 24 20 28 6d 61 70 20 2e 20 6d 61 70 29 20 66 20 6e 69 j.=.setDiag.1.$.(map...map).f.ni
5260 6a 00 2d 2d 20 6e 69 6a 20 3d 3d 20 33 20 3c 3d 3e 20 74 68 65 74 61 20 3d 20 70 69 2f 36 00 2d j.--.nij.==.3.<=>.theta.=.pi/6.-
5280 2d 20 6e 69 6a 20 3d 3d 20 32 20 3c 3d 3e 20 74 68 65 74 61 20 3d 20 70 69 2f 34 00 2d 2d 20 6e -.nij.==.2.<=>.theta.=.pi/4.--.n
52a0 69 6a 20 3d 3d 20 31 20 3c 3d 3e 20 74 68 65 74 61 20 3d 20 70 69 2f 33 00 2d 2d 20 6e 69 6a 20 ij.==.1.<=>.theta.=.pi/3.--.nij.
52c0 3d 3d 20 30 20 3c 3d 3e 20 74 68 65 74 61 20 3d 20 70 69 2f 32 00 2d 2d 20 75 73 69 6e 67 20 6e ==.0.<=>.theta.=.pi/2.--.using.n
52e0 69 6a 20 3d 20 34 20 63 6f 73 5e 32 20 74 68 65 74 61 5f 69 6a 00 2d 2d 20 67 69 76 65 6e 20 74 ij.=.4.cos^2.theta_ij.--.given.t
5300 68 65 20 44 79 6e 6b 69 6e 20 64 69 61 67 72 61 6d 20 6e 69 6a 2c 20 64 65 72 69 76 65 20 74 68 he.Dynkin.diagram.nij,.derive.th
5320 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6d 69 6a 20 6f 66 20 74 68 65 20 43 6f 78 65 74 65 e.coefficients.mij.of.the.Coxete
5340 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 6a 3e 20 r.group.<si.|.si^2,.(sisj)^mij>.
5360 28 73 6f 20 6d 69 69 20 3d 3d 20 31 29 00 00 64 79 6e 6b 69 6e 44 69 61 67 72 61 6d 20 74 20 6e (so.mii.==.1)..dynkinDiagram.t.n
5380 20 3d 20 64 79 6e 6b 69 6e 46 72 6f 6d 43 61 72 74 61 6e 20 24 20 63 61 72 74 61 6e 4d 61 74 72 .=.dynkinFromCartan.$.cartanMatr
53a0 69 78 20 74 20 6e 00 00 64 79 6e 6b 69 6e 46 72 6f 6d 43 61 72 74 61 6e 20 61 69 6a 20 3d 20 73 ix.t.n..dynkinFromCartan.aij.=.s
53c0 65 74 44 69 61 67 20 30 20 24 20 28 7a 69 70 57 69 74 68 20 2e 20 7a 69 70 57 69 74 68 29 20 28 etDiag.0.$.(zipWith...zipWith).(
53e0 2a 29 20 61 69 6a 20 28 4c 2e 74 72 61 6e 73 70 6f 73 65 20 61 69 6a 29 00 2d 2d 20 6e 69 6a 20 *).aij.(L.transpose.aij).--.nij.
5400 3d 20 41 69 6a 20 2a 20 41 6a 69 2c 20 6e 69 69 20 3d 20 30 00 2d 2d 20 67 69 76 65 6e 20 61 20 =.Aij.*.Aji,.nii.=.0.--.given.a.
5420 43 61 72 74 61 6e 20 6d 61 74 72 69 78 2c 20 64 65 72 69 76 65 20 74 68 65 20 63 6f 72 72 65 73 Cartan.matrix,.derive.the.corres
5440 70 6f 6e 64 69 6e 67 20 6d 61 74 72 69 78 20 64 65 73 63 72 69 62 69 6e 67 20 74 68 65 20 44 79 ponding.matrix.describing.the.Dy
5460 6e 6b 69 6e 20 64 69 61 67 72 61 6d 00 2d 2d 20 43 61 72 74 65 72 2c 20 53 65 67 61 6c 2c 20 4d nkin.diagram.--.Carter,.Segal,.M
5480 61 63 64 6f 6e 61 6c 64 20 70 31 37 2d 31 38 00 00 73 65 74 44 69 61 67 20 5f 20 5b 5b 5d 5d 20 acdonald.p17-18..setDiag._.[[]].
54a0 3d 20 5b 5b 5d 5d 00 73 65 74 44 69 61 67 20 63 20 6d 78 40 28 28 78 3a 78 73 29 3a 72 73 29 20 =.[[]].setDiag.c.mx@((x:xs):rs).
54c0 3d 20 28 63 3a 78 73 29 20 3a 20 7a 69 70 57 69 74 68 20 28 3a 29 20 28 6d 61 70 20 68 65 61 64 =.(c:xs).:.zipWith.(:).(map.head
54e0 20 72 73 29 20 28 73 65 74 44 69 61 67 20 63 20 24 20 6d 61 70 20 74 61 69 6c 20 72 73 29 00 2d .rs).(setDiag.c.$.map.tail.rs).-
5500 2d 20 73 65 74 20 74 68 65 20 64 69 61 67 6f 6e 61 6c 20 65 6e 74 72 69 65 73 20 6f 66 20 28 73 -.set.the.diagonal.entries.of.(s
5520 71 75 61 72 65 29 20 6d 61 74 72 69 78 20 6d 78 20 74 6f 20 63 6f 6e 73 74 61 6e 74 20 63 00 00 quare).matrix.mx.to.constant.c..
5540 2d 2d 20 28 53 6f 20 70 72 6f 62 61 62 6c 79 20 43 61 72 74 65 72 20 64 65 66 69 6e 65 73 20 74 --.(So.probably.Carter.defines.t
5560 68 65 20 72 6f 6f 74 73 20 6f 66 20 47 32 20 74 68 65 20 6f 74 68 65 72 20 77 61 79 20 72 6f 75 he.roots.of.G2.the.other.way.rou
5580 6e 64 20 74 6f 20 48 75 6d 70 68 72 65 79 73 29 00 2d 2d 20 54 68 65 79 20 61 67 72 65 65 20 77 nd.to.Humphreys).--.They.agree.w
55a0 69 74 68 20 6f 75 72 20 61 6e 73 77 65 72 73 20 65 78 63 65 70 74 20 66 6f 72 20 47 32 2c 20 77 ith.our.answers.except.for.G2,.w
55c0 68 69 63 68 20 69 73 20 74 68 65 20 74 72 61 6e 73 70 6f 73 65 00 2d 2d 20 43 61 72 74 65 72 2c hich.is.the.transpose.--.Carter,
55e0 20 53 69 6d 70 6c 65 20 47 72 6f 75 70 73 20 6f 66 20 4c 69 65 20 54 79 70 65 2c 20 70 34 34 2d .Simple.Groups.of.Lie.Type,.p44-
5600 35 20 67 69 76 65 73 20 74 68 65 20 65 78 70 65 63 74 65 64 20 61 6e 73 77 65 72 73 00 2d 2d 20 5.gives.the.expected.answers.--.
5620 54 68 6f 73 65 20 6f 66 20 42 2c 20 43 2c 20 46 2c 20 47 20 61 72 65 20 6e 6f 74 00 2d 2d 20 4e Those.of.B,.C,.F,.G.are.not.--.N
5640 6f 74 65 3a 20 54 68 65 20 43 61 72 74 61 6e 20 6d 61 74 72 69 63 65 73 20 66 6f 72 20 41 2c 20 ote:.The.Cartan.matrices.for.A,.
5660 44 2c 20 45 20 73 79 73 74 65 6d 73 20 61 72 65 20 73 79 6d 6d 65 74 72 69 63 2e 00 20 20 20 20 D,.E.systems.are.symmetric......
5680 77 68 65 72 65 20 72 6f 6f 74 73 20 3d 20 73 69 6d 70 6c 65 53 79 73 74 65 6d 20 74 20 6e 00 63 where.roots.=.simpleSystem.t.n.c
56a0 61 72 74 61 6e 4d 61 74 72 69 78 20 74 20 6e 20 3d 20 5b 5b 32 20 2a 20 28 61 69 20 3c 2e 3e 20 artanMatrix.t.n.=.[[2.*.(ai.<.>.
56c0 61 6a 29 20 2f 20 28 61 69 20 3c 2e 3e 20 61 69 29 20 7c 20 61 6a 20 3c 2d 20 72 6f 6f 74 73 5d aj)./.(ai.<.>.ai).|.aj.<-.roots]
56e0 20 7c 20 61 69 20 3c 2d 20 72 6f 6f 74 73 5d 00 00 2d 2d 20 43 41 52 54 41 4e 20 4d 41 54 52 49 .|.ai.<-.roots]..--.CARTAN.MATRI
5700 58 2c 20 44 59 4e 4b 49 4e 20 44 49 41 47 52 41 4d 2c 20 43 4f 58 45 54 45 52 20 53 59 53 54 45 X,.DYNKIN.DIAGRAM,.COXETER.SYSTE
5720 4d 00 00 00 2d 2d 20 68 6f 77 65 76 65 72 2c 20 72 65 66 6c 65 63 74 69 6f 6e 20 6d 61 74 72 69 M...--.however,.reflection.matri
5740 63 65 73 20 61 72 65 20 73 79 6d 6d 65 74 72 69 63 2c 20 73 6f 20 74 68 65 79 20 61 6c 73 6f 20 ces.are.symmetric,.so.they.also.
5760 66 6f 72 6d 20 74 68 65 20 72 6f 77 73 00 2d 2d 20 74 68 65 20 69 6d 61 67 65 73 20 6f 66 20 74 form.the.rows.--.the.images.of.t
5780 68 65 20 62 61 73 69 73 20 65 6c 74 73 20 66 6f 72 6d 20 74 68 65 20 63 6f 6c 75 6d 6e 73 20 6f he.basis.elts.form.the.columns.o
57a0 66 20 74 68 65 20 6d 61 74 72 69 78 00 20 20 20 20 20 20 20 20 20 20 65 20 3d 20 62 61 73 69 73 f.the.matrix...........e.=.basis
57c0 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 68 20 72 20 2d 2d 20 Elt.d.....where.d.=.length.r.--.
57e0 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 78 20 72 20 3d 20 6d dimension.of.the.space.wMx.r.=.m
5800 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 64 5d 5d 20 2d 2d 20 ap.(w.r).[e.i.|.i.<-.[1..d]].--.
5820 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 68 79 70 65 72 70 6c matrix.for.reflection.in.hyperpl
5840 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 65 20 57 65 79 6c 20 ane.orthogonal.to.r.--.The.Weyl.
5860 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 6e 67 20 74 6f 20 61 group.element.corresponding.to.a
5880 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 61 74 72 69 78 00 00 .root,.represented.as.a.matrix..
58a0 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 78 20 28 73 69 6d 70 weylMatrices.t.n.=.map.wMx.(simp
58c0 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 72 73 20 6f 66 20 74 leSystem.t.n).--.Generators.of.t
58e0 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 20 67 72 6f 75 70 00 he.Weyl.group.as.a.matrix.group.
5900 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 00 20 20 20 20 20 20 .........in.map.toPerm.rs.......
5920 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 78 2c 20 77 20 72 20 ..toPerm.r.=.fromPairs.[(x,.w.r.
5940 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 3d 20 63 6c 6f 73 75 x).|.x.<-.xs].........xs.=.closu
5960 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 53 79 73 74 65 6d 20 re.rs.....let.rs.=.simpleSystem.
5980 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 6e 65 72 61 74 6f 72 t.n.weylPerms.t.n.=.--.Generator
59a0 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 72 6d 75 74 61 74 69 s.of.the.Weyl.group.as.permutati
59c0 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 20 54 68 65 20 66 69 on.group.on.the.roots..--.The.fi
59e0 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 65 72 61 74 65 64 20 nite.reflection.group.generated.
5a00 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 4c 20 47 52 4f 55 50 by.the.root.system.--.WEYL.GROUP
5a20 00 7b 2d 2d 00 00 20 20 20 20 20 20 20 20 20 20 20 20 2d 2d 7d 00 20 20 20 20 20 20 20 20 20 20 .{--..............--}...........
5a40 20 20 69 6e 20 67 6f 20 70 72 27 20 6e 65 77 50 72 27 00 20 20 20 20 20 20 20 20 20 20 20 20 20 ..in.go.pr'.newPr'..............
5a60 20 20 20 6e 65 77 50 72 27 20 3d 20 53 2e 66 72 6f 6d 4c 69 73 74 20 5b 73 20 61 6c 70 68 61 20 ...newPr'.=.S.fromList.[s.alpha.
5a80 28 6e 65 67 61 74 65 20 3c 24 3e 20 62 65 74 61 29 20 7c 20 61 6c 70 68 61 20 3c 2d 20 73 73 2c (negate.<$>.beta).|.alpha.<-.ss,
5aa0 20 62 65 74 61 20 3c 2d 20 53 2e 74 6f 4c 69 73 74 20 6e 65 77 50 72 5d 20 53 2e 5c 5c 20 70 72 .beta.<-.S.toList.newPr].S.\\.pr
5ac0 27 00 20 20 20 20 20 20 20 20 20 20 20 20 6c 65 74 20 70 72 27 20 3d 20 53 2e 75 6e 69 6f 6e 20 '.............let.pr'.=.S.union.
5ae0 70 72 20 6e 65 77 50 72 00 20 20 20 20 20 20 20 20 7c 20 6f 74 68 65 72 77 69 73 65 20 3d 00 20 pr.newPr.........|.otherwise.=..
5b00 20 20 20 20 20 20 20 7c 20 53 2e 6e 75 6c 6c 20 6e 65 77 50 72 20 3d 20 70 72 00 20 20 20 20 67 .......|.S.null.newPr.=.pr.....g
5b20 6f 20 70 72 20 6e 65 77 50 72 00 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 73 73 20 3d 20 53 2e o.pr.newPr.positiveRoots.ss.=.S.
5b40 74 6f 4c 69 73 74 20 24 20 67 6f 20 53 2e 65 6d 70 74 79 20 28 53 2e 66 72 6f 6d 4c 69 73 74 20 toList.$.go.S.empty.(S.fromList.
5b60 73 73 29 20 77 68 65 72 65 00 7b 2d 2d 70 6f 73 69 74 69 76 65 52 6f 6f 74 73 20 3a 3a 20 53 69 ss).where.{--positiveRoots.::.Si
5b80 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 00 20 20 20 20 20 20 20 20 20 20 20 mpleSystem.->.[[Q]].............
5ba0 20 2d 2d 7d 00 20 20 20 20 20 20 20 20 20 20 20 20 69 6e 20 63 6c 6f 73 75 72 65 20 69 6e 74 65 .--}.............in.closure.inte
5bc0 72 69 6f 72 27 20 62 6f 75 6e 64 61 72 79 27 00 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 rior'.boundary'.................
5be0 62 6f 75 6e 64 61 72 79 27 20 3d 20 53 2e 66 72 6f 6d 4c 69 73 74 20 5b 73 20 61 6c 70 68 61 20 boundary'.=.S.fromList.[s.alpha.
5c00 62 65 74 61 20 7c 20 61 6c 70 68 61 20 3c 2d 20 73 73 2c 20 62 65 74 61 20 3c 2d 20 53 2e 74 6f beta.|.alpha.<-.ss,.beta.<-.S.to
5c20 4c 69 73 74 20 62 6f 75 6e 64 61 72 79 5d 20 53 2e 5c 5c 20 69 6e 74 65 72 69 6f 72 27 00 20 20 List.boundary].S.\\.interior'...
5c40 20 20 20 20 20 20 20 20 20 20 6c 65 74 20 69 6e 74 65 72 69 6f 72 27 20 3d 20 53 2e 75 6e 69 6f ..........let.interior'.=.S.unio
5c60 6e 20 69 6e 74 65 72 69 6f 72 20 62 6f 75 6e 64 61 72 79 00 20 20 20 20 20 20 20 20 7c 20 6f 74 n.interior.boundary.........|.ot
5c80 68 65 72 77 69 73 65 20 3d 00 20 20 20 20 20 20 20 20 7c 20 53 2e 6e 75 6c 6c 20 62 6f 75 6e 64 herwise.=.........|.S.null.bound
5ca0 61 72 79 20 3d 20 69 6e 74 65 72 69 6f 72 00 20 20 20 20 63 6c 6f 73 75 72 65 20 69 6e 74 65 72 ary.=.interior.....closure.inter
5cc0 69 6f 72 20 62 6f 75 6e 64 61 72 79 00 61 6c 6c 52 6f 6f 74 73 20 73 73 20 3d 20 53 2e 74 6f 4c ior.boundary.allRoots.ss.=.S.toL
5ce0 69 73 74 20 24 20 63 6c 6f 73 75 72 65 20 53 2e 65 6d 70 74 79 20 28 53 2e 66 72 6f 6d 4c 69 73 ist.$.closure.S.empty.(S.fromLis
5d00 74 20 73 73 29 20 77 68 65 72 65 00 7b 2d 2d 61 6c 6c 52 6f 6f 74 73 20 3a 3a 20 53 69 6d 70 6c t.ss).where.{--allRoots.::.Simpl
5d20 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 6f 73 75 72 65 20 eSystem.->.[[Q]].--.The.closure.
5d40 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 66 6c 65 63 74 69 of.a.set.of.roots.under.reflecti
5d60 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 6d 2c 20 72 65 74 on.--.Given.a.simple.system,.ret
5d80 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 64 79 6e 6b 69 6e urn.the.full.root.system..dynkin
5da0 49 6e 64 65 78 27 20 72 69 20 63 6d 20 3d 20 72 69 20 3c 2a 3e 3e 20 63 6d 00 64 79 6e 6b 69 6e Index'.ri.cm.=.ri.<*>>.cm.dynkin
5dc0 49 6e 64 65 78 27 20 3a 3a 20 5b 51 5d 20 2d 3e 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 51 5d 00 00 64 Index'.::.[Q].->.[[Q]].->.[Q]..d
5de0 79 6e 6b 69 6e 49 6e 64 65 78 20 72 20 73 20 3d 20 32 20 2a 20 28 72 20 3c 2e 3e 20 73 29 20 2f ynkinIndex.r.s.=.2.*.(r.<.>.s)./
5e00 20 28 72 20 3c 2e 3e 20 72 29 00 64 79 6e 6b 69 6e 49 6e 64 65 78 20 3a 3a 20 5b 51 5d 20 2d 3e .(r.<.>.r).dynkinIndex.::.[Q].->
5e20 20 5b 51 5d 20 2d 3e 20 51 00 00 64 79 6e 6b 69 6e 52 6f 77 20 72 20 73 73 20 3d 20 64 79 6e 6b .[Q].->.Q..dynkinRow.r.ss.=.dynk
5e40 69 6e 49 6e 64 65 78 20 72 20 3c 24 3e 20 73 73 00 64 79 6e 6b 69 6e 52 6f 77 20 3a 3a 20 5b 51 inIndex.r.<$>.ss.dynkinRow.::.[Q
5e60 5d 20 2d 3e 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 51 5d 00 00 69 73 4d 75 6c 74 ].->.SimpleSystem.->.[Q]..isMult
5e80 42 61 73 69 73 20 76 20 3d 20 28 6c 65 6e 67 74 68 20 24 20 66 69 6c 74 65 72 20 28 2f 3d 30 29 Basis.v.=.(length.$.filter.(/=0)
5ea0 20 76 29 20 3d 3d 20 31 00 69 73 4d 75 6c 74 42 61 73 69 73 20 3a 3a 20 5b 51 5d 20 2d 3e 20 42 .v).==.1.isMultBasis.::.[Q].->.B
5ec0 6f 6f 6c 00 00 63 61 72 74 61 6e 4d 61 74 72 69 78 20 73 73 20 3d 20 5b 64 79 6e 6b 69 6e 49 6e ool..cartanMatrix.ss.=.[dynkinIn
5ee0 64 65 78 20 72 20 3c 24 3e 20 73 73 20 7c 20 72 20 3c 2d 20 73 73 5d 00 63 61 72 74 61 6e 4d 61 dex.r.<$>.ss.|.r.<-.ss].cartanMa
5f00 74 72 69 78 20 3a 3a 20 53 69 6d 70 6c 65 53 79 73 74 65 6d 20 2d 3e 20 5b 5b 51 5d 5d 00 00 73 trix.::.SimpleSystem.->.[[Q]]..s
5f20 75 6d 56 20 3d 20 66 6f 6c 64 6c 31 20 28 3c 2b 3e 29 00 6d 61 78 56 20 78 73 20 3d 20 6d 61 78 umV.=.foldl1.(<+>).maxV.xs.=.max
5f40 69 6d 75 6d 20 3c 24 3e 20 28 74 72 61 6e 73 70 6f 73 65 20 78 73 29 00 6d 61 78 56 20 3a 3a 20 imum.<$>.(transpose.xs).maxV.::.
5f60 4f 72 64 20 61 20 3d 3e 20 5b 5b 61 5d 5d 20 2d 3e 20 5b 61 5d 00 00 20 20 20 20 20 20 20 20 65 Ord.a.=>.[[a]].->.[a]..........e
5f80 6c 73 65 20 6d 61 78 56 20 24 20 66 69 6c 74 65 72 20 69 73 4d 75 6c 74 42 61 73 69 73 20 5b 72 lse.maxV.$.filter.isMultBasis.[r
5fa0 6f 6f 74 49 6e 64 65 78 20 3c 2d 3e 20 72 20 7c 20 72 20 3c 2d 20 61 6c 6c 52 6f 6f 74 49 6e 64 ootIndex.<->.r.|.r.<-.allRootInd
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5fe0 78 00 20 20 20 20 69 66 20 69 73 4d 75 6c 74 42 61 73 69 73 20 72 6f 6f 74 49 6e 64 65 78 20 00 x.....if.isMultBasis.rootIndex..
6000 61 64 00 00 79 0e 00 00 b9 0e 00 00 00 10 00 00 09 00 00 00 00 00 00 00 d7 0f 00 00 8b 0f 00 00 ad..y...........................
6020 8a 0f 00 00 89 0f 00 00 62 0f 00 00 fb 0e 00 00 fa 0e 00 00 dc 0e 00 00 b9 0e 00 00 a9 0e 00 00 ........b.......................
6040 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6060 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6080 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
60a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
60c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
60e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6100 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
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 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6820 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6840 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6860 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6880 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
68a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
68c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
68e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6900 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6920 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6940 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6960 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6980 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
69a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
69c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
69e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6a00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6a20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6a40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6a60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6a80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6aa0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6ac0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6ae0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6b00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6b20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6b40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6b60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6b80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6ba0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6bc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6be0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6c00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6c20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6c40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6c60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6c80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6ca0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6cc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6ce0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6d00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6d20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6d40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6d60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6d80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6da0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6dc0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6de0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6e00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6e20 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6e40 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6e60 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6e80 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
6ea0 00 00 00 00 00 00 00 6d 49 6d 49 6e 64 65 78 20 72 6f 6f 74 49 6e 64 65 78 6d 49 6e 64 65 78 20 .......mImIndex.rootIndexmIndex.
6ec0 72 6f 6f 74 49 6e 64 65 78 20 61 6c 6c 52 6f 6f 74 49 6e 64 69 63 65 73 20 3d 20 00 6d 49 6e 64 rootIndex.allRootIndices.=..mInd
6ee0 65 78 20 3a 3a 20 5b 51 5d 20 2d 3e 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 51 5d 00 00 70 49 6e 64 65 ex.::.[Q].->.[[Q]].->.[Q]..pInde
6f00 78 20 61 6c 6c 52 6f 6f 74 49 6e 64 69 63 65 73 20 63 6d 20 72 6f 6f 74 49 6e 64 65 78 20 3d 20 x.allRootIndices.cm.rootIndex.=.
6f20 28 6d 49 6e 64 65 78 20 72 6f 6f 74 49 6e 64 65 78 20 61 6c 6c 52 6f 6f 74 49 6e 64 69 63 65 73 (mIndex.rootIndex.allRootIndices
6f40 29 20 3c 2d 3e 20 28 64 79 6e 6b 69 6e 49 6e 64 65 78 27 20 72 6f 6f 74 49 6e 64 65 78 20 63 6d ).<->.(dynkinIndex'.rootIndex.cm
6f60 29 00 70 49 6e 64 65 78 20 3a 3a 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 5b 51 5d 5d 20 2d 3e 20 5b 51 ).pIndex.::.[[Q]].->.[[Q]].->.[Q
6f80 5d 20 2d 3e 20 5b 51 5d 00 00 00 20 20 20 20 20 20 20 20 20 20 67 6f 20 28 78 3a 78 73 29 20 79 ].->.[Q].............go.(x:xs).y
6fa0 73 20 6b 20 3d 20 67 6f 20 78 73 20 28 69 66 20 78 20 3d 3d 20 30 20 74 68 65 6e 20 79 73 20 65 s.k.=.go.xs.(if.x.==.0.then.ys.e
6fc0 6c 73 65 20 79 73 20 2b 2b 20 5b 6b 5d 29 20 28 6b 20 2b 20 31 29 00 20 20 20 20 20 20 20 20 20 lse.ys.++.[k]).(k.+.1)..........
6fe0 20 67 6f 20 5b 5d 20 79 73 20 6e 20 3d 20 62 61 73 69 73 45 6c 74 20 6e 20 3c 24 3e 20 79 73 00 .go.[].ys.n.=.basisElt.n.<$>.ys.
7000 61 64 00 00 c6 00 00 00 5a 02 00 00 00 10 00 00 5e 00 00 00 00 00 00 00 ff 0f 00 00 e1 0f 00 00 ad......Z.......^...............
7020 cd 0f 00 00 cc 0f 00 00 a6 0f 00 00 73 0f 00 00 72 0f 00 00 57 0f 00 00 28 0f 00 00 27 0f 00 00 ............s...r...W...(...'...
7040 ff 0e 00 00 d9 0e 00 00 d8 0e 00 00 b9 0e 00 00 8d 0e 00 00 8c 0e 00 00 68 0e 00 00 48 0e 00 00 ........................h...H...
7060 47 0e 00 00 11 0e 00 00 df 0d 00 00 ba 0d 00 00 7b 0d 00 00 5d 0d 00 00 38 0d 00 00 22 0d 00 00 G...............{...]...8..."...
7080 ec 0c 00 00 7e 0c 00 00 53 0c 00 00 43 0c 00 00 42 0c 00 00 18 0c 00 00 d9 0b 00 00 c9 0b 00 00 ....~...S...C...B...............
70a0 ad 0b 00 00 97 0b 00 00 70 0b 00 00 01 0b 00 00 e4 0a 00 00 d4 0a 00 00 d3 0a 00 00 cf 0a 00 00 ........p.......................
70c0 c1 0a 00 00 85 0a 00 00 84 0a 00 00 42 0a 00 00 32 0a 00 00 14 0a 00 00 fc 09 00 00 c8 09 00 00 ............B...2...............
70e0 b3 09 00 00 ae 09 00 00 7c 09 00 00 4e 09 00 00 4d 09 00 00 02 09 00 00 a5 08 00 00 74 08 00 00 ........|...N...M...........t...
7100 5b 08 00 00 1c 08 00 00 d2 07 00 00 d1 07 00 00 d0 07 00 00 9f 07 00 00 9e 07 00 00 4d 07 00 00 [...........................M...
7120 2a 07 00 00 ea 06 00 00 cb 06 00 00 84 06 00 00 3f 06 00 00 ee 05 00 00 ed 05 00 00 ad 05 00 00 *...............?...............
7140 55 05 00 00 3f 05 00 00 3e 05 00 00 1b 05 00 00 c3 04 00 00 a7 04 00 00 56 04 00 00 55 04 00 00 U...?...>...............V...U...
7160 1d 04 00 00 1c 04 00 00 a4 03 00 00 84 03 00 00 67 03 00 00 4a 03 00 00 2d 03 00 00 10 03 00 00 ................g...J...-.......
7180 da 02 00 00 ad 02 00 00 ac 02 00 00 5a 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ............Z...................
71a0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
71c0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
71e0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
7200 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
7220 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ................................
7240 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 2d 2d 20 54 68 65 ..........................--.The
7260 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 78 65 74 65 72 .mij.coefficients.of.the.Coxeter
7280 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 6a 3e 2c 20 .group.<si.|.si^2,.(sisj)^mij>,.
72a0 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 3d 20 32 3b 20 as.a.matrix......where.f.0.=.2;.
72c0 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 6f 78 65 74 65 f.1.=.3;.f.2.=.4;.f.3.=.6.coxete
72e0 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 20 24 20 28 6d rFromDynkin.nij.=.setDiag.1.$.(m
7300 61 70 20 2e 20 6d 61 70 29 20 66 20 6e 69 6a 00 2d 2d 20 6e 69 6a 20 3d 3d 20 33 20 3c 3d 3e 20 ap...map).f.nij.--.nij.==.3.<=>.
7320 74 68 65 74 61 20 3d 20 70 69 2f 36 00 2d 2d 20 6e 69 6a 20 3d 3d 20 32 20 3c 3d 3e 20 74 68 65 theta.=.pi/6.--.nij.==.2.<=>.the
7340 74 61 20 3d 20 70 69 2f 34 00 2d 2d 20 6e 69 6a 20 3d 3d 20 31 20 3c 3d 3e 20 74 68 65 74 61 20 ta.=.pi/4.--.nij.==.1.<=>.theta.
7360 3d 20 70 69 2f 33 00 2d 2d 20 6e 69 6a 20 3d 3d 20 30 20 3c 3d 3e 20 74 68 65 74 61 20 3d 20 70 =.pi/3.--.nij.==.0.<=>.theta.=.p
7380 69 2f 32 00 2d 2d 20 75 73 69 6e 67 20 6e 69 6a 20 3d 20 34 20 63 6f 73 5e 32 20 74 68 65 74 61 i/2.--.using.nij.=.4.cos^2.theta
73a0 5f 69 6a 00 2d 2d 20 67 69 76 65 6e 20 74 68 65 20 44 79 6e 6b 69 6e 20 64 69 61 67 72 61 6d 20 _ij.--.given.the.Dynkin.diagram.
73c0 6e 69 6a 2c 20 64 65 72 69 76 65 20 74 68 65 20 63 6f 65 66 66 69 63 69 65 6e 74 73 20 6d 69 6a nij,.derive.the.coefficients.mij
73e0 20 6f 66 20 74 68 65 20 43 6f 78 65 74 65 72 20 67 72 6f 75 70 20 3c 73 69 20 7c 20 73 69 5e 32 .of.the.Coxeter.group.<si.|.si^2
7400 2c 20 28 73 69 73 6a 29 5e 6d 69 6a 3e 20 28 73 6f 20 6d 69 69 20 3d 3d 20 31 29 00 00 64 79 6e ,.(sisj)^mij>.(so.mii.==.1)..dyn
7420 6b 69 6e 44 69 61 67 72 61 6d 20 74 20 6e 20 3d 20 64 79 6e 6b 69 6e 46 72 6f 6d 43 61 72 74 61 kinDiagram.t.n.=.dynkinFromCarta
7440 6e 20 24 20 63 61 72 74 61 6e 4d 61 74 72 69 78 20 74 20 6e 00 00 64 79 6e 6b 69 6e 46 72 6f 6d n.$.cartanMatrix.t.n..dynkinFrom
7460 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 69 70 57 69 74 Cartan.aij.=.setDiag.0.$.(zipWit
7480 68 20 2e 20 7a 69 70 57 69 74 68 29 20 28 2a 29 20 61 69 6a 20 28 4c 2e 74 72 61 6e 73 70 6f 73 h...zipWith).(*).aij.(L.transpos
74a0 65 20 61 69 6a 29 00 2d 2d 20 6e 69 6a 20 3d 20 41 69 6a 20 2a 20 41 6a 69 2c 20 6e 69 69 20 3d e.aij).--.nij.=.Aij.*.Aji,.nii.=
74c0 20 30 00 2d 2d 20 67 69 76 65 6e 20 61 20 43 61 72 74 61 6e 20 6d 61 74 72 69 78 2c 20 64 65 72 .0.--.given.a.Cartan.matrix,.der
74e0 69 76 65 20 74 68 65 20 63 6f 72 72 65 73 70 6f 6e 64 69 6e 67 20 6d 61 74 72 69 78 20 64 65 73 ive.the.corresponding.matrix.des
7500 63 72 69 62 69 6e 67 20 74 68 65 20 44 79 6e 6b 69 6e 20 64 69 61 67 72 61 6d 00 2d 2d 20 43 61 cribing.the.Dynkin.diagram.--.Ca
7520 72 74 65 72 2c 20 53 65 67 61 6c 2c 20 4d 61 63 64 6f 6e 61 6c 64 20 70 31 37 2d 31 38 00 00 73 rter,.Segal,.Macdonald.p17-18..s
7540 65 74 44 69 61 67 20 5f 20 5b 5b 5d 5d 20 3d 20 5b 5b 5d 5d 00 73 65 74 44 69 61 67 20 63 20 6d etDiag._.[[]].=.[[]].setDiag.c.m
7560 78 40 28 28 78 3a 78 73 29 3a 72 73 29 20 3d 20 28 63 3a 78 73 29 20 3a 20 7a 69 70 57 69 74 68 x@((x:xs):rs).=.(c:xs).:.zipWith
7580 20 28 3a 29 20 28 6d 61 70 20 68 65 61 64 20 72 73 29 20 28 73 65 74 44 69 61 67 20 63 20 24 20 .(:).(map.head.rs).(setDiag.c.$.
75a0 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 67 6f 6e 61 6c map.tail.rs).--.set.the.diagonal
75c0 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 20 6d 78 20 74 .entries.of.(square).matrix.mx.t
75e0 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 6c 79 20 43 61 o.constant.c..--.(So.probably.Ca
7600 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 32 20 74 68 65 rter.defines.the.roots.of.G2.the
7620 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 79 73 29 00 2d .other.way.round.to.Humphreys).-
7640 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 72 73 20 65 78 -.They.agree.with.our.answers.ex
7660 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 72 61 6e 73 70 cept.for.G2,.which.is.the.transp
7680 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 73 20 6f 66 20 ose.--.Carter,.Simple.Groups.of.
76a0 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 78 70 65 63 74 Lie.Type,.p44-5.gives.the.expect
76c0 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 2c 20 46 2c 20 ed.answers.--.Those.of.B,.C,.F,.
76e0 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 61 6e 20 6d 61 G.are.not.--.Note:.The.Cartan.ma
7700 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 61 72 65 20 73 trices.for.A,.D,.E.systems.are.s
7720 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 20 73 69 6d 70 ymmetric......where.roots.=.simp
7740 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 20 6e 20 3d 20 leSystem.t.n.cartanMatrix.t.n.=.
7760 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 20 61 69 29 20 [[2.*.(ai.<.>.aj)./.(ai.<.>.ai).
7780 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 73 5d 00 00 2d |.aj.<-.roots].|.ai.<-.roots]..-
77a0 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 47 52 41 4d 2c -.CARTAN.MATRIX,.DYNKIN.DIAGRAM,
77c0 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 72 2c 20 72 65 .COXETER.SYSTEM...--.however,.re
77e0 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 74 72 69 63 2c flection.matrices.are.symmetric,
7800 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 00 2d 2d 20 74 .so.they.also.form.the.rows.--.t
7820 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 20 66 6f 72 6d he.images.of.the.basis.elts.form
7840 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 20 20 20 20 20 .the.columns.of.the.matrix......
7860 20 20 20 20 20 65 20 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 .....e.=.basisElt.d.....where.d.
7880 3d 20 6c 65 6e 67 74 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 =.length.r.--.dimension.of.the.s
78a0 70 61 63 65 00 77 4d 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 pace.wMx.r.=.map.(w.r).[e.i.|.i.
78c0 3c 2d 20 5b 31 2e 2e 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 <-.[1..d]].--.matrix.for.reflect
78e0 69 6f 6e 20 69 6e 20 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 ion.in.hyperplane.orthogonal.to.
7900 72 00 2d 2d 20 54 68 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 r.--.The.Weyl.group.element.corr
7920 65 73 70 6f 6e 64 69 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 esponding.to.a.root,.represented
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7960 20 6d 61 70 20 77 4d 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 .map.wMx.(simpleSystem.t.n).--.G
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generated by cgit v1.2.3 (git 2.25.1) at 2024-04-29 09:22:19 +0000