Reflection Groups, Root Systems, and SingularitiesBojko BakalovDepartment of MathematicsNorth Carolina State UniversityEnloe High School Future Scientists ClubOctober 22, 2009, Raleigh, NCLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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reflection with respect to the plane 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 flips 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 reflection with respect to the plane 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 flips 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Taking compositions we obtain all 6 permutations of the coordinates 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: 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: 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: 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They form the symmetric group 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Group: the composition of two permutations is a permutation, and the inverse of a permutation is a permutation.The symmetric group LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KVEiU0YnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYpUSJuRidGL0YyRjVGOEY6Rj1GLy9GPlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== has 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 elements.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial" foreground="[51,51,153]" size="16">Root Systems</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=A plane is determined by the two vectors of length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNictSSNtbkdGJDYlUSIyRicvJSVzaXplR1EjMTRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdRJmZhbHNlRidGO0Y4Rj5GQUY7 perpendicular to it.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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 the 3 planes 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 we obtain 6 vectors:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEmcGxvdHNGJ0YvRjJGOUY5RjUvJStleGVjdXRhYmxlR0Y9Rjk=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