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MATH 7364 Lecture 1 problems

MATH 7364 Lecture 1 problems - PROBLEMS ON SYMPLECTIC...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 1. Kleinian singularities Problem 1.1. Let G be a finite subgroup of SO 3 ( R ) . Consider its action on the unit sphere. Show that any non-unit element of G fixes a unique pair of opposite points and that the sta- bilizer of each point P is cyclic of some order, say, n P . Choose representatives P 1 , . . . , P k of orbits with non-trivial stabilizers, one in each orbit. Show that 2 ( 1 - 1 n ) = k i =1 ( 1 - 1 n P i ) . Use this to show that the finite subgroups of SO 3 ( R ) are precisely the following: (1) The cyclic group of order n – generated by a rotation by the angle of 2 π/n . (2) The dihedral group of order 2 n with n > 2 : the group of rotational symmetries of a regular n -gon on the plane inside of the 3D space (a regular 2 -gon= a segment). (3) The group of rotational symmetries of the regular tetrahedron isomorphic to the al- ternating group A 4 . (4) The group of rotational symmetries of the regular cube/octahedron isomorphic to the symmetric group S 4 .
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