Mon 2/21/00

 * went over S4 isomorphism with solid cube group, 
   numbering opposite corners same, 1-4

 * discussed Cn,m and pascal's triangle as part of
   analyzing S4

 * Went over matrix mulitplication, rotation of matrices, 
   3x3 matrices for face-centered 90 rotations of solid cubes.

   [  cos(theta)  sin(theta) 0 ]
   [ -sin(theta)  cos(theta) 0 ]
   [  0           0          1 ]
 
   and similar with theta=90.


assignment:

*  Prove or disprove: Any element (a) from a group "G" is conjugate
   to it's inverse (a'), i.e. there exists "b" such that
   b a b' = a'
   (Hint: think Abelian)

*  Show that Cn,(m-1) + Cn,m = C(n+1),m
   i.e. Pascal Triangle relation for Cn,m = n!/(m! (n-m)!)

*  Find a 3x3 matrix corresponding to one of the 120 degree
   corner rotations of the solid cube.  (Hint: use the
   90 face-centered ones we did in class as generators
   to get them.  Don't forget the right-to-left vs left-to-right issue.)