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.)