18.755 problems due 9/14/15 in class
The two boldface items are the actual problems. The long section in the middle
is just setting up notation for the second problem.
1. In class Friday we considered a group
H = Hx,ξ,z | x ∈ R, ξ ∈ R, z = eiζ ∈ S 1
with the multiplication law
Hx1 ,ξ1 ,z1 · Hx2 ,ξ2 ,z2 = Hx1 +x2 ,ξ1 +ξ2 ,z1 z2 eix1 ξ2 .
(This was a group of linear transformations of an infinite-dimensional
vector space of functions on R.) Can you find a collection of finite matrices
satisfying this multiplication law?
There is an answer in which all the matrices Hx,ξ,z are the identity. That’s not
so interesting: I’m asking for distinct matrices.
Here is a possible hint. Rewrite the multiplication law as
Hx1 ,ξ1 ,eiθ1 · Hx2 ,ξ2 ,eiθ2 = Hx1 +x2 ,ξ1 +ξ2 ,eiθ1 +θ2 +x1 ξ2 ,
or, more compactly,
(x1 , ξ1 , θ1 ) · (x2 , ξ2 , θ2 ) = (x1 + x2 , ξ1 + ξ2 , θ1 + θ2 + x1 ξ2 ).
Look at the matrices
Nx,ξ,θ

1 x

= 0 1
0 0

θ
ξ.
1
2. Same question with R and S 1 replaced by Z/pZ (integers mod a
prime p) and the group µp of complex pth roots of 1:
Hx1 ,ξ1 ,z1 · Hx2 ,ξ2 ,z2 = Hx1 +x2 ,ξ1 +ξ2 ,z1 z2 e2πix1 ξ2 /p
(xi ∈ Z/pZ, ξi ∈ Z/pZ, zi ∈ µp ).
(To make sense of this formula, notice that if m ∈ Z/pZ, then e2πim/p is
a well-defined pth root of 1. The formula defines a group of order p3 .)
Can you find a group of finite matrices satisfying this multiplication law?
How small can you make the matrices?
This problem is already interesting for p = 2; you can get partial credit just for
doing that case, and it might suggest a general solution to you.