Homework 6. Due March 21

(1) The pdf file

https://authors.library.caltech.edu/51770/7/1106.5473.pdf

contains the character table of the binary octahedral group on page 16.

Let c_1,c_2,...,c_8 be the rows of this character table (they represent the irreducible characters of our
group). Define a graph with vertices 1,2,...,8 in which i,j are joined if (c_i,c_jc_4) is nonzero. (Show that this
implies that i is different from j and (c_j,c_ic_4) is non-zero.)
Here (,) is the inner product of class functions.
Describe explicitly the resulting graph.

Hint: Let d_j=c_j(1).
Then
j with d_j=4 is joined with an i with d_i =3,3,2
each j with d_j=3 is joined with an i with d_i =4,2
two j with d_j=2 are joined with an i with d_i =3,1
one j with d_j=2 is joined with an i with d_i =4
each j with d_j=1 is joined with an i with d_i =2