notice that the trivial and sign representations are both 1-dimensional, and the geometric is 2-dimseional. 1-D representations are 1x1 matrices i.e. just a number, so the identity is just 1.
On the other hand, 2-D reps are 2x2 matrices. So the identity is the 2x2 identity matrix. This has trace 2 therefore the character is 2.
Why dont you mention the representations of (1)? Why is this got value 1 in the trivial and sign representations and 2 in the geometric representations?
to autumn16190:
notice that the trivial and sign representations are both 1-dimensional, and the geometric is 2-dimseional. 1-D representations are 1x1 matrices i.e. just a number, so the identity is just 1.
On the other hand, 2-D reps are 2x2 matrices. So the identity is the 2x2 identity matrix. This has trace 2 therefore the character is 2.
gauss1729 2 years ago
Why dont you mention the representations of (1)? Why is this got value 1 in the trivial and sign representations and 2 in the geometric representations?
autumn16190 2 years ago