Thread: Character/ Representation Theory

Can anybody help me with these questions as I am completely lost and haven't done any character theory or rep theory for a long time


(1) Let X be the matrix comprising the character table of a group.
Show that X is non-singular & hence prove that the number of real-valued characters is equal to the number
of self-inverse conjugacy classes.

(2) Show that in a group G of odd order, no element other than the identity
is conjugate to its inverse. Deduce that in a group of odd order, the trivial character is the only
irreducible character that is real-valued.

Originally Posted by Turloughmack

Can anybody help me with these questions as I am completely lost and haven't done any character theory or rep theory for a long time


(1) Let X be the matrix comprising the character table of a group.
Show that X is non-singular & hence prove that the number of real-valued characters is equal to the number
of self-inverse conjugacy classes.

This is a sketch of the proof borrowed from the book "Representations and characters of groups" by Liebeck (p264).

The irreducible character of G are linearly indepdent over as functions from G to , which follows that X is non-singular.

Let be the complex conjugate of matrix X. For each irreducible character , it is known that is also an irreducible character. Thus, we can construct a permutation matrix P induced from such that .

Similarly, for each conjugacy class , the entries in the column of X corresponding to are the complex conjugates of the entries in the column of X corresponding to , where denotes the conjugacy class of its representative , and the conjugacy class of . Similarly, we can construct a permutation matrix Q induced from such that . It follows that . Thus, we see that P and Q have the same trace. Recall that trace of a permutation matrix corresponds to the number of fixed points by the permutation. Therefore, the number of irreducible real-valued characters (corresponding to the fixed points of the first permutation) is the trace of P, and the number of self-inverse conjugacy classes (corresponding the the fixed points of the second permutation) is the trace of Q, which are equal.

(2) Show that in a group G of odd order, no element other than the identity
is conjugate to its inverse. Deduce that in a group of odd order, the trivial character is the only
irreducible character that is real-valued.

Note that iff each is conjugate to , i.e., for some . Then, , , etc. Since G is of odd order, we see that (verify this). Thus, only the identity is conjugate to its inverse. Hence, the self-inverse conjugacy class is {1}. It follows that the number of irreducible real-valued characters is only one by (1).

Alternatively, one can appeal to the amazing theorem that for a finite group if then the number of self-conjugate representations of is . Although, the proof of this comes startling close to the ideas that the previous paster mentioned.