Monday, 4 October 2010

gr.group theory - Faithful characters of finite groups

Here is a short proof of the weaker version of the statement from Question 1 (giving a polynomial with rational coefficients). Let's think of characters as functions on conjugacy classes. Then chi(1)=n=rmdim(V), and chi(g) for gne1 has smaller absolute value than n (since the representation is faithful and eigenvalues of g in chi are roots of 1). In particular, chi(g)nen. Now let P be the interpolation polynomial such that P(n)=|G| and P(x)=0 for any other value x of chi. Then P(chi) is the regular character, and it's easy to see that P has rational coefficients.



However, there seems to be a counterexample to the statement that P can be chosen to have integer coefficients. Namely, take G=A5, and chi the 5-dimensional character.
Its values are well known to be 5,0,1,1, so we can take P0=(x3x)/2, and any other
polynomial which works will be of the form P=P0Q, where Q is another polynomial (as P must vanish at 0,1,1). If P has integer coefficients, then Q/2=P/(x3x) must have integer coefficients, so values of Q at integers are even. On the other hand, we must have Q(5)=1, contradiction.

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