From: Robin Chapman Subject: Re: Artin's Theorem on Rational Characters Date: Mon, 13 Mar 2000 09:18:12 GMT Newsgroups: sci.math.research Summary: All group character integral combinations of permutation characters? In article <2as5e4l34p57@forum.swarthmore.edu>, emptyclass@yahoo.com (Cecil Andrew Ellard) wrote: > Artin's Theorem says that any rational character of a finite group can > be written as a linear combination, using rational coefficients, of > characters induced from the identity character on its cyclic > subgroups. I think the following shows that its statement becomes > false if "rational coefficients" is replaced with "integral > coefficients": Let G be Sym(3) of order 6, and let X(1), X(2), and > X(3) be the identity character, the linear non-identity character and > the non-linear irreducible character of G respectively. The transitive > permutation characters induced from the cyclic subgroups of orders 1, > 2, and 3 of G are X(1)+X(2)+2X(3), X(1)+X(3), and X(1)+X(2) > respectively, and these are linearly independent over the complex > numbers. X(1) can be expressed as a rational linear combination of > these permutation characters (using coefficients -1/2, 1, and 1/2 > respectively) and therefore (since the expression is unique, by linear > independence) X(1) can not be expressed as an integral-linear > combination of these permutation characters. The question remains: is > Artin's Theorem true if we require integral coefficients, but allow > linear combinations of all permutation characters (not just those > induced from cyclic subgroups)? This is the same as asking if each > rational character of a finite group lies in P(G). No. As a counterexample, take the quaternion group of order 8. This has an irreducible rational character of degree 2, which is not an integral linear combination of characters induced from trivial representations of subgroups. -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: serge bouc Subject: Re: Artin's Theorem on Rational Characters Date: Mon, 13 Mar 2000 12:09:13 +0100 Newsgroups: sci.math.research Cecil Andrew Ellard wrote: > > The question remains: is > Artin's Theorem true if we require integral coefficients, but allow > linear combinations of all permutation characters (not just those > induced from cyclic subgroups)? This is the same as asking if each > rational character of a finite group lies in P(G). > No. In general, the subgroup of the group of rational characters generated by permutation chararacters has finite index, but it can be a proper subgroup. If G is a p-group for some prime p, then the answer is yes. This was proved independently by Ritter and Segal in 1972. Best regards. -- Serge Bouc sbouc@nnx.com bouc@math.jussieu.fr www.math.jussieu.fr/~bouc ============================================================================== From: serge bouc Subject: Re: Artin's Theorem on Rational Characters Date: Mon, 13 Mar 2000 17:09:45 +0100 Newsgroups: sci.math.research serge bouc wrote: > If G is a p-group for some prime p, then the answer is yes. This > was proved independently by Ritter and Segal in 1972. > More precisely: any rational representation of G is a (virtual) permutation representation. -- Serge Bouc sbouc@nnx.com bouc@math.jussieu.fr www.math.jussieu.fr/~bouc