On Tue, 20 May 2003, Eugene Salamin wrote:
Are there any groups with this property that are small enough that you could show an example?
--- John Conway <conway@Math.Princeton.EDU> wrote:
There exist finite groups for which algebraic conjugacy acts differently on the rows (ie., irreducible representations) fromn the way it acts on the columns (ie., conjugacy classes). There is in fact a unique SIMPLE group with this property, the so-called Tits group. To my mind, this is the best proof that the rows and columns don'y really correspond.
JHC
I believe it first happens for some 2-groups, around order 64, but that may be out by a factor of 2. Personally, I think the Tits group is easier! What happens is that the relevent Galois group has a porition that's a 4-group, and there are 6 particular rows and columns I'll call a,b,c,d,e,f and A,B,C,D,E,F. On one set the 3-non-trivial elements of the 4-group act as (ab)(cd) , (ab)(ef), ((cd)(ef) while on the other they act as (AB)(CD), (AC)(BD), (AD)(BC). This has the effect that the number of irreducible characters whose values are rational differs by 2 from the number of conjugacy classes on which all character values are rational. There is, however, the Brauer trick, which shows that the two cycle-shapes of any element of the Galois group (on rows and on cols) are necessarily the same. From this it follows that you couldn't replace "rational" by "real" in the above statement. JHC