On Mon, 19 May 2003, John McCarthy wrote:
I think that while there isn't canonical 1-1 correspondence between a linear space X over a finite field and its dual X*, there does exist a canonical infinite class of correspondences.
Is there a canonical infinite class of correspondences between the classes and irreducible representations?
But if you're going to allow infinite classes, then trivially the set of all correspondences is a canonical one. 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