[math-fun] Re: Canonical 1-1 correspondences
Although for the symmetric group there is a canonical correspondence between the classes and irreducible representations (as has been pointed out) - my understanding is that it has been shown that for an arbitrary finite group no such correspondence exists. John McKAy
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?
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
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
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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
John Conway wrote But if you're going to allow infinite classes, then trivially the set of all correspondences is a canonical one. Yes, but consider the correspondences between X and X* you get when you choose a basis for X. A basis element b of X corresponds to the functional that is 1 on b and 0 on the other basis elements. It seems to me that these correspondences should be the canonical set of correspondences. However, I don't know of a general criterion for preferring them.
participants (4)
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Eugene Salamin -
John Conway -
John McCarthy -
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