From: Boudewijn Moonen
Subject: Direct correspondence between irreps and conjugacy classes in finite
Date: Tue, 19 Oct 1999 09:17:39 +0200
Newsgroups: sci.math.research
Let G be a finite group. It is then a fundamental well-known
result that the set of irreducible complex representations of
G and the set of conjugacy classes of G have the same cardinality.
It is therefore a natural question how to construct from a
given conjugacy class in G an irreducible representation thus
setting up a direct correspondence between conjugacy classes
and irreducible representations of G.
For the symmetric groups G = S_d this is indeed possible, using
Young symmetrizers or constructing the corresponding Specht modules;
for this, see the following book (among many others, of course)
William Fulton and Joe Harris
Representation Theory - A First Course
Graduate Texts in Mathematics 129
Springer 1991
On p. 46 of this book one finds the following remark:
Note also that the theorem gives a direct correspondence between
conjugacy classes in S_d and irreducible representations of S_d,
something which has never been achieved for general groups.
My question is if this is still the state of the art today. And
further, if there are other important classes of groups for which
such a direct correspondence is known; I could imagine that
something like that could hold e.g. for simple groups of Lie type.
Thanks,
--
Boudewijn Moonen
Institut fuer Photogrammetrie der Universitaet Bonn
Nussallee 15
D-53115 Bonn
GERMANY
e-mail: Boudewijn.Moonen@ipb.uni-bonn.de
Tel.: GERMANY +49-228-732910
Fax.: GERMANY +49-228-732712
==============================================================================
From: Torsten Ekedahl
Subject: Re: Direct correspondence between irreps and conjugacy classes in finite groups
Date: 19 Oct 1999 18:20:09 +0200
Newsgroups: sci.math.research
Boudewijn Moonen writes:
...
> On p. 46 of this book one finds the following remark:
>
> Note also that the theorem gives a direct correspondence between
> conjugacy classes in S_d and irreducible representations of S_d,
> something which has never been achieved for general groups.
>
Note that in one specific sense there is no such natural
correspondence, there are examples of finite groups where the action
of the group of outer automorphisms on the set of representations is
not equivalent to the action on the conjugacy classes (it is a fact
that the associated permutation representations are equivalent).
==============================================================================
From: greg@math.ucdavis.edu (Greg Kuperberg)
Subject: Re: Direct correspondence between irreps and conjugacy classes in finite
Date: 19 Oct 1999 11:38:56 -0700
Newsgroups: sci.math.research
In article <380C1B13.AEDE07D2@ipb.uni-bonn.de>,
Boudewijn Moonen wrote:
> Note also that the theorem gives a direct correspondence between
> conjugacy classes in S_d and irreducible representations of S_d,
> something which has never been achieved for general groups.
>
>My question is if this is still the state of the art today. And
>further, if there are other important classes of groups for which
>such a direct correspondence is known; I could imagine that
>something like that could hold e.g. for simple groups of Lie type.
As Torsten Ekedahl explained, it is sometimes the wrong question,
but in modified form, the answer is sometimes yes.
For example, consider A_5, or its central extension Gamma = SL(2,5).
The two 3-dimensional representations are Galois conjugates and there
is no way to choose one or the other in association with the conjugacy
classes. However, if you choose an embedding pi of Gamma in SU(2),
then there is a specific bijection given by the McKay correspondence.
The irreducible representations form an extended E_8 graph where
two representations are connected by an edge if you can get from one
to the other by tensoring with pi. The conjugacy classes also form
and E_8 graph if you resolve the singularity of the algebraic surface
C^2/Gamma. The resolution consists of 8 projective lines intersecting
in an E_8 graph. If you take the unit 3-sphere S^3 in C^2, then the
resolution gives you a surgery presentation of the 3-manifold S^3/Gamma.
The surgery presentation then gives you a presentation of Gamma itself
called the Wirtinger presentation. As it happens, each of the Wirtinger
generators lies in a different non-trivial conjugacy class.
In this way both conjugacy classes and irreps. are in bijection
with the vertices of E_8.
--
/\ Greg Kuperberg (UC Davis)
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