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Fusion maps and pre-fusion maps.

Let tex2html_wrap_inline3611 and tex2html_wrap_inline3613 be as above. Moreover, let tex2html_wrap_inline3615 . In order to approximate the map tex2html_wrap_inline3617 we will work with idempotent maps on tex2html_wrap_inline3619 . An approximation of j will be called a pre-fusion map. Moreover, any map on tex2html_wrap_inline3623 that describes which elements of tex2html_wrap_inline3625 map into the same conjugacy class of subgroups of G will be called a fusion map. If we can determine one such fusion map f we can identify tex2html_wrap_inline3631 and tex2html_wrap_inline3633 and thus have found j = f.

Def570

For any pre-fusion map tex2html_wrap_inline3669 on tex2html_wrap_inline3671 we denote the set of all tex2html_wrap_inline3673 -classes on tex2html_wrap_inline3675 by tex2html_wrap_inline3677 and denote by tex2html_wrap_inline3679 the canonical map from tex2html_wrap_inline3681 to tex2html_wrap_inline3683 . Thus, for any tex2html_wrap_inline3685 , its tex2html_wrap_inline3687 -class is tex2html_wrap_inline3689 . The relation between the pair tex2html_wrap_inline3691 and the sets tex2html_wrap_inline3693 and tex2html_wrap_inline3695 is illustrated by the following diagram.

displaymath3603

Due to (iii), we may regard tex2html_wrap_inline3697 as an equivalence relation on the set tex2html_wrap_inline3699 of all subgroups of G, with the property that tex2html_wrap_inline3703 whenever U and V are conjugate subgroups of G. Moreover, we may regard f as a function from the disjoint union of the sets of subgroups tex2html_wrap_inline3713 of the groups tex2html_wrap_inline3715 to tex2html_wrap_inline3717 via tex2html_wrap_inline3719 for any subgroup tex2html_wrap_inline3721 .

Note that, since tex2html_wrap_inline3723 is surjective and tex2html_wrap_inline3725 , also g must be surjective.

If, for example, tex2html_wrap_inline3729 is the identity map on tex2html_wrap_inline3731 and tex2html_wrap_inline3733 is the global equivalence on tex2html_wrap_inline3735 defined by tex2html_wrap_inline3737 for all tex2html_wrap_inline3739 then tex2html_wrap_inline3741 is a pre-fusion map since tex2html_wrap_inline3743 and, vacuously, tex2html_wrap_inline3745 implies tex2html_wrap_inline3747 .

The following lemma provides the termination condition of the approximation process.

  La588

proof591

The approximation process is guided by a partial order on the set of pre-fusion maps on tex2html_wrap_inline3771 .

Let tex2html_wrap_inline3773 and tex2html_wrap_inline3775 be pre-fusion maps on tex2html_wrap_inline3777 . We call tex2html_wrap_inline3779 stronger than tex2html_wrap_inline3781 (and write tex2html_wrap_inline3783 ) if there are maps tex2html_wrap_inline3785 and tex2html_wrap_inline3787 such that tex2html_wrap_inline3789 and tex2html_wrap_inline3791 , i.e. if the following diagram is commutative.

displaymath3604

Let tex2html_wrap_inline3793 and tex2html_wrap_inline3795 be conjugate subgroups of G. Then tex2html_wrap_inline3799 . This allows us to define tex2html_wrap_inline3801 for tex2html_wrap_inline3803 . Thus, if we define an equivalence tex2html_wrap_inline3805 on tex2html_wrap_inline3807 by tex2html_wrap_inline3809 if tex2html_wrap_inline3811 , then tex2html_wrap_inline3813 is a pre-fusion map. (Because then, tex2html_wrap_inline3815 and tex2html_wrap_inline3817 implies tex2html_wrap_inline3819 .) Moreover, tex2html_wrap_inline3821 is stronger that tex2html_wrap_inline3823 , since, with tex2html_wrap_inline3825 for all tex2html_wrap_inline3827 and tex2html_wrap_inline3829 the identity map on tex2html_wrap_inline3831 we have tex2html_wrap_inline3833 .

Every group contains one trivial subgroup. There are r elements tex2html_wrap_inline3837 in tex2html_wrap_inline3839 corresponding to the trivial subgroups of the r maximal subgroups of G. Define

displaymath3605

then tex2html_wrap_inline3845 is a pre-fusion map and tex2html_wrap_inline3847 is stronger than tex2html_wrap_inline3849 .

  La619

proof622


next up previous
Next: Subgroups of subgroups. Up: Approximating the fusion map. Previous: Approximating the fusion map.

Götz Pfeiffer Wed Oct 30 09:52:08 GMT 1996