On the “longest element”

MV cycles are the components of a variety defined as the intersection of two infinite and coinfinite subsets of the affine grassmanian. In Kamnitzer the second item in the intersection is described in terms of the “longest” element of the Weyl group. This is troublesome because the affine Weyl group doesn’t seem to have a longest element.

The papers by Anderson and Mirkovic-Vilonen however describe the two in terms of and . In retrospect the failure of the (affine)Weyl group to fully parametrize our world is not new as we saw already it fixes imaginary roots.

For , I have an inkling at this time to augment the affinte Weyl group with an articial “longest element” which acts by rotating 180 degrees around the axis.

• In terms of chamber weights this element will swap the “two” imaginary chamber weights and will create a second parabola, downward pointing and with opposite “c”-value.
• In terms of polytopes this will make our Pseudo Weyl polytopes in intersections of opposing parabolas, something like those I posted early on.
• In terms of Kamnitzer’s plucker relations this new element may categorize as a third simple reflection (though, unfortunately not fixing the real fundamental weights) possibly bringing some of his equations to bear.

That last point was one of the problems I mentioned in my last post to which I will now add detail. Kamnitzer’s inequalities restrict the values of M’s, for example, whenever we have triple satisfying ,, and either or or . If there are only two simple reflections then only two triples will be considered (e,1,2) and (e,2,1) in both cases .

Then Kamnitzer’s treatment imposes no restriction and allows all PW-polytopes to be MV-polytopes. This may be the case, indeed it is the case for . But even if this particular issue is in fact a non-issue it at least illustrates how offcolour things seem.