Last update: 17 February 2013
In this section we review the formulas for the Bott-Samelson classes as established in, for example, [HKi0903.3926, CPZ0905.1341, BEv0968883, BEv1044959]. Though some of these references are not considering the equivariant case, the same machinery applies to define these classes in In particular, this is the place in the theory where the BGG/Demazure operators are derived from the geometry. These operators play a fundamental role in the combinatorial study of
Using the notation for parabolic subgroups and partial flag varieties as in (2.3), if and
Then, in the setting on Theorem 3.1,
and and correspond to
where is given by the operator in the nil affine Hecke algebra given by
with the set of positive roots for A special case is when for which
is the BGG-Demazure operator (see [BEv0968883, Cor.-Def. 1.9]). The calculus of the operators is controlled via the identities in Section 8.
For a sequence with define the Bott-Samelson class
where, in the notation of (3.17),
Theorem 4.1. [BEv1044959, Prop. 1], [HKi0903.3926, Prop. 3.1], [KKu0895705, Lemma 3.15], see also [HHH0409305, Proposition 4.1]) The generalized cohomology
where, for each is a fixed reduced word for
Let us explain where this comes from. Let be a Following [Ful1644323, Example 1.9.1], or [CGi1433132, §5.5], a cellular decomposition of is a filtration
by closed subvarieties such that are isomorphic to a disjoint union of affine spaces for The "cells" of are the
Theorem 4.2. (see [Gro1958, Prop. 7]; [Ful1644323, Example 1.9.1] who refers to [Cho0078006]; [CGi1433132, Lemma 5.5.1]; [BEv1044959, Proposition 1]; [HKi0903.3926, Theorem 2.5]) Let be a with a cellular decomposition. Then has an given by resolutions of cell closures (choose one resolution for each cell).
For the Bruhat decomposition
and the Schubert varieties are the closures of the Schubert cells. Let be the minimal parabolics of (with and and let be the corresponding simple reflections in The group is generated by Let be a reduced word for Then the Bott-Samelson variety provides a resolution of
Then following, for example, the proof of [BEv1044959, Prop. 2], since the diagram
it is a pullback square. Thus
The following result then follows by induction.
Theorem 4.3. ([HKi0903.3926, Theorem 3.2], [BEv1044959, Proposition 2]) If is a sequence in and is as in (4.5) then
Theorem 4.3 says that the values on the vertices of the element on the moment graph of are exactly the coefficients of the terms in the expansion of
For example, in type
provides the expansion of in the basis An example of the pushpull in (4.6) in the case of type
has moment graphs as in Figure 1, and the computation in (4.7) for this example is
In the same way that Theorem 3.1 provides one can obtain
and, if is the inclusion then the change of group homomorphisms
are given, combinatorially, by
with the set of positive roots for The pushforward is similar to the pushforward operator appearing in (4.1) except acting on the left factor of (see, for example, the definition of in [Kaj2010, §7]).
This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.