Last update: 17 February 2013
Now we consider the inclusions of the Schubert varieties into the flag variety. For define the Schubert classes
If is not smooth then, as discussed further below, it is not clear that is well defined. Though we consider various approaches to the analysis of below, w have not yet found a definition of which is fully satisfying (at least to us) in the singular case.
In generalized cohomology
for a reduced word of although, in equivariant cohomology and equivariant if is a reduced word for We consider various approaches to the analysis of
(a) Is given by (3.3)? As pointed out in [Tym0604578, Proposition 2.7], since is filtered by Schubert cells with and has even real dimension, the Schubert variety has no odd-dimensional cohomology, and thus, by [GKM1489894, Theorem 14], the Schubert variety is ‘equivariantly formal’ (i.e., is a GKM-space) and the moment graph theory applies. The moment graph of is the subgraph of the moment graph of with vertices If is smooth then there are no challenges in defining the pushforward and the pushforward formula in (3.3) gives that
as found, for example, in [BLa1782635, Theorem 7.2.1] (the notation for elements of is as (3.17)). For example, the inclusion for corresponds to the inclusion of moment graphs
The following example illustrates that this procedure does not work well when is not smooth. From [Kum1383959, Prop. 6.1], the singular Schubert varieties for of rank 2 are
The inclusion for (Type corresponds to the inclusion of moment graphs
but the direct "naive" application of the pushforward formula in (3.3) produces
which cannot be correct for since the right hand side does not satisfy the condition to be in im (the difference across the edge is not divisible by This answer needs to be corrected by finding so that
where the correction factor appears on vertices corresponding to the singular locus.
In the example in (5.4) we see that the moment graph knows that is not smooth! It is interesting to contrast (5.4) with the same analysis for where the pushforward formula gives
which is in im (this case works out well since is smooth).
(b) Using (5.2) to compare and Working in rank 2, use notations and as in (7.2), so that (see (4.8) and Figure 1)
Since is smooth it is reasonable to apply the pushforward formula in (3.3) which gives
Using these and computing with the formulas (3.10)-(3.12) gives the formula
which is reflected in [CPZ0905.1341, 17.3, first equation] and [HKi0903.3926, §5.2]. Similarly, with our conjectured correction factor as in (7.3), we get a formula which would provide in cohomology and K-theory but have in complex or algebraic cobordism:
(c) Combinatorial forcing: The Schubert classes satisfy
These properties do not characterize the Schubert classes; the Bott-Samelson classes also satisfy these properties. As observed, for example, in [HHH0409305, Proposition 4.3], in equivariant cohomology a degree condition can be imposed to get uniqueness. It is not clear to us how to generalize the degree \ condition to equivariant K-theory and/or equivariant cobordism. It seems plausible that in generalized equivariant cohomology the Schubert classes might be characterized by positivity properties, or by using the structure of and as in the theory of Soergel bimodules (see [Soe2329762] and [EWi2012]).
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.