Up until now, all the Schubert calculus I’ve posted about has been on the Grassmannian . But that isn’t where the story ends. The theory continues to be in development in at least two ways:
- the space of interest might be the flag variety, or more general , for a linear algebraic group and a parabolic or Borel subgroup;
- the ring structure (cohomology theory) might vary: equivariant cohomology, quantum cohomology, K-theory.
So I’ll end this series of posts with a few words on these generalizations: what works, and what — so far — remains mysterious.
Flags and or
The geometric story carries over to the variety of complete flags in in its entirety. The space has a stratification into Schubert cells, each isomorphic to an affine space. The Schubert varieties describe, again, the dimensions of the intersections of the given flag with a fixed auxiliary flag. And, again, all the computations work cleanly in coordinates (using a “forward” or “backwards” flag). Instead of being indexed by partitions, the cells of are indexed by permutations .
So, by the theorem on affine stratifications, the cohomology and K-theory of the flag manifold have additive bases coming from the Schubert varieties, and so can be described using some sort of “ring structure on permutations”, with positive structure constants corresponding to certain triple products,
Unfortunately, the combinatorial story is not nearly as nice. There is a polynomial model similar to that of , using “Schubert polynomials” to multiply these classes. But it is not a positive rule: it computes the coefficient as an alternating sum. Thus it is not a priori clear that the coefficients are positive. Another way of putting it is that it isn’t a “combinatorial” rule: it does not express as the cardinality of some set (of suitably-defined combinatorial objects).
Certain partial combinatorial rules are known: analogs of the Pieri rule, for example; and for multiplying a flag Schubert class by (the pullback of) a Grassmannian Schubert class. This story is clearly unfinished.
In the general case of , the geometric story still works cleanly, with an affine stratification by Schubert cells indexed by the Weyl group. The structure constants are still known to be positive — in fact, even in K-theory, the structure constants are known to alternate ‘correctly’ with the codimension. This phenomenon seems to be called “positivity” (reference: M. Brion, Positivity in the Grothendieck group of complex flag varieties.) However, as far as I know there are very few general combinatorial rules out there. I believe there are some conjectural rules involving root systems, but I think these still only cover special cases.
Other cohomology theories
First, for K-theory, some of the story is understood. In particular, there are combinatorial rules for K-theoretic Schubert calculus on Grassmannians.
One such rule (due to Thomas and Yong ’06) involves “K-theoretic growth diagrams”, which are similar in spirit to Fomin’s growth diagrams. They use “increasing tableaux” rather than standard tableaux, and have a “diagonal strip switching rule” rather than the diagonal box switching rule used in cohomology.
This rule, like many others, was proved in two conceptual steps:
- They showed that the rule is associative, using structural properties of their switching procedure — one of which, the uniqueness of rectifications, was not only a challenge to prove, but turned out only to be true for a certain class of tableaux!
- They showed that their rule agrees with the known (K-theoretic) Pieri rule, by direct computation. This was a bit easier to do because multiplying by a single-row partition is fundamentally a simpler thing to do (both in geometry and in combinatorics).
This two-step procedure is in pretty common use in Schubert calculus. It’s nice, since it never requires us to actually compute arbitrary products, just the Pieri rule. Associativity takes care of the rest for us.
Beyond K-theory — I don’t know much about these myself — torus-equivariant cohomology was used by Knutson and Tao to establish the puzzle rule, and I have seen other research on equivariant Schubert calculus on varieties other than the Grassmannian. Equivariant cohomology makes use of the action of the torus on the planes and flags of our vector space. Somehow it reduces cohomological computations, which a priori involve integrals over the whole space, to finite computations taking place at the fixed points of the torus action. (In there are such points; in the flag variety there are such points.)
Quantum cohomology I know even less about. But there is such a thing as “quantum Schubert calculus”, and I think it somehow involves a polynomial parameter , such that setting recovers ordinary cohomology, while setting or other values “deforms” the theory in interesting ways. But I’m way out of my depth on this one.