Schubert Products and
Last update: 2 September 2012
The nil affine Hecke algebra has
Writing these classes in terms of sections on the moment graph
making it easy to see that
Here we will use
In this case
Moment graph pictures:
the first few Bott-Samelson classes are
In this case, all the Schubert varieties are smooth so that
are reflected in [CPZ, 17.3 first equation] and [HK, §5.2].
Note that in ,
and, in ,
so that, in and , one does have
Pieri-Chevalley formulas: Using
which is not very pleasing.
A more pleasing derivation of the last Pieri-Chevalley formular for Type is
is a decomposition of into Schubert classes.
Schubert products: The moment graph sections provide fairly quick computations of the products
An example of a check of one of these products is:
provides a quick check that these formulas agree with the computations for
equivariant cohomology and equivariant K-theory which are appear in [GR, §5].
More specifically, in [GR] we have
Notes and References
These notes are part of an ongoing project with Nora Ganter studying generalised cohomologies of flag varieties.
[CPZ] B. Calmès, V. Petrov, K. Zainoulline, Invariants, torsion indeices and oriented cohomology of complete flags, arxiv:0905.1341
The elliptic Weyl character formula, arXiv:1206.0528.
[GR] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combinatorics 25 (2004) 1263-1283.
[HK] J. Hornbostel and V. Kiritchenko, Schubert caculus foralgebraic cobordism, arxiv:0903.3936v3.