Last update: 21 February 2013
Let us explain how our results in are related to the cohomology and Schubert polynomials. The transfer is by way of the Chern character ch.
If is a finite CW complex then the Chern character (see [Mac1968, Ch. 10], [Hir1995, §23-24], [Har1977, App. A])
is a natural ring isomorphism, i.e. if is continuous then If is a morphism of nonsingular projective varieties then the Grothendieck-Riemann-Roch theorem [Har1977, App. A Theorem 5.3], henceforth G-R-R, says
where is Todd class of the relative tangent sheaf of If is a line bundle on with first Chern class then the Chern character and the Todd class of are the elements of given by
The expression is finite sum since in whenever
as the quotient of a polynomial ring
Let Let be the Lie algebra of and let be the ring of polynomials on (over This is a polynomial ring in the variables (the simple roots). Let be the homomorphism given by for all If then is the constant term of It is a classical theorem of Borel (see [BGG1973, Prop. 1.3]) that
where is the ideal of generated by Proposition 1.5 is the K-theory analogue of Borel’s theorem.
Let The element determines a character of denoted by (see section 1). Let be the first Chern class of the line bundle where is the map in (1.2). Because of the isomorphism in (5.1) we often abuse notation and write All of the maps in the following commutative diagram are isomorphisms. (Recall that we can identify and
The left hand ch map is obtained by viewing and as subsets of and respectively, where is the classifying space of
The relation between and
Let be a simple root and let be the parabolic subgroup with Lie algebra spanned by and the root space Let corresponding
|A proof of Proposition 3.3:|
Since is torsion free, we can check the identity by applying ch to both sides and checking the result in cohomology.
The G-R-R for the map says
where is the bundle of tangents along the fibres, which is the line bundle associated to the Since this module has weight and so (5.3) becomes
The BGG operator is explicitly given by the formula
see [Dem1974]. Thus, by applying to both sides of (5.4) we obtain
The strategy now is to manipulate the right hand side of (5.6) using the “skew-Leibniz” rule satisfied by to obtain For convenience, let Then the right side of (5.6) is
and we claim
The first equality is trivial and the second can be proved by formal computation (carefully done!) or by applying the G-R-R again. Using (a) and (b) we obtain
Now cancelling the in the second term on the right and recalling that we find
We see that
which relates to to in an explicit way. In fact, if one wished one could reverse the argument and derive the formula (5.5) for from the formula for
A proof of Proposition 1.6
Proposition 1.6. If is the natural projection then the induced map is an injection.
Since is torsion free there is a natural injection and so it suffices to check that the pull-back in rational cohomology is injective. Since the odd cohomology groups of the base and fiber are zero the Serre spectral sequence of the bundle shows that is injective even for
Dictionary between and
In summary, the Chern character gives an isomorphism
Let be the element which is Poincaré dual to the fundamental cycle of in This element is called a Schubert polynomial. Then
From a general fact (see [Ful1984, Ex. 15.2.16] or [CGi1433132, 5.8.13(i) and p. 289])
If is a simple root and is the corresponding then
in and respectively. As illustrated in (5.7) each of these two formulas can be derived from the other via the use of the Grothendieck-Riemann-Roch Theorem. This means that the following formulas (Proposition 3.4 and [BGG1973, Th. 3.14])
Our new Pieri-Chevalley formula in Theorem 4.3, and Chevalley’s classical Pieri formula in [Che1994, Prop. 10], are
where the sum is over all such that and there is a root such that Chevalley’s formula can be obtained from ours formula by subtracting from each side, applying the Chern character ch, and comparing the lowest degree terms on each side.
This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).
Research supported in part by National Science Foundation grant DMS-9622985.