Last update: 21 February 2013
In this section we set up notations and recall how one is able to work with the K-theory of “purely combinatorially”. This is possible because (in analogy with the cohomology and the combinatorial theory of Schubert polynomials) the K-theory of is isomorphic to a quotient of which is a ring of Laurent polynomials. In section 5 we shall explain how our results about relate to and generalize well known results about Schubert polynomials and
The K-theory K(G/B)
If is a quasi-projective variety let
If is a morphism of projective varieties then we have maps
See [CGi1433132, §5.2] for K-theory background. If is smooth then the isomorphism between and is given by assigning to the class of a sheaf the alternating sum of the sheaves in a locally free resolution, see [BSe1958, §4] and [Har1977, III Ex. 6.9]. There is always a map which assigns to a locally free sheaf the underlying vector bundle and the results of [Pit1972] imply that this map is an isomorphism when see Proposition 1.5 below. Thus, in the case which we wish to consider in this paper, all three K-theories are isomorphic.
Let be a complex connected simply connected semisimple Lie group. Fix a maximal torus and a Borel subgroup such that The Bruhat decomposition says that is a disjoint union of double cosets of indexed by the elements of the Weyl group The flag variety is the projective variety formed by the coset space and the Bruhat decomposition of induces a cell decomposition of For each the subset is the Schubert cell and its closure is the Schubert variety. The formulas
define the length of and the Bruhat-Chevalley order on the Weyl group, respectively. It follows from the Bruhat decomposition (see lecture 4 by Grothendieck in [Che1958]) that
where are the Schubert varieties in and is the structure sheaf of extended to by defining it to be 0 outside
For any group let be the Grothendieck group of complex representations of Let where the are the fundamental weights of the Lie algebra of We shall use the “geometric” convention (see [CGi1433132, 6.1.9(ii)]) and let be the element of corresponding to the character determined by Then
and Weyl group action determined by for and In this way is a Laurent polynomial ring and is the subalgebra of "symmetric functions" in
Suppose that is a Since where is the unipotent radical of we can extend to be a by defining the action of to be trivial. Define a vector bundle
so that is the set of pairs modulo the equivalence relation This construction induces a ring homomorphism
If is a then in This is because the map
is an isomorphism between and the trivial bundle Define by for Then the map in (1.2) gives an isomorphism (see Proposition 1.5 below)
where is the ideal generated by Equivalently, where acts on by if is a
K(G/P) for a parabolic subgroup P
A similar setup works when is replaced by any parabolic subgroup containing The coset space is a projective variety and the Bruhat decomposition takes the form where is the subgroup of given by where is the Lie algebra of The Schubert varieties are the closures of the Schubert cells in
Write where is the unipotent radical of and is a Levi subgroup. The Weyl group of is and
is the subring of functions in The same construction as in (1.2) with replaced by and replaced by gives a ring homomorphism
Proposition 1.5. Let be a connected simply connected semisimple Lie group and let be a maximal torus of Let be a parabolic subgroup of with Levi decomposition and let be the Weyl group of Then
where is the ideal generated by and is the map given by for Equivalently,
Let be the Grothendieck group of vector bundles on and let be the map which assigns to a locally free sheaf its underlying vector bundle. Let be the composition
Since which is free abelian by the Bruhat decomposition. Since the unipotent radical of is contractible the projection is a homotopy equivalence. Thus is free abelian and we may apply the results of [Pit1972] to conclude that is surjective, is projective over with rank and is a free of the same rank. (Note: The results of [Pit1972] can be applied since and are the complexifications of compact groups.)
Since is surjective the map is also surjective. Then, since and are both free of rank it follows that must be an isomorphism. This means two things: (1) that we can identify and and (2) that is surjective.
The kernel of is identified by using (1.3).
Transfer from K(G/B) to K(G/P)
Although we will work primarily with it is standard to transfer results from to results on This can be accomplished with the following proposition. The proof will be given in section 5.
Proposition 1.6. If is the natural projection then the induced map is an injection. This map is given explicitly by
where is the unique element of longest length in the coset
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.