Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last update: 21 February 2013
The Pieri-Chevalley formula
In this section we shall inductively apply formula (3.1d) to obtain an expansion of the product
in terms of the basis
We use the path model of P. Littelmann to keep track of the combinatorics involved in iterating formula (3.1d).
The path model
be the weight lattice and let
A path in is a piecewise linear map
such that Let
be a path, let be a simple root and let
be the function given by
At this function gives the position of in the
Let be the minimal value of
and define functions
The root operators (see [Lit1997] Definitions 2.1 and 2.2) are operators on the paths given by
where we use 0 to denote the “null path”.
Fix a dominant weight Let
be the path given by
be the set of all paths obtained by applying sequences of root operators
This is the set of Lakshmibai-Seshadri paths of shape P. Littelmann [Lit1994] has shown that this set of
paths is finite and can be characterized in terms of an integrality condition. We shall not need this alternative characterization.
Let be the stabilizer of The cosets in
are partially ordered by the Bruhat-Chevalley order. Use a pair of sequences
to encode the path given by
We shall write
Every path is of this form. Littelmann introduced this set of paths
as a model for the Weyl character formula. He proved that
is the half-sum of the positive roots.
Application of the path model
Fix a dominant weight and let
The initial direction of is
A maximal lift of with respect to is a choice of representatives
of the cosets
each is maximal in Bruhat order such that
final direction of with respect to is
is a maximal lift of
with respect to
For each such that
of is the set of paths
where is maximal such that
Statement (a) is [Lit1995] Lemma 2.1a, statement (b) is [Lit1994] Lemma 5.3b, and statement (c) follows from [Lit1994] Lemma 5.3c and [Lit1995] Lemma 2.1e. All of
these facts are really coming from the explicit form of the action of the root operators on the Lakshmibai-Seshadri paths which is given in [Lit1994] Proposition 4.2.
The consequence of (a) and (c) is that
Let be such that
It follows from (a) that
Suppose that and
are maximal lifts of with respect to and respectively.
If then is not divisible by
It follows that
and thus that
If then all the are divisible by
and it follows that
if We conclude that
Let be a dominant integral weight and let Then
as operators on
The proof is by induction on The base case
is formula (3.1d). Let
Let be a dominant integral weight. In K(G/B)
Since the sheaf is supported on the single point
where is a vector bundle on is the class of a bundle supported on the single point
Thus, since is the class of a line bundle we have
By Corollary 3.6,
and so the result follows from Theorem 4.2.
The following example illustrates how one computes the product
in for the group of type
In this case
and the starting path is the straight line path from the origin to the point
The paths in the set
are the paths in the following diagrams.
These paths yield the following data:
and thus we get
Notes and References
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.