4 1. INTRODUCTION
k-shape λ, we then define the cohomology k-shape function
to be the weight
denotes the set of reverse-maximal k-shape tableaux of shape λ.
We show the k-shape functions are symmetric and that when λ is a k + 1-core,
mod Ik−1 ,
(see Proposition 4.11). We give a combinatorial expansion of any k-shape function
in terms of dual (k − 1)-Schur functions.
Theorem 1.4. For λ ∈ Πk, the cohomology k-shape function
symmetric function with the decomposition
It is from this theorem that we deduce Theorem 1.2. Letting λ ∈
using (1.7), (1.11) for k − 1, (1.16), and Theorem 1.4.
A (homology) k-shape function can also be defined for each k-shape μ by
and its ungraded version is
1]. We trivially have from this def-
when μ ∈
Further, from (1.18) at t = 1,
Theorem 1.2 and (1.7), we have that
for μ ∈
The Pieri rule for ungraded homology k-shape functions is given by
Theorem 1.5. For λ ∈ Πk and r ≤ k − 1, one has
where the sum is over maximal strips ν/λ of rank r.
When λ is a k-core, Theorem 1.5 implies the Pieri rule for (k−1)-Schur functions
proven in .
Here we have introduced the cohomology k-shape functions as the generating
function of tableaux that generalize k-tableaux (those defining the dual k-Schur
functions). There is another family of “strong k-tableaux” whose generating func-
tions are k-Schur functions . The generalization of this family to give a direct