4 1. INTRODUCTION

k-shape λ, we then define the cohomology k-shape function

Sλk) (

to be the weight

generating function

Sλk)[X] (

=

T ∈Tabk

λ

xwt(T )

, (1.15)

where Tabλ

k

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,

(1.16)

Weakλk)[X] (

=

Sλk)[X] (

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

Sλk)[X] (

is a

symmetric function with the decomposition

Sλk)[X] (

=

μ∈Ck

|Pk(λ,

μ)|

Weakμk−1)[X] (

. (1.17)

It is from this theorem that we deduce Theorem 1.2. Letting λ ∈

Ck+1

and

μ ∈

Ck,

we have

bμλ

(k)

=

Weakλk)[X] (

,

sμk−1)[X] (

k

=

Weakλk)[X] (

,

sμk−1)[X] (

k−1

=

Sλk)[X] (

,

sμk−1)[X] (

k−1

=

|Pk(λ,

μ)|

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

(1.18)

sμk)[X; (

t] =

λ∈Ck+1 [p]∈Pk(λ,μ)

tcharge(p)

sλk)[X; (

t] ,

and its ungraded version is

sμk)[X] (

:=

sμk)[X; (

1]. We trivially have from this def-

inition that

sμk)[X] (

=

sμk)[X] (

when μ ∈

Ck+1.

Further, from (1.18) at t = 1,

Theorem 1.2 and (1.7), we have that

(1.19)

sμk)[X] (

=

sμk−1)[X] (

for μ ∈

Ck.

The Pieri rule for ungraded homology k-shape functions is given by

Theorem 1.5. For λ ∈ Πk and r ≤ k − 1, one has

hr[X]

sλk)[X] (

=

ν∈Πk

sνk)[X](

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 [14].

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 [9]. The generalization of this family to give a direct