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
Previous Page Next Page