1.3. k-SHAPE FUNCTIONS 3
Hereafter, we will label k-Schur functions by cores rather than k-bounded partitions
using the bijection between
given by the map rs.
Theorem 1.2. For all λ ∈
and μ ∈
We conjecture that the charge statistic on paths gives the branching polyno-
Conjecture 1.3. For all λ ∈ Ck+1 and μ ∈ Ck,
1.3. k-shape functions
The proof of Theorem 1.2 relies on the introduction of a new family of symmet-
ric functions indexed by k-shapes. These functions generalize the dual (aﬃne/weak)
k-Schur functions studied in [15, 6, 9].
The images of the dual k-Schur functions
form a basis for
Λ/Ik where Ik = mλ : λ1 k (1.10)
of the space Λ of symmetric functions over Z, while the ungraded k-Schur functions
form a basis for the subring
= Z[h1, . . . , hk] of Λ. The Hall
inner product · , · : Λ × Λ → Z is defined by mλ , hμ = δλμ. For each k there is
an induced perfect pairing · , ·k : Λ/Ik ×
→ Z, and it was shown in  that
Moreover, it was shown in  that
represents Schubert classes
in the cohomology of the aﬃne Grassmannian GrSLk+1 of SLk+1.
The original characterization of
was given in  using k-tableaux.
A k-tableau encodes a sequence of k + 1-cores
(1.12) ∅ =
⊂ · · · ⊂
= λ ,
are certain horizontal strips. The weight of a k-tableau T is
(1.13) wt(T ) = (a1,a2,...,aN ) where ai =
For λ a k + 1-core, the dual k-Schur function is defined as the weight generating
is the set of k-tableaux (or weak tableaux) of shape λ.
Here we consider k-shape tableaux. These are defined similarly, but now we
allow the shapes in (1.12) to be k-shapes and
are certain reverse-maximal
strips (defined in Chapter 4). The weight is again defined by (1.13) and for each