1.3. k-SHAPE FUNCTIONS 3

so that

sμk−1)[X] (

=

λ∈Ck+1

bμλ

(k) sλk)[X].(

(1.7)

Hereafter, we will label k-Schur functions by cores rather than k-bounded partitions

using the bijection between

Ck+1

and

Bk

given by the map rs.

Theorem 1.2. For all λ ∈

Ck+1

and μ ∈

Ck,

(1.8) bμλ

(k)

= |P

k

(λ, μ)|.

We conjecture that the charge statistic on paths gives the branching polyno-

mials.

Conjecture 1.3. For all λ ∈ Ck+1 and μ ∈ Ck,

(1.9) bμλ

(k)

(t) =

[p]∈Pk(λ,μ)

tcharge(p).

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

{Weakλk)[X]}λ∈Ck+1 (

form a basis for

the quotient

Λ/Ik where Ik = mλ : λ1 k (1.10)

of the space Λ of symmetric functions over Z, while the ungraded k-Schur functions

{s(k)[X]}λ∈Ck+1

λ

form a basis for the subring

Λ(k)

= 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 ×

Λ(k)

→ Z, and it was shown in [15] that

(1.11)

Weakλk)[X] (

,

sμk)[X] (

k

= δλμ

Moreover, it was shown in [7] that

{Weakλk)[X]}λ∈Ck+1 (

represents Schubert classes

in the cohomology of the aﬃne Grassmannian GrSLk+1 of SLk+1.

The original characterization of

Weakλk)[X] (

was given in [15] using k-tableaux.

A k-tableau encodes a sequence of k + 1-cores

(1.12) ∅ =

λ(0)

⊂

λ(1)

⊂ · · · ⊂

λ(N)

= λ ,

where

λ(i)/λ(i−1)

are certain horizontal strips. The weight of a k-tableau T is

(1.13) wt(T ) = (a1,a2,...,aN ) where ai =

|∂λ(i)|

−

|∂λ(i−1)|

.

For λ a k + 1-core, the dual k-Schur function is defined as the weight generating

function

(1.14)

Weak(k)[X]

λ

=

T ∈WTabk

λ

xwt(T )

,

where WTabλ

k

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

λ(i)/λ(i−1)

are certain reverse-maximal

strips (defined in Chapter 4). The weight is again defined by (1.13) and for each