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 = : 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 , = 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
Previous Page Next Page