1.3. k-SHAPE FUNCTIONS 3 so that s(k−1)[X] μ = λ∈Ck+1 b(k)s(k)[X]. μλ λ (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) b (k) μλ = |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) b(k)(t) μλ = [p]∈P k (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 {Weak(k)[X]} λ 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 λ∈Ck+1 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) Weak(k)[X] λ , sμk)[X] ( k = δλμ Moreover, it was shown in [7] that {Weak(k)[X]} λ λ∈Ck+1 represents Schubert classes in the cohomology of the aﬃne Grassmannian GrSL k+1 of SLk+1. The original characterization of Weak(k)[X] λ 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

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