2 1. INTRODUCTION the polynomials ˜(k→k ) μλ (t) can be expressed positively in terms of the branching polynomials ˜(k)(t) μλ := ˜(k−1→k)(t) μλ , via iteration (tables of branching polynomials are given in Appendix A). It has also come to light that ungraded k-Schur functions (the case when t = 1) are intimately tied to problems in combinatorics, geometry, and representation theory beyond the theory of Macdonald polynomials. Thus, understanding the branching coefficients, ˜(k) μλ := ˜(k)(1) μλ gives a step-by-step approach to problems in areas such as affine Schubert calculus and K-theory (see Section 1.4 for more details). Our work here gives a combinatorial description for the branching coefficients, proving Conjecture 1.1 when t = 1. We use the ungraded k-Schur functions s(k)[X] λ defined in [14], which coincide with those defined in [9] terms of strong k-tableaux. Moreover, we conjecture a formula for the branching polynomials in general. The combinatorics behind these formulas involves a certain poset of k-shapes. 1.2. The poset of k-shapes A key development in our work is the introduction of a new family of partitions called k-shapes and a poset on these partitions (see Chapter 2 for full details and examples). Our formula for the branching coefficients is given in terms of path enumeration in the poset of k-shapes. For any partition λ identified by its Ferrers diagram, we define its k-boundary ∂λ to be the cells of λ with hook-length no greater than k. ∂λ is a skew shape, to which we associate compositions rs(λ) and cs(λ), where rs(λ)i (resp. cs(λ)i) is the number of cells in the i-th row (resp. column) of ∂λ. A partition λ is said to be a k-shape if both rs(λ) and cs(λ) are partitions. The rank of k-shape λ is defined to be |∂λ|, the number of cells in its k-boundary. Πk denotes the set of all k-shapes. We introduce a poset structure on Πk where the partial order is generated by distinguished downward relations in the poset called moves (Definition 2.14). The set of k-shapes contains the set Ck of all k-cores (partitions with no cells of hook- length k) and the set Ck+1 of k + 1-cores. Moreover, the maximal elements of Πk are given by Ck+1 and the minimal elements by Ck. In Definition 3.1 we give a charge statistic on moves from which we obtain an equivalence relation on paths (sequences of moves) in Πk roughly speaking, two paths are equivalent if they are related by a sequence of charge-preserving diamonds (see Eqs. (3.1)-(3.3)). Charge is thus constant on equivalence classes of paths. For λ, μ Πk, Pk(λ, μ) is the set of paths in Πk from λ to μ and P k (λ, μ) is the set of equivalence classes in Pk(λ, μ). Our main result is that the branching coefficients enumerate these equivalence classes. To be precise, for λ Ck+1 and μ Ck, set b(k)(t) μλ := ˜(k) rs(μ)rs(λ) (t) (1.5) b(k) μλ := ˜(k) rs(μ)rs(λ) (1.6)
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