CHAPTER 1 Introduction 1.1. k-Schur functions and branching coeﬃcients The theory of k-Schur functions arose from the study of Macdonald polynomials and has since been connected to quantum and aﬃne Schubert calculus, K-theory, and representation theory [2, 3, 4, 7, 10, 15]. The origin of the k-Schur functions is related to Macdonald’s positivity conjecture, which asserted that in the expansion (1.1) Hμ[X q, t] = λ Kλμ(q, t) sλ , the coeﬃcients Kλμ(q, t), called q, t-Kostka polynomials, belong to Z≥0[q, t]. Al- though the final piece in the proof of this conjecture was made by Haiman [5] using representation theoretic and geometric methods, the long study of this conjecture brought forth many further problems and theories. The study of the q, t-Kostka polynomials remains a matter of great interest. It was conjectured in [11] that by fixing an integer k 0, any Macdonald polynomial indexed by λ ∈ Bk (the set of partitions such that λ1 ≤ k) could be decomposed as: (1.2) Hμ[X q, t ] = λ∈Bk K(k)(q, λμ t) s(k)[X λ t ] where K(k)(q, λμ t) ∈ Z≥0[q, t] , for some symmetric functions s (k) λ [X t] associated to sets of tableaux called atoms. Conjecturally equivalent characterizations of s(k)[X λ t] were later given in [12, 9] and the descriptions of [11, 12, 9] are now all generically called (graded) k-Schur functions. A basic property of the k-Schur functions is that (1.3) s(k)[X λ t] = sλ for k ≥ |λ| , and it thus follows that Eq. (1.2) significantly refines Macdonald’s original conjec- ture since the expansion coeﬃcient K(k)(q, λμ t) reduces to Kλμ(q, t) for large k. Furthermore, it was conjectured that the k-Schur functions satisfy a highly structured filtration, which is our primary focus here. To be precise: Conjecture 1.1. For k k and partitions μ ∈ Bk and λ ∈ Bk , there are polynomials ˜(k→k ) μλ (t) ∈ Z≥0[t] such that (1.4) sμk)[X ( t] = λ∈Bk ˜(k→k ) μλ (t) s(k ) λ [X t]. In particular, the Schur function expansion of a k-Schur function is obtained from (1.3) and (1.4) by letting k → ∞. The remarkable property described in Conjec- ture 1.1 provides a step-by-step approach to understanding k-Schur functions since 1

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