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\(k\)-Schur expansions of Catalan functions. (English) Zbl 1443.05181
Summary: We make a broad conjecture about the \(k\)-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the \(k\)-Schur expansion of (1) Hall-Littlewood polynomials, proving the \(q = 0\) case of the strengthened Macdonald positivity conjecture from [L. Lapointe et al., Duke Math. J. 116, No. 1, 103–146 (2003; Zbl 1020.05069)]; (2) the product of a Schur function and a \(k\)-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) \(k\)-split polynomials, solving a substantial special case of a problem of B. Broer [Prog. Math. 123, 1–19 (1994; Zbl 0855.22015)] and M. Shimozono and J. Weyman [Eur. J. Comb. 21, No. 2, 257–288 (2000; Zbl 0956.05100)] on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that the \(k\)-Schur functions defined via \(k\)-split polynomials [L. Lapointe and J. Morse, J. Comb. Theory, Ser. A 101, No. 2, 191–224 (2003; Zbl 1018.05101)] agree with those defined in terms of strong tableaux [T. Lam et al., Affine insertion and Pieri rules for the affine Grassmannian. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1208.14002)].

MSC:
05E10 Combinatorial aspects of representation theory
14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
05E05 Symmetric functions and generalizations
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