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 of Lapointe, Lascoux, and Morse; (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, proving a substantial case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that $k$-Schur functions defined in terms of $k$-split polynomials agree with strong tableau $k$-Schur functions.

中文翻译：

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Catalan 函数的 k-Schur 展开

我们对加泰罗尼亚函数的 $k$-Schur 正性进行了广泛的猜想，对称函数可推广（抛物线）霍尔-利特伍德多项式。在解决 (1) Hall-Littlewood 多项式的 $k$-Schur 展开的情况下，我们用正组合公式解决了这个猜想，证明了 Lapointe、Lascoux 和 Morse 的强化 Macdonald 正性猜想的 $q=0$ 情况；(2) 当索引分区连接到一个分区时，Schur 函数和 $k$-Schur 函数的乘积，描述了完整标志变体的量子上同调的一类 Gromov-Witten 不变量；(3) $k$-split polynomials，证明了抛物线 Hall-Littlewood 多项式上 Broer 和 Shimozono-Weyman 问题的一个实质性案例。此外，