球Hecke代数的预典范基的正性
Positivity of Pre-Canonical Bases for Spherical Hecke Algebras
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中文总结 AI 辅助
该研究提出计算Kostka-Foulkes多项式的新算法，证明球Hecke代数预典范基间转换矩阵系数非负，确立正性猜想，还得到典范基原子分解算法，对应新的Schur正对称函数类。
中文摘要 AI 辅助
我们引入一种计算Kostka-Foulkes多项式的新算法。该算法源于一种组合过程，用于计算Libedinsky、Patimo与第一作者引入的球Hecke代数的相继预典范基之间的转换系数，以在球Hecke代数的标准基与典范基之间进行插值。利用此算法，我们证明了从任意预典范基到下一预典范基的转换矩阵中多项式的系数均非负。这确立了预典范基的正性猜想，且给出了一个严格更强的结果，揭示了超出原猜想范围的额外正性现象。作为副产品，该方法得到了计算典范基元素的Lascoux原子分解的组合算法。最后，我们的结果表明，在应用Satake同构并关于Hall内积对偶化后，预典范基对应于一类新的Schur正对称函数。
英文摘要
We introduce a new algorithm for computing Kostka-Foulkes polynomials. It arises from a combinatorial procedure that computes the transition coefficients between successive pre-canonical bases for spherical Hecke algebras, introduced by Libedinsky, Patimo, and the first author, in order to interpolate between the standard and canonical bases of the spherical Hecke algebras. Using this algorithm, we prove that the coefficients of the polynomials in the transition matrix from any pre-canonical basis to the next one has nonnegative coefficients. This establishes the positivity conjecture for pre-canonical bases and, moreover, yields a strictly stronger result that uncovers additional positivity phenomena beyond the original scope of the conjecture. As a by-product, this method yields a combinatorial algorithm for computing Lascoux's atomic decomposition of canonical basis elements. Finally, our results show that, after applying the Satake isomorphism and dualizing with respect to the Hall inner product, pre-canonical bases correspond to a new family of Schur-positive symmetric functions.
