发表机构
Université du Québec à Montréal（蒙特利尔魁北克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出通过简单齐次化过程将$q$-模拟转化为对称$(q,t)$-模拟的双射对应方法，发展了非齐次扩展，结合$\u03b3$-正性等领域最新进展阐释该方法，提出新构造与猜想并探索更多研究方向。
AI 中文摘要
通过一种简单的齐次化过程，人们可以将一个$q$-模拟转化为对称的$(q,t)$-模拟。这种双射对应虽然简单，却能让许多概念和恒等式变得更加自然，且相关证明可直接由双变量对称函数的经典性质推导得出。本文还发展了一种更为复杂的非齐次扩展。我们通过重新审视$\boldsymbol{\u03b3}$-正性、Lucas模拟以及分圆生成函数幺半群研究中的最新进展，对该方法进行了阐释，这也自然催生了新的构造和猜想。我们进一步探索了其他研究方向，包括分次与等变$\boldsymbol{\u03b3}$-正性以及$\boldsymbol{\u03b3}$-反正性（也称为交替$\boldsymbol{\u03b3}$-正性）。
英文摘要
By a simple homogenization process, one may turn a $q$-analog into a symmetric $(q,t)$-analog. Although simple, this bijective correspondence makes many concepts and identities become more natural, and proofs follow readily from classical properties of symmetric functions in two variables. A more intricate non-homogeneous extension is also developed. We illustrate this approach, revisiting recent developments in the study of $γ$-positivity, Lucas analogues, and the monoid of Cyclotomic generating functions. This also leads naturally to new constructions and conjectures. We further explore other avenues of investigations, included graded and equivariant $γ$-positivity and $γ$-anti-positivity (also known as alternatingly $γ$-positive).
Comments39 pages, 6 figures
