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arXiv · 2608.30979

A $(q,t)$-Overview of $q$-Analogs

François Bergeron
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Abstract

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 $\gamma$-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 $\gamma$-positivity and $\gamma$-anti-positivity (also known as alternatingly $\gamma$-positive).

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François Bergeron. 2026-08-31. A $(q,t)$-Overview of $q$-Analogs. https://arxiv.org/abs/2608.30979

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