PreprintCC BY 4.0arXiv
A (q,t)-Overview of q-Analogs
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 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).
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BibTeX
@misc{bergeron2026qtoverview,
title = {{A $(q,t)$-Overview of $q$-Analogs}},
author = {François Bergeron},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30979},
eprint = {2608.30979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30979},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A (q,t)-Overview of q-Analogs
AU - François Bergeron
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30979
UR - https://arxiv.org/abs/2608.30979
AB - 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).
ER -