Year: 2026
Type: preprint
Abstract: In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].
Source: arXiv (Cornell University)
Authors: Elhamdadi, Mohamed, Ta, Luc, Virgin, Bryce
Institutions: University of Pittsburgh, University of South Florida
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: University of South Florida
OpenAlex ID: W7208723682