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arXiv · 2510.13717

Algebraic Constructions of Universal Cycles on Grassmannians G_q(2,n)

Chen Yu Chi·Ming Hsuan Kang·Yu Hsuan Hsieh
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Abstract

We study universal cycles on the Grassmannian $G_q(2,n)$, the set of $2$-dimensional $\mathbb{F}_q$-subspaces of $\mathbb{F}_q^n$. While their existence is known from inductive and Eulerian graph methods, we give a direct algebraic construction when $n$ is odd under the coprimality condition $\gcd(n,\,q(q^2-1))=1$, using a projective-ratio decomposition and a global product condition. We also present explicit examples where a single cycle is simultaneously universal for both $G_q(2,5)$ and $G_q(3,5)$, realizing Grassmannian duality $|G_q(k,n)|=|G_q(n-k,n)|$ at the level of universal cycles.

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Chen Yu Chi, Ming Hsuan Kang, Yu Hsuan Hsieh. 2025-10-15. Algebraic Constructions of Universal Cycles on Grassmannians G_q(2,n). https://arxiv.org/abs/2510.13717

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