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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.13717 |
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| _version_ | 1866911212194758656 |
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| author | Chi, Chen Yu Kang, Ming Hsuan Hsieh, Yu Hsuan |
| author_facet | Chi, Chen Yu Kang, Ming Hsuan Hsieh, Yu Hsuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13717 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic Constructions of Universal Cycles on Grassmannians G_q(2,n) Chi, Chen Yu Kang, Ming Hsuan Hsieh, Yu Hsuan Combinatorics 05E18, 11T99 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. |
| title | Algebraic Constructions of Universal Cycles on Grassmannians G_q(2,n) |
| topic | Combinatorics 05E18, 11T99 |
| url | https://arxiv.org/abs/2510.13717 |