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

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 $\mathrm{\mid }{G}_q(k,n)\mathrm{\mid }=\mathrm{\mid }{G}_q(n-k,n)\mathrm{\mid }$ at the level of universal cycles.