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

Chen Yu Chi, Ming Hsuan Kang, Yu Hsuan Hsieh

TL;DR

The paper develops an explicit algebraic framework to construct universal cycles on the Grassmannian $G_q(2,n)$ when $n$ is odd and $ exists$ a common divisor with $q(q^2-1)$, by mapping $2$-subspaces to projective-ratio classes and exploiting a non-collapsing condition. It introduces a Global Product Construction that pairs PGL$_2(F)$-orbits with Galois orbits, producing a representative system whose cyclic product traverses all subspaces exactly once, yielding a universal cycle with the property that $W_{i+r}=\alpha W_i$. The approach yields explicit instances, notably for $G_2(2,5)$, and demonstrates that, in special cases, a single universal cycle can realize Grassmann duality for both $G_q(2,n)$ and $G_q(n-2,n)$. The work provides a concrete, purely algebraic method that complements existence results and offers flexibility to impose additional structure or dual universality in targeted parameter regimes.

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.

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

TL;DR

The paper develops an explicit algebraic framework to construct universal cycles on the Grassmannian when is odd and a common divisor with , by mapping -subspaces to projective-ratio classes and exploiting a non-collapsing condition. It introduces a Global Product Construction that pairs PGL-orbits with Galois orbits, producing a representative system whose cyclic product traverses all subspaces exactly once, yielding a universal cycle with the property that . The approach yields explicit instances, notably for , and demonstrates that, in special cases, a single universal cycle can realize Grassmann duality for both and . The work provides a concrete, purely algebraic method that complements existence results and offers flexibility to impose additional structure or dual universality in targeted parameter regimes.

Abstract

We study universal cycles on the Grassmannian , the set of -dimensional -subspaces of . While their existence is known from inductive and Eulerian graph methods, we give a direct algebraic construction when is odd under the coprimality condition , using a projective-ratio decomposition and a global product condition. We also present explicit examples where a single cycle is simultaneously universal for both and , realizing Grassmannian duality at the level of universal cycles.
Paper Structure (9 sections, 1 theorem, 25 equations)

This paper contains 9 sections, 1 theorem, 25 equations.

Table of Contents

  1. Introduction
  2. Algebraic Decomposition of $G_q(2,n)$
  3. Projective Ratios
  4. Fibers and Non-Collapsing Condition
  5. The Global Product Construction
  6. Constructing the Cycle
  7. Remark (Even $n$).
  8. Example: $G_2(2,5)$
  9. Additional Remark.

Key Result

Theorem 4.1

Let $n$ be odd and $q$ a prime power such that $\gcd(n,\, q(q^2-1))=1$. Then the sequence constructed from the cyclic product system where $\{c_1,\dots,c_r\}$ are chosen as in Section sec:algebraic-decomposition and the indices of $c_i$ are taken modulo $r$, is a universal cycle on $G_q(2,n)$. That is, every 2-dimensional subspace of $\mathbb{F}_q^n$ appears exactly once in the sequence, with per

Theorems & Definitions (1)

  • Theorem 4.1: Universal Cycles for Odd $n$