arXiv subjects

Yu Hsuan Hsieh

Publications and source records attributed to Yu Hsuan Hsieh.

4 recordsLinked to original sources

arXiv2026

Universal Cycles for Affine Planes and 3-Subspaces over Finite Fields

We construct universal cycles for affine planes in $\mathbb F_q^n$ for all prime powers $q$ and all $n\ge4$, using sliding windows of length three. The construction is local-to-global: explicit local cycles are built on frame configurations, the linear $2$-subspaces are organized by a layered frame decomposition, and the resulting cycles are assembled by gluing along shared directions. The universal cycle obtained has direction set containing all $1$-subspaces. We also extend the construction to universal cycles for $3$-subspaces of $\mathbb F_q^n$.

math.CO↗

arXiv2026

From Characters to Matrices: An Elementary Construction of Irreducible Representations of Finite Groups

Let $G$ be a finite group and let $χ$ be an ordinary irreducible character. We give an elementary algorithm which constructs explicit matrices affording $χ$. The regular representation provides a canonical ambient representation, and the usual central idempotent projects onto the $χ$-isotypic component. The main step is then to maximize the squared norm of a diagonal matrix coefficient on the unit sphere of this component. The maximum is $1/χ(1)$, and it is attained precisely by vectors whose cyclic span is an irreducible subrepresentation affording $χ$. Thus the construction reduces the passage from characters to matrices to a concrete optimization problem. The same extraction method applies inside any smaller ambient representation containing $χ$, such as an induced representation from a subgroup. We complement the theory with a discussion on dimension reduction, robust numerical implementation, and an explicit $S_4$ example.

math.RT↗

arXiv2025

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 $|G_q(k,n)|=|G_q(n-k,n)|$ at the level of universal cycles.

math.CO↗

arXiv2025

De Bruijn Tori Without Zeros: A Field-Theoretic Perspective

We present an algebraic construction of trace-based De Bruijn tori over finite fields, focusing on the nonzero variant that omits the all-zero pattern. The construction arranges nonzero field elements on a toroidal grid using two multiplicatively independent generators, with values obtained by applying a fixed linear map, typically the field trace. We characterize sampling patterns as subsets whose associated field elements form an $ \mathbb{F}_p $-basis, and show that column structures correspond to cyclic shifts of De Bruijn sequences determined by irreducible polynomials over subfields. Recursive update rules based on multiplicative translations enable efficient computation.

math.CO↗