M. A. Khrystik | Searcharxiv

arXiv2026

Combinatorics on finite words and the length of a finite-dimensional associative algebra

Let $f_W(n)$ be the number of different factors of length $n$ appearing in $W$. A classical result of Morse and Hedlund, stated in 1938, asserts that an infinite word $W$ is ultimately periodic if and only if $f_W(n)\leq n$ for some $n\in \mathbb N$. In this paper, we describe the form of finite words that satisfy the condition $f_W(n)\leq n$. We study relations between power avoidance and subword complexity of a finite word. We apply our combinatorial results to study the interrelations between various numerical invariants of finite-dimensional associative algebras.

math.RA↗

arXiv2024

An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group

Let $\mathcal A$ be an $\mathbb F$-algebra and let $\mathcal S$ be its generating set. The length of $\mathcal S$ is the smallest number $k$ such that $\mathcal A$ equals the $\mathbb F$-linear span of all products of length at most $k$ of elements from $\mathcal S$. The length of $\mathcal A$, denoted by $l(\mathcal A)$, is defined to be the maximal length of its generating set. In this paper, it is shown that the $l(\mathcal A)$ does not exceed the maximum of $\dim \mathcal A / 2$ and $m(\mathcal A)-1$, where $m(\mathcal A)$ is the largest degree of the minimal polynomial among all elements of the algebra $\mathcal A$. For arbitrary odd $n$, it is proven that the length of the group algebra of the dihedral group of order $2n$ equals $n$.

math.RA↗