An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group
M. A. Khrystik
TL;DR
The paper establishes a general, sharp upper bound on the length of a finite-dimensional algebra: $l({\cal A}) \le \max\{ \dim({\cal A})/2,\ m({\cal A})-1 \}$. It then applies this bound to group algebras of dihedral groups by introducing a bicirculant algebra ${\cal C}_n({\mathbb F})$ and a bicirculant representation of ${\mathbb F}{\cal D}_n$, deriving concrete length estimates. In particular, for odd $n$ the length of the dihedral group algebra satisfies $l({\mathbb F}{\cal D}_n)=n$, while for even $n$ the bound yields $l({\mathbb F}{\cal D}_n) \le n+1$, with a strong lower bound $l({\mathbb F}{\cal D}_n) \ge n$, supporting the conjecture that $l({\mathbb F}{\cal D}_n)=n$ for all $n$. The bicirculant approach also provides explicit bounds on $m({\mathbb F}{\cal D}_n)$, illustrating the method's utility for numerical characteristics of algebras and its potential to inform Paz-type questions in broader classes. Overall, the work blends structural algebra with representation-theoretic techniques to obtain precise, broadly applicable bounds for the length of algebras and their group algebras."
Abstract
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 $n$.