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

TL;DR

This paper establishes an upper bound on the length of an algebra based on its dimension and minimal polynomial degree, and applies this to determine the length of dihedral group algebras.

Contribution

It introduces a new upper bound for algebra length and explicitly computes the length of dihedral group algebras for odd orders.

Findings

The algebra length does not exceed max(dim A/2, m(A)-1).

The length of the dihedral group algebra of order 2n is n for odd n.

Provides a method to bound algebra length using minimal polynomial degrees.

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 $n$.