An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912378254262272 |
|---|---|
| author | Khrystik, M. A. |
| author_facet | Khrystik, M. A. |
| contents | 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group Khrystik, M. A. Rings and Algebras Representation Theory 16S34 (Primary) 20C05, 20C30 (Secondary) 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 $n$. |
| title | An Upper Bound on the Length of an Algebra and Its Application to the Group Algebra of the Dihedral Group |
| topic | Rings and Algebras Representation Theory 16S34 (Primary) 20C05, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2412.06123 |