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AN UPPER BOUND ON THE LENGTH OF AN ALGEBRA AND ITS APPLICATION TO THE GROUP ALGEBRA OF THE DIHEDRAL GROUP

Part of: Representation theory of groups Rings and algebras arising under various constructions

Published online by Cambridge University Press:  10 February 2025

M. A. KHRYSTIK Open the ORCID record for M. A. KHRYSTIK [Opens in a new window]
M. A. KHRYSTIK*
Affiliation:
Faculty of Computer Science, HSE University, Moscow 101000, Russia and Moscow Center of Fundamental and Applied Mathematics, Moscow 119991, Russia
*
e-mail: good_michael@mail.ru
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Abstract

Let ${\mathcal {A}}$ be a unital ${\mathbb {F}}$-algebra and let ${\mathcal {S}}$ be a generating set of ${\mathcal {A}}$. 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 sets. We show that $l({\mathcal {A}})$ 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 ${\mathcal {A}}$. As an application, we show that for arbitrary odd n, the length of the group algebra of the dihedral group of order $2n$ equals n.

Keywords

finite-dimensional algebraslength of an algebragroup algebrasdihedral grouprepresentations of dihedral groups

MSC classification

Primary: 16S34: Group rings , Laurent polynomial rings
Secondary: 20C05: Group rings of finite groups and their modules 20C30: Representations of finite symmetric groups

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 112 , Issue 1 , August 2025 , pp. 139 - 148
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by Russian Science Foundation, grant 20-11-20203.

References

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