Automating the stable rank computation for special biserial algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914856019427328 |
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| author | Srivastava, Suyash Kuber, Amit |
| author_facet | Srivastava, Suyash Kuber, Amit |
| contents | Given a special biserial algebra $Λ$ over an algebraically closed field, let $\mathrm{rad}_Λ$ denote the radical of its module category. The authors showed with Sinha that the stable rank of a special biserial algebra $Λ$, i.e., the least ordinal $γ$ satisfying $\mathrm{rad}_Λ^γ=\mathrm{rad}_Λ^{γ+1}$, is strictly bounded above by $ω^2$. We use finite automata to give simple algorithmic proofs, complete with their time complexity analyses, of two key ingredients in the proof of this result--the first one states that certain linear orders called hammocks associated with such algebras are \emph{finite description linear orders}, i.e., they lie in the smallest class of linear orders that contains finite linear orders and $ω$, and that is closed under isomorphisms, order-reversals, binary sums, co-lexicographic products and finitary shuffles. We also document a complete proof of the result that the class of order types(=order-isomorphism classes) of finite description linear orders coincides with that of languages of finite automata under inorder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automating the stable rank computation for special biserial algebras Srivastava, Suyash Kuber, Amit Representation Theory 16S90, 68Q45, 06A05, 16G20 Given a special biserial algebra $Λ$ over an algebraically closed field, let $\mathrm{rad}_Λ$ denote the radical of its module category. The authors showed with Sinha that the stable rank of a special biserial algebra $Λ$, i.e., the least ordinal $γ$ satisfying $\mathrm{rad}_Λ^γ=\mathrm{rad}_Λ^{γ+1}$, is strictly bounded above by $ω^2$. We use finite automata to give simple algorithmic proofs, complete with their time complexity analyses, of two key ingredients in the proof of this result--the first one states that certain linear orders called hammocks associated with such algebras are \emph{finite description linear orders}, i.e., they lie in the smallest class of linear orders that contains finite linear orders and $ω$, and that is closed under isomorphisms, order-reversals, binary sums, co-lexicographic products and finitary shuffles. We also document a complete proof of the result that the class of order types(=order-isomorphism classes) of finite description linear orders coincides with that of languages of finite automata under inorder. |
| title | Automating the stable rank computation for special biserial algebras |
| topic | Representation Theory 16S90, 68Q45, 06A05, 16G20 |
| url | https://arxiv.org/abs/2407.02326 |