$τ$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras
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| Format: | Preprint |
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2024
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| _version_ | 1866911082669408256 |
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| author | Børve, Erlend D. Grevstad, Jacob Fjeld Rundsveen, Endre S. |
| author_facet | Børve, Erlend D. Grevstad, Jacob Fjeld Rundsveen, Endre S. |
| contents | We prove that any $τ$-tilting finite incidence algebra of a finite poset is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of $τ$-tilting finite incidence algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are $τ$-tilting infinite, and to prove the latter, we show that wild concealed algebras are not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a $τ$-tilting reduction of a concealed algebra of hyperbolic type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $τ$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras Børve, Erlend D. Grevstad, Jacob Fjeld Rundsveen, Endre S. Representation Theory We prove that any $τ$-tilting finite incidence algebra of a finite poset is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of $τ$-tilting finite incidence algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are $τ$-tilting infinite, and to prove the latter, we show that wild concealed algebras are not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a $τ$-tilting reduction of a concealed algebra of hyperbolic type. |
| title | $τ$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2407.17965 |