Quasi-Hereditary Orderings of Nakayama Algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913899502108672 |
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| author | Zhang, Yuehui Zhong, Xiaoqiu |
| author_facet | Zhang, Yuehui Zhong, Xiaoqiu |
| contents | Let $A$ be an algebra with iso-class of simple modules $\mathcal{S}$ of cardinality $n$. A total ordering on $\mathcal{S}$ making every Weyl module Schurian and every indecomposable projective module filtered by the Weyl modules is called to be a quasi-hereditary ordering or $q$-ordering on $A$ and $A$ is a quasi-hereditary algebra under this ordering. The number of $q$-orderings on $A$ is denoted by $q(A)$. To determine whether an ordering on $\mathcal{S}$ is a $q$-ordering is a hard problem. A famous result due to Dlab and Ringel is that $A$ is hereditary if and only if every ordering is a $q$-ordering, equivalently, $q(A)=n!$. The twenty-years old $q$-ordering conjecture claims that $q(A)\le\dfrac{2}{3}n!$. The present paper proves a very simple criterion for $q$-orderings when $A$ is a Nakayama algebra. This criterion is applied to getting a full classification of all $q$-orderings of $A$ and an explicit iteration formula for $q(A)$, and also a positive proof of the $q$-ordering conjecture for Nakayama algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-Hereditary Orderings of Nakayama Algebras Zhang, Yuehui Zhong, Xiaoqiu Representation Theory Category Theory Rings and Algebras 16G20 16G10 16D10 Let $A$ be an algebra with iso-class of simple modules $\mathcal{S}$ of cardinality $n$. A total ordering on $\mathcal{S}$ making every Weyl module Schurian and every indecomposable projective module filtered by the Weyl modules is called to be a quasi-hereditary ordering or $q$-ordering on $A$ and $A$ is a quasi-hereditary algebra under this ordering. The number of $q$-orderings on $A$ is denoted by $q(A)$. To determine whether an ordering on $\mathcal{S}$ is a $q$-ordering is a hard problem. A famous result due to Dlab and Ringel is that $A$ is hereditary if and only if every ordering is a $q$-ordering, equivalently, $q(A)=n!$. The twenty-years old $q$-ordering conjecture claims that $q(A)\le\dfrac{2}{3}n!$. The present paper proves a very simple criterion for $q$-orderings when $A$ is a Nakayama algebra. This criterion is applied to getting a full classification of all $q$-orderings of $A$ and an explicit iteration formula for $q(A)$, and also a positive proof of the $q$-ordering conjecture for Nakayama algebras. |
| title | Quasi-Hereditary Orderings of Nakayama Algebras |
| topic | Representation Theory Category Theory Rings and Algebras 16G20 16G10 16D10 |
| url | https://arxiv.org/abs/2405.02860 |