Homological algebra of Nakayama algebras and 321-avoiding permutations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916757745172480 |
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| author | Chavli, Eirini Marczinzik, Rene |
| author_facet | Chavli, Eirini Marczinzik, Rene |
| contents | Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321-avoiding bijections via the Billey-Jockusch-Stanley bijection. Thus to every 321-avoiding permutation $π$ we can associate in a natural way a linear Nakayama algebra $A_π$. We give a homological interpretation of the fixed points statistic of 321-avoiding permutations using Nakayama algebras with a linear quiver. We furthermore show that the space of self-extension for the Jacobson radical of a linear Nakayama algebra $A_π$ is isomorphic to $K^{\mathfrak{s}(π)}$, where $\mathfrak{s}(π)$ is defined as the cardinality $k$ such that $π$ is the minimal product of transpositions of the form $s_i=(i,i+1)$ and $k$ is the number of distinct $s_i$ that appear. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_13764 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homological algebra of Nakayama algebras and 321-avoiding permutations Chavli, Eirini Marczinzik, Rene Combinatorics Representation Theory 16G10, 18G20 Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321-avoiding bijections via the Billey-Jockusch-Stanley bijection. Thus to every 321-avoiding permutation $π$ we can associate in a natural way a linear Nakayama algebra $A_π$. We give a homological interpretation of the fixed points statistic of 321-avoiding permutations using Nakayama algebras with a linear quiver. We furthermore show that the space of self-extension for the Jacobson radical of a linear Nakayama algebra $A_π$ is isomorphic to $K^{\mathfrak{s}(π)}$, where $\mathfrak{s}(π)$ is defined as the cardinality $k$ such that $π$ is the minimal product of transpositions of the form $s_i=(i,i+1)$ and $k$ is the number of distinct $s_i$ that appear. |
| title | Homological algebra of Nakayama algebras and 321-avoiding permutations |
| topic | Combinatorics Representation Theory 16G10, 18G20 |
| url | https://arxiv.org/abs/2204.13764 |