Dimensions of $τ$-tilting modules over path algebras and preprojective algebras of Dynkin type
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909528227840000 |
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| author | Aoki, Toshitaka Mizuno, Yuya |
| author_facet | Aoki, Toshitaka Mizuno, Yuya |
| contents | In this paper, we introduce a new generating function called $d$-polynomial for the dimensions of $τ$-tilting modules over a given finite dimensional algebra. Firstly, we study basic properties of $d$-polynomials and show that it can be realized as a certain sum of the $f$-polynomials of the simplicial complexes arising from $τ$-rigid pairs. Secondly, we give explicit formulas of $d$-polynomials for preprojective algebras and path algebras of Dynkin quivers by using a close relation with $W$-Eulerian polynomials and $W$-Narayana polynomials. Thirdly, we consider the ordinary and exponential generating functions defined from $d$-polynomials and give closed-form expressions in the case of preprojective algebras and path algebras of Dynkin type $\mathbb{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13527 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimensions of $τ$-tilting modules over path algebras and preprojective algebras of Dynkin type Aoki, Toshitaka Mizuno, Yuya Representation Theory Combinatorics Rings and Algebras In this paper, we introduce a new generating function called $d$-polynomial for the dimensions of $τ$-tilting modules over a given finite dimensional algebra. Firstly, we study basic properties of $d$-polynomials and show that it can be realized as a certain sum of the $f$-polynomials of the simplicial complexes arising from $τ$-rigid pairs. Secondly, we give explicit formulas of $d$-polynomials for preprojective algebras and path algebras of Dynkin quivers by using a close relation with $W$-Eulerian polynomials and $W$-Narayana polynomials. Thirdly, we consider the ordinary and exponential generating functions defined from $d$-polynomials and give closed-form expressions in the case of preprojective algebras and path algebras of Dynkin type $\mathbb{A}$. |
| title | Dimensions of $τ$-tilting modules over path algebras and preprojective algebras of Dynkin type |
| topic | Representation Theory Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2402.13527 |