Dimensions of τ-tilting modules over path algebras and preprojective algebras of Dynkin type

• DOI: 10.1016/j.jalgebra.2024.12.022
• Publication Date: 2025-01-04T10:43:30Z
• Authors: Toshitaka Aoki; Yuya Mizuno
• Keywords: Rings and Algebras (math.RA); FOS: Mathematics; Mathematics - Combinatorics; Mathematics - Rings and Algebras; Combinatorics (math.CO); Representation Theory (math.RT); Mathematics - Representation Theory

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 $\mathbb{A}$.<br/>35 pages<br/>