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

Toshitaka Aoki, Yuya Mizuno

TL;DR

The paper introduces the $d$-polynomial $d(A;t)$ as a dimension-weighted variant of the $f$-polynomial for $ au$-tilting theory and establishes its reduction-based expression as a sum of $f$-polynomials of reduced algebras. It then derives explicit formulas for $d$-polynomials of preprojective algebras and Dynkin-type path algebras, connecting them to $W$-Eulerian and $W$-Narayana polynomials; these results reveal orientation-independence in several cases and yield concrete dimension counts for indecomposable $\tau$-rigid objects. A comprehensive generating-function framework is developed, giving closed forms for exponential generating functions in type $A$ and linking $d$-polynomials to Narayana polynomials through derivatives of Narayana generating functions. Collectively, these contributions provide a new numerical invariant for $\tau$-tilting modules and a rich interplay between representation theory and classical combinatorial polynomials. The results enhance computational tools for tau-tilting theory in Dynkin-type algebras and suggest avenues for further exploration of generating-function methods in this area.

Abstract

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 $\mathbb{A}$.

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

TL;DR

The paper introduces the -polynomial as a dimension-weighted variant of the -polynomial for -tilting theory and establishes its reduction-based expression as a sum of -polynomials of reduced algebras. It then derives explicit formulas for -polynomials of preprojective algebras and Dynkin-type path algebras, connecting them to -Eulerian and -Narayana polynomials; these results reveal orientation-independence in several cases and yield concrete dimension counts for indecomposable -rigid objects. A comprehensive generating-function framework is developed, giving closed forms for exponential generating functions in type and linking -polynomials to Narayana polynomials through derivatives of Narayana generating functions. Collectively, these contributions provide a new numerical invariant for -tilting modules and a rich interplay between representation theory and classical combinatorial polynomials. The results enhance computational tools for tau-tilting theory in Dynkin-type algebras and suggest avenues for further exploration of generating-function methods in this area.

Abstract

In this paper, we introduce a new generating function called -polynomial for the dimensions of -tilting modules over a given finite dimensional algebra. Firstly, we study basic properties of -polynomials and show that it can be realized as a certain sum of the -polynomials of the simplicial complexes arising from -rigid pairs. Secondly, we give explicit formulas of -polynomials for preprojective algebras and path algebras of Dynkin quivers by using a close relation with -Eulerian polynomials and -Narayana polynomials. Thirdly, we consider the ordinary and exponential generating functions defined from -polynomials and give closed-form expressions in the case of preprojective algebras and path algebras of Dynkin type .
Paper Structure (23 sections, 41 theorems, 151 equations, 2 figures, 6 tables)

This paper contains 23 sections, 41 theorems, 151 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Preliminaries
  5. $\tau$-tilting theory
  6. Preprojective algebras
  7. $d$-polynomials in $\tau$-tilting theory
  8. Simplicial complexes of $\tau$-rigid pairs
  9. A connection between $d$-polynomials and $f$-polynomials
  10. $d$-polynomials of preprojective algebras
  11. Reduction at $\tau$-rigid submodules of $e_{\ell}\Pi$
  12. Type $\mathbb{A}$
  13. Type $\mathbb{D}$
  14. Case $\ell = \pm1$
  15. Case $\ell\neq \pm1$
  16. ...and 8 more sections

Key Result

Proposition 1.2

We have where $M$ runs over all indecomposable $\tau$-rigid $A$-modules up to isomorphisms, and $C_M$ is the algebra obtained by the reduction at $M$.

Figures (2)

  • Figure 1: The $g$-simplicial complexes in Example \ref{['ex:d-polynomial']}.
  • Figure 2: The $g$-simplicial complexes in Example \ref{['example:BTA']}.

Theorems & Definitions (81)

  • Definition 1.1: Subsection \ref{['Enumeration']}
  • Proposition 1.2: Proposition \ref{['prop:d and f']}
  • Theorem 1.3: Theorem \ref{['thm:orbit ppalg']}
  • Theorem 1.4: Theorem \ref{['thm:OOpathalg']}
  • Definition 1.5
  • Proposition 1.6: Proposition \ref{['prop:Euler-ppaA']}
  • Theorem 1.7: Theorem \ref{['thm:gen ppaA']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 71 more