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On Zeta functions and $μ$-series of string algebras

Rohun Easwar, Amit Kuber, Mihir Mittal

Abstract

Let $\overlineμ_Λ(t):=\sum\limits_{m\geq1}μ_Λ(m)t^m$ be the \emph{$μ$-series} of a finite-dimensional tame algebra $Λ$ over an algebraically closed field, where $μ_Λ(m)$ denotes the minimal number of one-parameter families of $Λ$-modules with total dimension $m$. When $Λ$ is a string algebra with $\mathrm{Ba}(Λ)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $ζ_Λ(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(Λ)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $μ$-series is rational.

On Zeta functions and $μ$-series of string algebras

Abstract

Let be the \emph{-series} of a finite-dimensional tame algebra over an algebraically closed field, where denotes the minimal number of one-parameter families of -modules with total dimension . When is a string algebra with as its set of bands up to cyclic permutation, define the \emph{zeta function} , where is the length of . We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its -series is rational.

Paper Structure

This paper contains 7 sections, 16 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strings and bands for zero-relation algebras
  4. Zeta Functions for zero-relation algebras
  5. Prime number theorem for string algebras
  6. Proof of Theorem \ref{['main']}
  7. Extension to special biserial algebras

Key Result

Theorem A

For a string algebra $\Lambda$, there are positive integers $C_\Lambda, L_\Lambda$ such that where $R(A_\Lambda)$ is the spectral radius of $A_\Lambda$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: An example of a directed graph
  • Figure 2: $\mathrm{GP}_{2,3}$ with $\rho = \{a^2, b^3, ab, ba\}$
  • Figure 3: The state graph of $GP_{2,3}$

Theorems & Definitions (54)

  • Theorem A
  • Corollary B
  • Theorem C
  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: Perron and Frobenius
  • Remark 2.7
  • ...and 44 more