Data

• Title: Growth problems in diagram categories
• Authors: Jonathan Gruber and Daniel Tubbenhauer
• Status: preprint. Last update: Wed, 5 Feb 2025 03:03:57 UTC
• Code and (possibly empty) Erratum: Click
• ArXiv link: http://arxiv.org/abs/2503.00685

Abstract

In the semisimple case, we derive (asymptotic) formulas for the growth rate of the number of summands in tensor powers of the generating object in diagram/interpolation categories.

A few extra words

We begin with the following table, the meaning of which we will explain shortly:

Let \(C\) be an additive Krull--Schmidt monoidal category. Let \(X\in C\) be an object of \(C\). We define \[ b_{n}=b_{n}^{C,X}:=\#\text{indecomposable summands in \(X^{\otimes n}\) counted with multiplicities}. \] We identify the sequence \(b_{n}\) with its function \(n\mapsto b_{n}\).
The function \(b_{n}\) has been the subject of extensive study. In particular, in well-behaved categories, such as finite-dimensional representations of a group one has \[ \lim_{n\to\infty}\sqrt[n]{b_{n}}\in\mathbb{R}_{\geq 1}, \quad\text{exponential growth} \] which shows that \(b_{n}\) grows exponentially. In contrast, some still well-structured categories exhibit superexponential growth, meaning that \[ \sqrt[n]{b_{n}}\text{ is unbounded,} \quad\text{superexponential growth}. \] We study the asymptotic behavior of \(\sqrt[n]{b_{n}}\) in the above diagram categories, all in the semisimple situation (all parameters are generic) and over the complex numbers.