Growth Problems of Quantum Groups

TL;DR

This paper investigates the asymptotic behavior of tensor power decompositions of tilting modules for quantum groups, revealing universal laws and specific results in type A1 related to random walks.

Contribution

It provides a universal asymptotic law for the size of tensor power decompositions of tilting modules, applicable across different types of quantum groups.

Findings

In type A1, the number of indecomposable summands follows a sharp asymptotic pattern.

The dominant asymptotic behavior depends only on the module's dimension, not on specific modules.

The correction term in the asymptotics is determined solely by the root system.

Abstract

We study the asymptotic size of decompositions of tensor powers of tilting modules for quantum groups (mostly at a complex root of unity). In type A1 we obtain a sharp result for the number of indecomposable summands, explained by a one dimensional half-line random walk with a periodic congruence constraint. In general type we prove a universal law: the dominant part is governed only by the dimension of the module, while the correction depends only on the root system, so the asymptotic size is largely independent of the specific tilting module.