Research paper

Growth problems of quantum groups

Tensor powers for tilting modules at roots of unity: the main exponential term is robust, while the correction remembers the root system.

quantum groupstilting modulesrandom walksgrowth
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Data

  • Title: Growth problems of quantum groups
  • Authors: Jensen O'Sullivan and Daniel Tubbenhauer
  • Status: preprint. Last update: Mon, 10 Nov 2025 05:59:04 UTC
  • Code / errata: Click
  • arXiv: https://arxiv.org/abs/2511.06737

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 largely independent of the specific tilting module.

What is the point?

Tensor powers quickly become too complicated to decompose exactly. The point here is to understand the stable large-scale law: what grows like the characteristic-zero answer, what changes at a root of unity, and which part is controlled only by the root system.

Input:
a tilting module T.
Question:
how many summands in T^⊗n?
Answer:
a universal asymptotic law.

More details

To set the stage, let

\[ b_n=b_n(T)=\#\{\text{indecomposable summands of }T^{\otimes n}\ \text{counted with multiplicity}\}, \]
for an object \(T\) in a monoidal category where this count makes sense (e.g. a finite dimensional representation of a group \(G\)). Exact multiplicities are extremely sensitive to the precise setting and often completely out of reach, but \(b_n\) often has a stable, representation–theoretically natural asymptotic, and even a general form. As an analogy: exact prime counts are subtle (at best), yet their growth is governed by the simple formula \(n/\ln n\); likewise, our focus is the coarse law for \(b_n\).
The contributions of this paper are as follows. Let \(U_q(\mathfrak{g})\) be a divided power quantum group at a complex root \(q\) of unity (e.g. \(q=\exp(\pi i/\ell)\)) and \(T\) a tilting \(U_q(\mathfrak{g})\)-module. Let \(b_n=b_n(T)\) and let \(a_n\) be the characteristic zero analog. We study the growth of \(b_n\) as \(n\to\infty\), and show that it is asymptotically of the form \(\Theta(a_n)\), where \(a_n\) is the characteristic zero analog of \(b_n\). For \(\mathfrak{g}=\mathfrak{sl}_2\) and its defining representation we prove a sharper result using random walks.
In the picture \(a_n\) and \(b_n\) correspond to the sum over all \(m\):