Fractal behavior of tensor powers in prime characteristic
Tensor powers of the two-dimensional representation of SL2 exhibit fractal/Mahler-type asymptotics in positive characteristic.
Data
- Title: Fractal behavior of tensor powers of the two dimensional space in prime characteristic
- Authors: Kevin Coulembier, Pavel Etingof, Victor Ostrik and Daniel Tubbenhauer
- Status: Modern algebra. Vol. 1. Representation theory, 85–138, Contemp. Math., 829. Last update: Wed, 18 Dec 2024 03:39:38 UTC
- Code / errata: GitHub
- arXiv: https://arxiv.org/abs/2405.16786
Abstract
We study the number of indecomposable summands in tensor powers of the vector representation of SL2. Our main focus is on positive characteristic where this sequence of numbers and its generating function show fractal behavior akin to Mahler functions.
What is the point?
Tensor powers are among the simplest operations in representation theory. In characteristic zero they already have interesting asymptotics. In prime characteristic the same question develops a fractal flavour: arithmetic, representation theory and analytic methods start talking to each other.
tilting modules for SL2.
Mahler functions and Tauberian methods.
fractal-looking growth laws.
Pictures
The characteristic zero plots first, followed by the prime characteristic behaviour.
