Citation: Coulembier, Kevin, Victor Ostrik, and Daniel Tubbenhauer. "Growth rates of the number of indecomposable summands in tensor powers." arXiv preprint arXiv:2301.00885 (2023).
Web: https://arxiv.org/abs/2301.00885
Tags: Mathematical, Monoidal-categories, Non-finite/semisimple,
This paper discusses an interesting way of thinking about the dimension of a representation. Namely, the dimension of a representation counts asymptotically how many factors will appear in the direct sum decomposition of repeated tensor powers.
This formula was already known, proved earlier by Benson and Symonds:
> Benson, Dave, and Peter Symonds. "The non-projective part of the tensor powers of a module." Journal of the London Mathematical Society 101.2 (2020): 828-856.
The insight of this paper was asking to what extent this result generalizes. It was found that result is remarkably stable, and applies to a large number of situations. Thus, this is a very robust way of thinking about dimension.
Another good takeaway from this paper is its first line, which gives nice representation theory philosophy: "A central, yet hard, problem in representation theory is the decomposition of tensor products of representations into indecomposable summands. Computations of these decomposition numbers are often major unsolved problems in representation theory".
The question of the asymptotic behavior of direct summand decompositions has continued since this paper was published, most notably in
> Coulembier, Kevin, Pavel Etingof, and Victor Ostrik. "Asymptotic properties of tensor powers in symmetric tensor categories." arXiv preprint arXiv:2301.09804 (2023).