Generality of the dimension–root-system growth heuristic

Determine in what generality the heuristic that exponential growth is governed by module dimension while subexponential growth is governed by the root system and the characteristic applies across semisimple and nonsemisimple reductive-group-like settings.

Background

The paper places its results in a broader heuristic for tensor-power growth: the exponential base is expected to be determined by the dimension of the object, while the subexponential factor is expected to depend on the root system and the characteristic. The authors note that this principle is supported in characteristic zero and by their results for quantum groups at roots of unity, but its scope across reductive-group-like settings, especially in positive characteristic and other nonsemisimple contexts, is not established.

References

Together these advances support the heuristic ``exponential from dimension, and subexponential from root system and $p$'' as an organizing principle for reductive-group-like settings across semisimple and nonsemisimple contexts. We do not know in what generality this works, but:

— Growth Problems of Quantum Groups  (2511.06737 - O'Sullivan et al., 10 Nov 2025) in Section 1, Introduction