Given a presentably symmetric monoidal -category C and an E_-monoid M, we introduce and classify twisted graded categories, which generalize the Day convolution structure on Fun (M, C). These are characterized by a braiding encoded in symmetric group actions on tensor powers, whose character we show depends only on the T-equivariant monoidal dimension. We analyze the T-action on the dimension of invertible objects and identify it with the T-transfer map. Finally, we compute braiding characters in examples arising from higher cyclotomic extensions, such as the (S, n+1) -oriented extension of Mod₄₍^ at all primes and heights, and of the cyclotomic closure of Vectⁿ at low heights.
Keidar et al. (Thu,) studied this question.