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We consider a pivotal monoidal functor whose domain is a modular tensorcategory (MTC). We show that the trace of such a functor naturally extends to arepresentation of the corresponding tube category. As irreduciblerepresentations of the tube category are indexed by pairs of simple objects inthe underlying MTC, the simple multiplicities of this representation form acandidate modular invariant matrix. In general, this matrix will not be modularinvariant, however it will always commute with the T-matrix. Furthermore, undercertain additional conditions on the original functor, it is shown that thecorresponding representation of the tube category is a haploid, symmetric,commutative Frobenius algebra. Such algebras are known to be connected tomodular invariants, in particular a result of Kong and Runkel implies that thematrix of simple multiplicities commutes with the S-matrix if and only if thedimension of the algebra is equal to the dimension of the underlying MTC.Finally, we apply these techniques to certain pivotal monoidal functors arisingfrom module categories over the Temperley-Lieb category and the associated MTC.This provides a novel explanation of the A-D-E pattern appearing in theclassification of $ A_1^{(1)} $ modular invariants.

Extending the trace of a pivotal monoidal functor