Non-invertible symmetries and modular invariance in lattice models

We consider classical 2d lattice models with face interactions defined in terms of a fusion category.

The symmetries of such models typically include an algebra of topological operators sitting on a closed path in the lattice.

In the case when the face interactions obey the Temperley-Lieb (TL) relations, we present a generic algorithm to determine the decomposition of the transfer-matrix space of states as a direct sum of simple TL modules.

We apply this approach to several examples, and analyse the action of topological operators.

As an application, we compute the modular transformation of the irreducible TL characters at primitive roots of unity.