  The boundary seam algebras $\mathsf{b}_{n,k}(\beta=q+q^{-1})$ were introduced
by Morin-Duchesne, Ridout and Rasmussen to formulate algebraically a large
class of boundary conditions for two-dimensional statistical loop models. The
representation theory of these algebras $\mathsf{b}_{n,k}(\beta=q+q^{-1})$ is
given: their irreducible, standard (cellular) and principal modules are
constructed and their structure explicited in terms of their composition
factors and of non-split short exact sequences. The dimensions of the
irreducible modules and of the radicals of standard ones are also given. The
methods proposed here might be applicable to a large family of algebras, for
example to those introduced recently by Flores and Peltola, and Cramp\'e and
Poulain d'Andecy.
