SciPost Phys. 8, 019 (2020) ·
published 5 February 2020

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The boundary seam algebras $\mathsf{b}_{n,k}(\beta=q+q^{1})$ were introduced
by MorinDuchesne, Ridout and Rasmussen to formulate algebraically a large
class of boundary conditions for twodimensional 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 nonsplit 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.