The representation theory of seam algebras

TL;DR

This paper develops the representation theory of boundary seam algebras, constructing their modules and analyzing their structure, which advances understanding of algebraic boundary conditions in statistical loop models.

Contribution

It explicitly constructs irreducible, standard, and principal modules for boundary seam algebras and details their structure, including composition factors and exact sequences.

Findings

Dimensions of irreducible modules determined

Structure of modules in terms of composition factors elucidated

Methods applicable to other algebra families

Abstract

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.