# The representation theory of seam algebras

### Submission summary

 As Contributors: Alexis Langlois-Rémillard Arxiv Link: https://arxiv.org/abs/1909.03499v2 Date submitted: 2019-11-29 Submitted by: Langlois-Rémillard, Alexis Submitted to: SciPost Physics Discipline: Mathematics Subject area: Mathematical Physics Approach: Theoretical

### 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.

###### Current status:
Editor-in-charge assigned

Dear editor,

We thank the referees for their careful reading of the manuscript. Please see below a list of changes in this new version.

Sincerely,
Alexis Langlois-Rémillard and Yvan Saint-Aubin

### List of changes

1) Typos noted by referees (and two others) corrected: superscript "(k)", an old notation, removed in equation (2.9); $TL(\lambda)$ changed to $TL(\beta)$ after (3.4), and two small grammar mistakes.

2) Third paragraph of introduction extended to introduce Jacobsen's and Saleur's contribution. (Second paragraph also slightly modified.)

3) Citations added when the Wenzl-Jones projector is defined.

4) Paragraph added at the end of section 2.2 to make precise the relationship between blob and seam algebras.

5) Figure 3, a graphical representation of the Lemma 3.8 inserted between this lemma and proposition 3.9.

6) Thanks to the referees added to the Acknowledgements.

### Submission & Refereeing History

Resubmission 1909.03499v02 on 29 November 2019
Submission 1909.03499v01 on 13 September 2019