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The representation theory of seam algebras
by Alexis Langlois-Rémillard, Yvan Saint-Aubin
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Alexis Langlois-Rémillard |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1909.03499v2 (pdf) |
| Date accepted: | 2020-01-22 |
| Date submitted: | 2019-11-29 01:00 |
| Submitted by: | Langlois-Rémillard, Alexis |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Mathematics |
| Specialties: |
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| 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.
Author comments upon resubmission
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.
Published as SciPost Phys. 8, 019 (2020)