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The representation theory of seam algebras
by Alexis Langlois-Rémillard, Yvan Saint-Aubin
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Submission summary
| Authors (as registered SciPost users): | Alexis Langlois-Rémillard |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1909.03499v1 (pdf) |
| Date submitted: | Sept. 13, 2019, 2 a.m. |
| 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.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2019-11-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1909.03499v01, delivered 2019-11-12, doi: 10.21468/SciPost.Report.1313
Strengths
Weaknesses
Report
The paper is well written, interesting and clearly publishable. However, it would be helpful to the readers if the authors could clarify its connections to some earlier literature. First, the defining relations (2.7) are reminiscent of the cabling construction described in arXiv:math-ph/0611078, in the first lines of section 5 and figure 16. Second, is the determinant of the Gram matrix given here in proposition 2.4 related to any one of the constructions given in section 6.2 of arXiv:0709.0812, and can the Gram matrix be formulated in similar graphical terms?
Requested changes
1-Please comment on the relations to the papers cited in the report.
Report #1 by Anonymous (Referee 1) on 2019-11-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1909.03499v01, delivered 2019-11-11, doi: 10.21468/SciPost.Report.1311
Strengths
1/ Clear and pedagogical ;
Weaknesses
1/ Technical and specific subject ;
Report
The paper is well-written and very pedagogical and can be used as a reference to learn about cellular algebra. Even though the subject is very specific and technical, the result and the method are interesting and deserve to be published.
Requested changes
1- in equation (2.9), the object with superscript " (k) " is not defined. 2- page 9 three lines after (3.4) lambda must be changed to beta