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The representation theory of seam algebras
by Alexis LangloisRémillard, Yvan SaintAubin
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Submission summary
As Contributors:  Alexis LangloisRémillard 
Arxiv Link:  https://arxiv.org/abs/1909.03499v1 (pdf) 
Date submitted:  20190913 02:00 
Submitted by:  LangloisRémillard, Alexis 
Submitted to:  SciPost Physics 
Academic field:  Mathematics 
Specialties: 

Approach:  Theoretical 
Abstract
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.
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Reports on this Submission
Anonymous Report 2 on 20191112 Invited Report
 Cite as: Anonymous, Report on arXiv:1909.03499v01, delivered 20191112, doi: 10.21468/SciPost.Report.1313
Strengths
1Provides an exhaustive account on the representation theory of the socalled boundary seam algebras, both in the generic and the root of unity cases.
Weaknesses
1The discussion of some related previous results could be better discussed.
Report
This paper considers a class of cellular algebras which were introduced under the name of boundary seam algebras by MorinDuchesne, Ridout and Rasmussen. The authors recall results on the related TemperleyLieb algebra and go on to set up the representation theory of the boundary seam algebras (going beyond MorinDuchesne et al), both in the generic and the root of unity cases. The results are clearly stated in theorem 2.5, in the form of a complete set of irreducible standard modules in the generic case, and exact short sequences in the root of unity cases.
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:mathph/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
1Please comment on the relations to the papers cited in the report.
Anonymous Report 1 on 20191111 Invited Report
 Cite as: Anonymous, Report on arXiv:1909.03499v01, delivered 20191111, doi: 10.21468/SciPost.Report.1311
Strengths
1/ Clear and pedagogical ;
Weaknesses
1/ Technical and specific subject ;
Report
The authors study the representations of an algebra recently introduced and called seam algebras when q (the parameter of the algebra) is generic or root of unity. They recall the representation theory of the TemperleyLieb algebra then they state the main theorem about the representation theory of the seam algebra which mimics the ones of the TemperleyLieb algebra. The rest of the paper is devoted to prove this result and is based on the notion of cellular algebra.
The paper is wellwritten 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