The representation theory of seam algebras
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Bibliographic record
Abstract
The boundary seam algebras \mathsf{b}_{n,k}(\beta=q+q^{-1}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖻</mml:mi> </mml:mstyle> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖻</mml:mi> </mml:mstyle> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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 Crampé and Poulain d’Andecy.
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| Category | Codex | Gemma |
|---|---|---|
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| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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