Generalised Temperley-Lieb algebras of type $G(r,p,n)$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912169490120704 |
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| author | Lehrer, Gus Lyu, Mengfan |
| author_facet | Lehrer, Gus Lyu, Mengfan |
| contents | In an earlier work, we defined a ``generalised Temperley-Lieb algebra'' $TL_{r,1,n}$ corresponding to the imprimitive reflection group $G(r,1,n)$ as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley-Lieb algebra $TL_{r,p,n}$ which corresponds to the complex reflection group $G(r,p,n)$. Our definition identifies $TL_{r,p,n}$ as the fixed-point subalgebra of $TL_{r,1,n}$ under a certain automorphism $σ$. We prove the cellularity of $TL_{r,p,n}$ by proving that $σ$ induces a special shift automorphism with respect to the cellular structure of $TL_{r,1,n}$. We also give a description of the cell modules of $TL_{r,p,n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09377 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalised Temperley-Lieb algebras of type $G(r,p,n)$ Lehrer, Gus Lyu, Mengfan Representation Theory 16G20, 20C08 (Primary), 20G42 (Secondary) In an earlier work, we defined a ``generalised Temperley-Lieb algebra'' $TL_{r,1,n}$ corresponding to the imprimitive reflection group $G(r,1,n)$ as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley-Lieb algebra $TL_{r,p,n}$ which corresponds to the complex reflection group $G(r,p,n)$. Our definition identifies $TL_{r,p,n}$ as the fixed-point subalgebra of $TL_{r,1,n}$ under a certain automorphism $σ$. We prove the cellularity of $TL_{r,p,n}$ by proving that $σ$ induces a special shift automorphism with respect to the cellular structure of $TL_{r,1,n}$. We also give a description of the cell modules of $TL_{r,p,n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory. |
| title | Generalised Temperley-Lieb algebras of type $G(r,p,n)$ |
| topic | Representation Theory 16G20, 20C08 (Primary), 20G42 (Secondary) |
| url | https://arxiv.org/abs/2211.09377 |