Idempotents and homology of diagram algebras
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914003068911616 |
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| author | Boyde, Guy |
| author_facet | Boyde, Guy |
| contents | This paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the `invertible-parameter' cases of the Temperley-Lieb and Brauer results of Boyd-Hepworth and Boyd-Hepworth-Patzt. We are also able to give a new proof of Sroka's theorem that the homology of an odd-strand Temperley-Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham-Lehrer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_01826 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Idempotents and homology of diagram algebras Boyde, Guy Algebraic Topology Representation Theory 20J06, 16E40 (Primary), 20B30 (Secondary) This paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the `invertible-parameter' cases of the Temperley-Lieb and Brauer results of Boyd-Hepworth and Boyd-Hepworth-Patzt. We are also able to give a new proof of Sroka's theorem that the homology of an odd-strand Temperley-Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham-Lehrer. |
| title | Idempotents and homology of diagram algebras |
| topic | Algebraic Topology Representation Theory 20J06, 16E40 (Primary), 20B30 (Secondary) |
| url | https://arxiv.org/abs/2212.01826 |