TY - UNPB

T1 - Idempotents and homology of diagram algebras

AU - Boyde, Guy

N1 - 40 pages, comments welcome. Added comments (esp Rmks 1.5, 1.15) which relate this paper to cellular algebras in the sense of Graham-Lehrer

PY - 2022/12/4

Y1 - 2022/12/4

N2 - 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.

AB - 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.

KW - math.AT

KW - math.RT

KW - 20J06, 16E40 (Primary), 20B30 (Secondary)

U2 - 10.48550/arXiv.2212.01826

DO - 10.48550/arXiv.2212.01826

M3 - Preprint

SP - 1

EP - 40

BT - Idempotents and homology of diagram algebras

PB - arXiv

ER -