Abstract
AbstractThis 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.
Funder
H2020 European Research Council
Publisher
Springer Science and Business Media LLC
Reference25 articles.
1. Benson, D.J.: Representations and Cohomology. I. Second. Vol. 30. Cambridge Studies in Advanced Mathematics. Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge University Press, Cambridge (1998)
2. Boyd, R., Hepworth, R.: The homology of the Temperley-Lieb algebras. Geom. Topol. 28(3), 1437–1499 (2024)
3. Boyd, R., Hepworth, R.: Combinatorics of injective words for Temperley–Lieb algebras. J. Comb. Theory Ser. A 181(27), 105446 (2021)
4. Boyd, R., Hepworth, R., Patzt, P.: The homology of the Brauer algebras. Selecta Math. (N.S.) 27.5(85), 31 (2021)
5. Boyd, R., Hepworth, R., Patzt, P.: The homology of the partition algebras. Pacific J. Math. 327(1), 1–27 (2023)