Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras
APA
(2024). Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras. SciVideos. https://youtube.com/live/5GafdDLg4ZA
MLA
Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras. SciVideos, Dec. 13, 2024, https://youtube.com/live/5GafdDLg4ZA
BibTex
@misc{ scivideos_ICTS:30479, doi = {}, url = {https://youtube.com/live/5GafdDLg4ZA}, author = {}, keywords = {}, language = {en}, title = {Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras}, publisher = {}, year = {2024}, month = {dec}, note = {ICTS:30479 see, \url{https://scivideos.org/index.php/icts-tifr/30479}} }
Abstract
We study a variant of the Iwahori-Hecke algebra of a Coxeter group, whose generators T_i satisfy the braid relations but are assumed to be nilpotent (in parallel to Coxeter groups where the T_i are involutions, and 0-Hecke algebras where they are idempotent). Motivated by Coxeter (1957) and Broue-Malle-Rouquier (1998), we classify the finite-dimensional among these "generalized nil-Coxeter algebras". These turn out to be the usual nil-Coxeter algebras, and exactly one other type-A family of algebras, which have a finite "word basis" in the T_i and a unique longest word.
In the remaining time I will present joint work with Sutanay Bhattacharyya, in which we explore the "Temperley-Lieb" variant of the above, wherein all sufficiently long braid words are also killed. Now the finite-dimensional algebras obtained include ones with bases indexed by:
(a) words with a unique reduced expression (any Coxeter type),
(b) fully commutative words (counted by Stembridge),
(c) Catalan numbers (via the XYX algebras of Postnikov), and
(d) the \bar{12} avoiding signed permutations (in type B=C).