ICTS:30479

Video URL

Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras

Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras

(2024). Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras. SciVideos. https://youtube.com/live/5GafdDLg4ZA

Finite-dimensional generalized nil-Coxeter and nil-Temperley-Lieb algebras. SciVideos, Dec. 13, 2024, https://youtube.com/live/5GafdDLg4ZA 

          @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}}
          }
Apoorva Khare
Talk numberICTS:30479
Source RepositoryICTS-TIFR
Collection

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).